Datasets:
Duplicate from prometheus-eval/into-the-unknown
Browse filesCo-authored-by: Seungone Kim <seungone@users.noreply.huggingface.co>
This view is limited to 50 files because it contains too many changes. See raw diff
- .gitattributes +231 -0
- LICENSE +201 -0
- README.md +110 -0
- benchmark/index.csv +16 -0
- benchmark/paper1/images/image1.jpg +3 -0
- benchmark/paper1/images/image2.jpg +3 -0
- benchmark/paper1/images/image3.jpg +3 -0
- benchmark/paper1/paper.md +293 -0
- benchmark/paper1/paper.pdf +3 -0
- benchmark/paper1/task_spec.md +289 -0
- benchmark/paper10/images/image1.jpg +3 -0
- benchmark/paper10/images/image10.jpg +3 -0
- benchmark/paper10/images/image11.jpg +3 -0
- benchmark/paper10/images/image12.jpg +3 -0
- benchmark/paper10/images/image13.jpg +3 -0
- benchmark/paper10/images/image14.jpg +3 -0
- benchmark/paper10/images/image15.jpg +3 -0
- benchmark/paper10/images/image16.jpg +3 -0
- benchmark/paper10/images/image17.jpg +3 -0
- benchmark/paper10/images/image18.jpg +3 -0
- benchmark/paper10/images/image19.jpg +3 -0
- benchmark/paper10/images/image2.jpg +3 -0
- benchmark/paper10/images/image20.jpg +3 -0
- benchmark/paper10/images/image21.jpg +3 -0
- benchmark/paper10/images/image22.jpg +3 -0
- benchmark/paper10/images/image23.jpg +3 -0
- benchmark/paper10/images/image24.jpg +3 -0
- benchmark/paper10/images/image25.jpg +3 -0
- benchmark/paper10/images/image26.jpg +3 -0
- benchmark/paper10/images/image27.jpg +3 -0
- benchmark/paper10/images/image28.jpg +3 -0
- benchmark/paper10/images/image29.jpg +3 -0
- benchmark/paper10/images/image3.jpg +3 -0
- benchmark/paper10/images/image30.jpg +3 -0
- benchmark/paper10/images/image31.jpg +3 -0
- benchmark/paper10/images/image4.jpg +3 -0
- benchmark/paper10/images/image5.jpg +3 -0
- benchmark/paper10/images/image6.jpg +3 -0
- benchmark/paper10/images/image7.jpg +3 -0
- benchmark/paper10/images/image8.jpg +3 -0
- benchmark/paper10/images/image9.jpg +3 -0
- benchmark/paper10/paper.md +0 -0
- benchmark/paper10/paper.pdf +3 -0
- benchmark/paper10/task_spec.md +224 -0
- benchmark/paper11/images/image1.jpg +3 -0
- benchmark/paper11/images/image10.jpg +3 -0
- benchmark/paper11/images/image11.jpg +3 -0
- benchmark/paper11/images/image12.jpg +3 -0
- benchmark/paper11/images/image13.jpg +3 -0
- benchmark/paper11/images/image14.jpg +3 -0
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*.7z filter=lfs diff=lfs merge=lfs -text
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saved_model/**/* filter=lfs diff=lfs merge=lfs -text
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# Audio files - uncompressed
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# Audio files - compressed
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# Image files - uncompressed
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benchmark/paper1/paper.pdf filter=lfs diff=lfs merge=lfs -text
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README.md
ADDED
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|
| 1 |
+
---
|
| 2 |
+
license: other
|
| 3 |
+
pretty_name: Into the Unknown
|
| 4 |
+
license_name: mixed
|
| 5 |
+
license_link: LICENSE
|
| 6 |
+
tags:
|
| 7 |
+
- agents
|
| 8 |
+
- ai-scientist
|
| 9 |
+
- scientific-discovery
|
| 10 |
+
- benchmark
|
| 11 |
+
task_categories:
|
| 12 |
+
- text-generation
|
| 13 |
+
configs:
|
| 14 |
+
- config_name: benchmark
|
| 15 |
+
data_files: benchmark/index.csv
|
| 16 |
+
- config_name: outcomes
|
| 17 |
+
data_files: outcomes/index.csv
|
| 18 |
+
- config_name: trajectories
|
| 19 |
+
data_files: trajectories/index.csv
|
| 20 |
+
---
|
| 21 |
+
|
| 22 |
+
# Into the Unknown
|
| 23 |
+
|
| 24 |
+
**Dataset Summary**
|
| 25 |
+
|
| 26 |
+
Into the Unknown tests whether AI agents can build on a scientist's work and write a follow-up paper. Each of its 15 tasks gives an agent a published (or recently submitted) paper and a follow-up research direction proposed by the paper's own author; the agent must carry out the research and write a follow-up paper that is honest, complete, and meets the scientist's bar.
|
| 27 |
+
|
| 28 |
+
This dataset holds the 15 tasks and everything nine agents produced on them: 135 submissions, with each agent's paper, full trajectory, and the files it wrote. Code, rubrics and graders are on [GitHub](https://github.com/prometheus-eval/into-the-unknown); the paper is on [arXiv](https://arxiv.org/abs/2610.19159).
|
| 29 |
+
|
| 30 |
+
**Supported Tasks and Leaderboards**
|
| 31 |
+
|
| 32 |
+
The task is open-ended research: given the paper and the direction, produce the follow-up paper. The [leaderboard](https://prometheus-eval.github.io/into-the-unknown/viewer/#/) shows every agent's results and every rubric verdict, and you can [submit your agent's paper](https://prometheus-eval.github.io/into-the-unknown/viewer/#/submit) to be evaluated.
|
| 33 |
+
|
| 34 |
+
**Subsets**
|
| 35 |
+
|
| 36 |
+
| subset | content |
|
| 37 |
+
|---|---|
|
| 38 |
+
| `benchmark/` | The 15 tasks: `paper<i>/task_spec.md`, the original paper (`paper.pdf`, its text `paper.md`, figures `images/`) and `index.csv`. A task's code and data are downloaded by `prepare_tasks.py` in the GitHub repository. |
|
| 39 |
+
| `outcomes/` | The papers the agents wrote, `<agent>/paper<i>.pdf` (the `proposal/report.pdf` the judge read): 131 PDFs, since four GLM-5.3 runs produced none. |
|
| 40 |
+
| `trajectories/` | Each run's `<agent>/paper<i>.json` (run metadata and every agent event), plus the harnesses' own logs (`*.harness_logs.tar`; DeepSeek-V4-Pro's full conversation is here) and the runners' console logs (`*.runner_logs.tar`). |
|
| 41 |
+
| `file_systems/` | Each submission's files as the judge saw them: `proposal/` and the rest of the working directory (`workdir_outside_proposal/`), without environments, caches or the final transcript. `view.tar.part-*` with a SHA-256 `manifest.json`; nine runs add a `supplement.tar` of files the judge's copy lacked. |
|
| 42 |
+
|
| 43 |
+
Agents (`<agent>`): `opus-4.8`, `opus-5`, `opus-5.5`, `fable-5`, `sol-5.6`, `astra-6`, `deepseek-v4-pro`, `kimi-k3`, `glm-5.3`.
|
| 44 |
+
|
| 45 |
+
**Dataset Structure**
|
| 46 |
+
|
| 47 |
+
An example of a `benchmark` datum is as follows:
|
| 48 |
+
|
| 49 |
+
```
|
| 50 |
+
paper: (str) - The task id, P1 to P15.
|
| 51 |
+
group: (str) - A: published 2024 or earlier; B: submitted 2026.
|
| 52 |
+
title: (str) - The original paper's title.
|
| 53 |
+
domain, venue, lab: (str) - Its field, venue and the laboratory of its authors.
|
| 54 |
+
citations: (int) - Citations as of September 14, 2026.
|
| 55 |
+
original_paper: (str) - A link to the original paper.
|
| 56 |
+
follow_up_direction: (str) - The research direction the paper's author proposed.
|
| 57 |
+
task_spec, paper_pdf, paper_md: (str) - Paths to the task specification and the original paper in this dataset.
|
| 58 |
+
```
|
| 59 |
+
|
| 60 |
+
`outcomes/index.csv` and `trajectories/index.csv` list one row per submission (agent, paper, file, and run metadata).
|
| 61 |
+
|
| 62 |
+
**Evaluation**
|
| 63 |
+
|
| 64 |
+
Run an agent and judge its submission with the [GitHub repository](https://github.com/prometheus-eval/into-the-unknown):
|
| 65 |
+
|
| 66 |
+
```bash
|
| 67 |
+
bash src/solver/run.sh <agent> <paper_id>
|
| 68 |
+
python src/grader/run_codex_judge.py prepare --agents <agent> # then rescope, launch
|
| 69 |
+
```
|
| 70 |
+
|
| 71 |
+
To unpack one submission's files:
|
| 72 |
+
|
| 73 |
+
```bash
|
| 74 |
+
huggingface-cli download prometheus-eval/into-the-unknown --repo-type dataset \
|
| 75 |
+
--include "file_systems/astra-6/paper3/*" --local-dir .
|
| 76 |
+
cd file_systems/astra-6/paper3 && cat view.tar.part-* > view.tar && tar -xf view.tar
|
| 77 |
+
```
|
| 78 |
+
|
| 79 |
+
**Sample Usage**
|
| 80 |
+
|
| 81 |
+
Load the benchmark subset directly with the `datasets` library:
|
| 82 |
+
|
| 83 |
+
```python
|
| 84 |
+
from datasets import load_dataset
|
| 85 |
+
|
| 86 |
+
tasks = load_dataset('prometheus-eval/into-the-unknown', 'benchmark', split='train')
|
| 87 |
+
```
|
| 88 |
+
|
| 89 |
+
A few notes specific to this dataset:
|
| 90 |
+
|
| 91 |
+
- `outcomes/fable-5/paper8.pdf` has a real person's name, affiliation and e-mail blacked out on its title page; every other PDF is byte-identical to the submission.
|
| 92 |
+
- Files written on the compute cluster can contain absolute paths of the cluster's file system.
|
| 93 |
+
- In `file_systems/opus-5.5/paper12`, two R help pages (`qsreg.html`, `spam.html`) differ only in case from other pages; the archive has the originals, the others are in `supplement.tar`, and `manifest.json` lists both under `checked_against_judge_record.sha256_mismatch`.
|
| 94 |
+
|
| 95 |
+
**Citation**
|
| 96 |
+
|
| 97 |
+
If you find our work helpful, please consider citing it:
|
| 98 |
+
|
| 99 |
+
```bibtex
|
| 100 |
+
@article{kim2026into,
|
| 101 |
+
title={Into the Unknown: Benchmarking AI agents on open scientific research problems},
|
| 102 |
+
author={Kim, Seungone and Geng, Jiayi and Lee, Young-Jun and Liu, Bo and Suk, Juyoung and Lubbers, Nicholas and Wood, Mitchell and Noack, Marcus and Yao, Yong-Xin and Shi, Yuan and others},
|
| 103 |
+
journal={arXiv preprint arXiv:2610.19159},
|
| 104 |
+
year={2026}
|
| 105 |
+
}
|
| 106 |
+
```
|
| 107 |
+
|
| 108 |
+
**License**
|
| 109 |
+
|
| 110 |
+
Task specifications, rubrics and documentation are released under the Apache-2.0 license of the GitHub repository. The original papers in `benchmark/` remain under their authors' and publishers' terms and are included for research use; agent outputs may contain third-party code and data that keep their own licenses.
|
benchmark/index.csv
ADDED
|
@@ -0,0 +1,16 @@
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 |
+
paper,group,title,domain,venue,citations,lab,original_paper,follow_up_direction,task_spec,paper_pdf,paper_md
|
| 2 |
+
P1,A,Hierarchical modeling of molecular energies using a deep neural network,Computational chemistry,J. Chem. Phys.,428,LANL,https://arxiv.org/abs/1710.00017,Use HIP-NN's hierarchy-breakdown signal to identify failure motifs and calibrate uncertainty estimates that transfer across conformations.,benchmark/paper1/task_spec.md,benchmark/paper1/paper.pdf,benchmark/paper1/paper.md
|
| 3 |
+
P2,A,Extending the accuracy of the SNAP interatomic potential form,Materials science,J. Chem. Phys.,311,SNL,https://arxiv.org/abs/1711.11131,"Replace SNAP bispectrum components with a learned descriptor and test its accuracy, data efficiency, and cross-material transfer.",benchmark/paper2/task_spec.md,benchmark/paper2/paper.pdf,benchmark/paper2/paper.md
|
| 4 |
+
P3,A,Convolutional Dictionary Learning: A Comparative Review and New Algorithms,Computational imaging,IEEE TCI,282,LANL,https://arxiv.org/abs/1709.02893,Unify CDL dictionary-update algorithms within one multichannel derivation framework and use it to reveal a new method and equivalence.,benchmark/paper3/task_spec.md,benchmark/paper3/paper.pdf,benchmark/paper3/paper.md
|
| 5 |
+
P4,A,A Kriging-based approach to autonomous experimentation with applications to X-ray scattering,Scientific ML,Sci. Rep.,136,LBNL,https://doi.org/10.1038/s41598-019-48114-3,Develop an acquisition strategy that reconstructs synthetic functions faster than SMART and gSMART.,benchmark/paper4/task_spec.md,benchmark/paper4/paper.pdf,benchmark/paper4/paper.md
|
| 6 |
+
P5,A,Adaptive variational quantum dynamics simulations,Quantum,PRX Quantum,203,Ames,https://arxiv.org/abs/2011.00622,Make AVQDS compatible with fixed nearest-neighbor qubit topology while measuring the resulting accuracy and resource overhead.,benchmark/paper5/task_spec.md,benchmark/paper5/paper.pdf,benchmark/paper5/paper.md
|
| 7 |
+
P6,A,Simulations of relativistic-quantum plasmas using real-time lattice scalar QED,Plasma physics,Phys. Rev. E,32,PPPL,https://arxiv.org/abs/1802.00524,"Stress-test variational lattice scalar QED in new valid and fundamentally invalid regimes, then diagnose and repair its failures.",benchmark/paper6/task_spec.md,benchmark/paper6/paper.pdf,benchmark/paper6/paper.md
|
| 8 |
+
P7,A,The non-Riemannian nature of perceptual color space,Color science,PNAS,51,LANL,https://doi.org/10.1073/pnas.2119753119,Test whether perceptual diminishing returns are organized by cone channels rather than global scaling or generic anisotropy.,benchmark/paper7/task_spec.md,benchmark/paper7/paper.pdf,benchmark/paper7/paper.md
|
| 9 |
+
P8,A,Parameterized Physics-informed Neural Networks for Parameterized PDEs,Scientific ML,ICML,113,,https://arxiv.org/abs/2408.09446,"Extend P²INN into a problem-specification encoder that generalizes jointly across coefficients, initial conditions, and boundary conditions.",benchmark/paper8/task_spec.md,benchmark/paper8/paper.pdf,benchmark/paper8/paper.md
|
| 10 |
+
P9,B,An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations,Quantum,J. Comput. Phys.,6,LLNL,https://arxiv.org/abs/2501.11146,Improve the mathematical LCHS algorithm by substituting a kernel with better asymptotic resource scaling.,benchmark/paper9/task_spec.md,benchmark/paper9/paper.pdf,benchmark/paper9/paper.md
|
| 11 |
+
P10,B,Efficient Berry Phase Calculation via Adaptive Variational Quantum Computing Approach,Quantum,APL Quantum,5,Ames,https://arxiv.org/abs/2506.19150,Determine how much realistic hardware noise the AVQDS Berry-phase method can tolerate before its topological output fails.,benchmark/paper10/task_spec.md,benchmark/paper10/paper.pdf,benchmark/paper10/paper.md
|
| 12 |
+
P11,B,Projected Hessian Learning: Fast Curvature Supervision for Accurate Machine-Learning Interatomic Potentials,Computational chemistry,Unpublished,3,LANL,https://arxiv.org/abs/2603.04523,Reduce the stochastic-probe requirement of Projected Hessian Learning without sacrificing estimator or downstream model quality.,benchmark/paper11/task_spec.md,benchmark/paper11/paper.pdf,benchmark/paper11/paper.md
|
| 13 |
+
P12,B,gp2Scale: Compactly-Supported Non-Stationary Kernels and Distributed Computing for Exact Gaussian Processes on 10 Million Data Points,Scientific ML,ICML,2,LBNL,https://arxiv.org/abs/2512.06143,Apply exact large-scale Gaussian processes to a new domain using a domain-specific kernel or sparsity construction.,benchmark/paper12/task_spec.md,benchmark/paper12/paper.pdf,benchmark/paper12/paper.md
|
| 14 |
+
P13,B,Particle-in-cell simulations of laser crossbeam energy transfer via magnetized ion-acoustic wave,Plasma physics,Physics,2,LLNL,https://arxiv.org/abs/2511.03090,Map the boundary where Vlasov theory ceases to predict MagCBET PIC behavior and identify the first invalid assumption.,benchmark/paper13/task_spec.md,benchmark/paper13/paper.pdf,benchmark/paper13/paper.md
|
| 15 |
+
P14,B,Simulating plasma wave propagation on a superconducting quantum chip,Quantum,Phys. Rev. App.,1,LLNL,https://arxiv.org/abs/2507.09479,Add ZZ interactions to the spin-chain model and test whether it quantitatively reproduces nonlinear plasma dynamics.,benchmark/paper14/task_spec.md,benchmark/paper14/paper.pdf,benchmark/paper14/paper.md
|
| 16 |
+
P15,B,Beyond Classical Molecular Dynamics with Layered Interatomic Potentials,Materials science,Unpublished,0,SNL,https://www.researchsquare.com/article/rs-9077430/v1,Replace the Sommerfeld potential's heuristic sigmoid with a physically principled switching function.,benchmark/paper15/task_spec.md,benchmark/paper15/paper.pdf,benchmark/paper15/paper.md
|
benchmark/paper1/images/image1.jpg
ADDED
|
Git LFS Details
|
benchmark/paper1/images/image2.jpg
ADDED
|
Git LFS Details
|
benchmark/paper1/images/image3.jpg
ADDED
|
Git LFS Details
|
benchmark/paper1/paper.md
ADDED
|
@@ -0,0 +1,293 @@
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| 1 |
+
# Hierarchical modeling of molecular energies using a deep neural network
|
| 2 |
+
|
| 3 |
+
Nicholas Lubbers [ nlubbers@lanl.gov ] Justin S. Smith [ Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA ] Kipton Barros [ Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA ]
|
| 4 |
+
|
| 5 |
+
###### Abstract
|
| 6 |
+
|
| 7 |
+
We introduce the Hierarchically Interacting Particle Neural Network (HIP-NN) to model molecular properties from datasets of quantum calculations. Inspired by a many-body expansion, HIP-NN decomposes properties, such as energy, as a sum over hierarchical terms. These terms are generated from a neural network—a composition of many nonlinear transformations—acting on a representation of the molecule. HIP-NN achieves state-of-the-art performance on a dataset of 131k ground state organic molecules, and predicts energies with 0.26 kcal/mol mean absolute error. With minimal tuning, our model is also competitive on a dataset of molecular dynamics trajectories. In addition to enabling accurate energy predictions, the hierarchical structure of HIP-NN helps to identify regions of model uncertainty.
|
| 8 |
+
|
| 9 |
+
## I Introduction
|
| 10 |
+
|
| 11 |
+
Models of chemical properties have wide-ranging applications in fields such as materials science, chemistry, molecular biology, and drug design. Commonly, one treats the nuclei positions as fixed (the Born-Oppenheimer approximation), and molecular properties follow from the quantum-mechanical state of electrons. The many-body Schrödinger equation is extremely difficult to solve fully, and in practice computational quantum chemistry involves some level of approximation. Common choices are, e.g., Coupled Cluster (CC) [1; 2] and Density Functional Theory (DFT). [3; 4] Such ab initio methods typically exhibit cubic or worse scaling in the number of electrons. Faster calculations are crucial in contexts such as molecular dynamics (MD) simulation or high-throughput molecular screening.
|
| 12 |
+
|
| 13 |
+
To improve efficiency, one may sacrifice accuracy. For example, the effective interactions between nuclei may be modeled with local classical potentials of fixed form. Such potentials may be parameterized to match given experimental data or quantum calculations. Classical potentials are extremely fast, and enable MD simulations of systems with 10^{6}–10^{9} atoms. However, the parameterization process is empirical and the resulting potentials may not transfer to new systems or new dynamical processes. For example, it is notoriously difficult to model the energetic barriers of bond breaking in a transferrable way. [5; 6] Force fields are also known to lack transferability to chemical environments that differ from those used in the fitting process. [7] One may also compromise between ab initio and empirical methodologies; e.g., Density Functional Tight Binding [8; 9] enables MD simulations of 10^{3}–10^{5} atoms, [10] but brings its own challenges in parameterization and transferability.
|
| 14 |
+
|
| 15 |
+
Recently there has been tremendous interest in using machine learning (ML) to automatically construct potentials based upon large datasets of quantum calculations. [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22] This approach aims for the best of both worlds: the accuracy of full quantum calculations and efficiency comparable to empirical classical potentials. An especially promising direction builds upon recent advances in computer vision. [23; 24; 25; 26] Convolutional neural nets are designed for translation-invariant processing of an image plane via convolutional filters. Similar architectural principles allow us to design neural nets that process molecules while respecting translation, rotation, and permutation invariances. [27; 28; 29] Modern neural net architectures automatically learn representations of local atomic environments without requiring any feature engineering, and achieve state of the art performance in predictions of molecular properties. [30; 31; 32; 33] An advantage of neural nets (compared to, e.g., kernel ridge regression and Gaussian process regression [34; 35]) is that the training time scales linearly in the number of data points, making it practical to learn from databases of millions of quantum calculations. [36]
|
| 16 |
+
|
| 17 |
+
In this paper, we introduce the Hierarchically Interacting Particle Neural Network (HIP-NN), which takes inspiration from the many-body expansion (MBE). Following common practice, [34; 35; 37] we assume that the ab initio total energy $E$ of a molecule may be modeled as a sum over local contributions at each atom $i$,
|
| 18 |
+
|
| 19 |
+
$E\approx\hat{E}=\sum_{i=1}^{N_{\text{atom}}}\hat{E}_{i}.$ (1)
|
| 20 |
+
|
| 21 |
+
HIP-NN further decomposes the local energy model $\hat{E}_{i}$ in contributions over orders $n$,
|
| 22 |
+
|
| 23 |
+
$\hat{E}_{i}=\sum_{n=0}^{N_{\text{interaction}}}\hat{E}_{i}^{n}.$ (2)
|
| 24 |
+
|
| 25 |
+
The MBE, commonly employed in classical potentials, [38; 39] would use $\hat{E}^{n}$ to represent $(n+1)$-body contributions to the energy, i.e., interactions between atom $i$ and up to $n$ of its neighbors. Integration of the MBE into ML models of molecular energies has been suggested in Refs. [35] and [40]. This prior work employed separate ML models for each expansion order $n$. A key aspect of
|
| 26 |
+
|
| 27 |
+
|
| 28 |
+
Figure 1: HIP-NN processes a molecule from left to right, building successive atomic features to describe the local chemical environments. This processing occurs through interaction and on-site layers (green and red boxes, respectively). Interaction layers collect information from a local neighborhood (green circles). The total molecular energy $\tilde{E}$ includes contributions $\tilde{E}_{i}^{n}$ at all sites $i$ and hierarchical levels $n$.
|
| 29 |
+
|
| 30 |
+
HIP-NN is that a single network produces $\tilde{E}_{i}^{n}$ at all orders, allowing these terms to be simultaneously learned in coherent way. Furthermore, the HIP-NN ansatz is more general than the MBE, in that the terms $E_{i}^{n}$ may incorporate many-body interactions at higher order than $n$. The decomposition is non-unique, but should be designed such that $\tilde{E}_{i}^{n}$ rapidly vanishes with increasing order $n$. To pursue this, our training procedure utilizes a hierarchical regularization term to encourage the outputs $\tilde{E}_{i}^{n}$ to decay with $n$. After training, if decay with $n$ is not observed for a given input molecule, then the HIP-NN energy prediction is less likely to be accurate. That is, HIP-NN can estimate the reliability of its own energy predictions.
|
| 31 |
+
|
| 32 |
+
We detail the HIP-NN architecture and training procedure in the next section. Section III demonstrates that HIP-NN effectively learns molecular energies for various benchmark datasets. On the QM9 dataset of organic molecules, [41] HIP-NN predicts energies with a groundbreaking mean absolute error of 0.26 kcal/mol. HIP-NN also performs well on datasets of MD trajectories with minimal tuning. Variants of HIP-NN achieve good performance with parameter counts ranging from $\sim 10^{3}$ to $10^{5}$. In addition to enabling robust predictions, the hierarchical structure of HIP-NN provides a built-in measure of model uncertainty. In Sec. IV we further discuss and interpret our numerical results, and we conclude in Sec. V.
|
| 33 |
+
|
| 34 |
+
## II HIP-NN Methodology
|
| 35 |
+
|
| 36 |
+
Figure 1 illustrates HIP-NN, our neural network for predicting molecular properties and energies. The molecular configuration is input on the left using a simple representation, discussed below, that is symmetric with respect to translation, rotation, and permutation of atoms. As the molecule is processed from left to right, HIP-NN builds consecutive sets of atomic features to characterize the chemical environment of each atom. Blue boxes denote hierarchical contributions to the total energy—the final output of HIP-NN. Green boxes denote interaction layers, which mix information between pairs of atoms within some radius (illustrated for a single carbon atom using green circles). Red boxes denote on-site layers, which process the atomic features of a single atom. These components are described mathematically in the subsections below.
|
| 37 |
+
|
| 38 |
+
### II.1 Molecular representation
|
| 39 |
+
|
| 40 |
+
A molecular configuration $\mathcal{C}=\{(Z_{i},\mathbf{r}_{i})\}$ is defined by the atomic numbers $Z_{i}$ and coordinates $\mathbf{r}_{i}$ of atoms $i=1\ldots N_{\text{atom}}$. We seek a representation of $\mathcal{C}$ suitable for input to HIP-NN.
|
| 41 |
+
|
| 42 |
+
To achieve a representation of the molecular geometry that is invariant under rigid transformations (i.e., translations, rotations, and reflections) we work with pairwise distances $r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|$ rather than coordinates $\mathbf{r}_{i}$. Furthermore, we keep only distances satisfying $r_{ij}<R_{\text{cut}}$. In our energy model, we apply a smooth radial cutoff to ensure smoothness with respect to atomic positions.
|
| 43 |
+
|
| 44 |
+
We represent the atomic numbers $Z_{i}$ using a one-hot encoding,
|
| 45 |
+
|
| 46 |
+
$z_{i,a}^{0}=\delta_{Z_{i},\mathcal{Z}(a)},$ (3)
|
| 47 |
+
|
| 48 |
+
where $\delta_{ij}$ is the Kronecker delta and $\mathcal{Z}$ enumerates the atomic numbers under consideration. We benchmark on datasets of organic molecules containing atomic species [H, C, N, O, F] for which $\mathcal{Z}=[1,6,7,8,9]$. By construction, HIP-NN will sum over atomic and feature indices ($i$ and $a$), and is thus invariant to their permutation.
|
| 49 |
+
|
| 50 |
+
### II.2 Atomic features and energies
|
| 51 |
+
|
| 52 |
+
HIP-NN generalizes $z_{i,a}^{0}$ to real-valued, dimensionless atomic features $z_{i,a}^{\ell}$ (i.e., neural network activations) over layers indexed by $\ell=0\ldots N_{\text{layer}}$. [27] Suppressing the feature index $a=1\ldots N_{\text{feature}}^{\ell}$, we call $z_{i}^{\ell}$ the feature vector. The input feature vector $z_{i}^{0}$ represents the species of atom $i$. At subsequent layers, HIP-NN generates successively more abstract, “dressed” representations $z_{i}^{\ell+1}$ of the chemical environment of atom $i$ based upon information $(z_{j}^{\ell},r_{ij})$ from neighboring atoms $j$.
|
| 53 |
+
|
| 54 |
+
The key challenge in HIP-NN is to learn good features $z_{i,a}^{\ell}$ that faithfully capture the chemical environment of atom $i$. Once known, HIP-NN uses linear regression (blue boxes in Fig. 1) on the atomic features to model the local hierarchical energies,
|
| 55 |
+
|
| 56 |
+
$\hat{E}_{i}^{n}=\sum_{a=1}^{N_{\text{feature}}}w_{a}^{n}z_{i,a}^{\ell_{n}}+b^{n},$ (4)
|
| 57 |
+
|
| 58 |
+
where $w_{a}^{n}$ and $b^{n}$ are learned parameters with dimensions of energy. The total HIP-NN energy $\hat{E}$ is then given by Eqs. (1) and (2). Note that only certain network layers $\ell_{n}$ contribute to the energy.
|
| 59 |
+
|
| 60 |
+
### II.3 On-site layers
|
| 61 |
+
|
| 62 |
+
The on-site layers (red squares in Fig. 1) operate on the features $z_{i,a}^{\ell}$ of a single atom,
|
| 63 |
+
|
| 64 |
+
$\hat{z}_{i,a}^{\ell+1}$ $=$ $\!\!{}_{\text{on-site}}f\left(\sum_{b}W_{ab}^{\ell}z_{i,b}^{\ell}+B_{a}^{\ell}\right),$ (5)
|
| 65 |
+
|
| 66 |
+
where $W_{a,b}^{\ell}$ and $B_{a}^{\ell}$ are learned parameters. Various choices of activation function $f(x)$ are possible. Rectifiers (i.e., functions saturating for $x\rightarrow-\infty$ and increasing indefinitely when $x\rightarrow\infty$) are often preferred because they help mitigate the so-called vanishing gradient problem. [42; 43] For HIP-NN, we select the softplus activation function, [33; 44]
|
| 67 |
+
|
| 68 |
+
$f(x)=\log(1+e^{x}).$ (6)
|
| 69 |
+
|
| 70 |
+
To obtain the final atomic features at layer $\ell+1$, we apply a residual network (ResNet) transformation, [26]
|
| 71 |
+
|
| 72 |
+
$z_{i,a}^{\ell+1}=\sum_{b}\left(\tilde{W}_{ab}^{\ell}\hat{z}_{b}^{\ell+1}+\tilde{M}_{ab}^{\ell}z_{i,b}^{\ell}\right)+\tilde{B}_{a}^{\ell},$ (7)
|
| 73 |
+
|
| 74 |
+
where $\tilde{W}_{ab}^{\ell}$, $\tilde{M}_{ab}^{\ell}$, and $\tilde{B}_{a}^{\ell}$ are again learned parameters. Following the suggestion of the ResNet authors, if layers $\ell$ and $\ell+1$ have the same number of features, we instead make $\tilde{M}_{ab}$ unlearnable, and fix $\tilde{M}_{ab}^{\ell}=\delta_{ab}$. Empirically, the ResNet architecture further mitigates the vanishing gradients problem, allowing training of deeper networks.
|
| 75 |
+
|
| 76 |
+
### II.4 Interaction layers
|
| 77 |
+
|
| 78 |
+
The interaction layers (green boxes in Fig. 1) operate similarly to on-site layers, Eq. (5), and additionally transmit information between atoms. The transformation rule for interaction layers is
|
| 79 |
+
|
| 80 |
+
$\hat{z}_{i,a}^{\ell+1}$ $=$ $\!\!{}_{\text{inter.}}f\left(\sum_{j,b}v_{ab}^{\ell}(r_{ij})z_{j,b}^{\ell}+\sum_{b}W_{ab}^{\ell}z_{i,b}^{\ell}+B_{a}^{\ell}\right),$ (8)
|
| 81 |
+
|
| 82 |
+
|
| 83 |
+
Figure 2: Within an interaction layer, the spatial sensitivity functions $s_{\nu}(r_{ij})$ modulate communication between pairs of atoms separated by distance $r_{ij}$. Blue curves: The initial sensitivities $s_{\nu}(r_{ij})$ for $\nu=1\ldots 20$. Dashed black curve: All sensitivities are scaled by the factor $\varphi_{\text{cut}}(r_{ij})$, which introduces a hard spatial cutoff of 15 Bohr.
|
| 84 |
+
|
| 85 |
+
where $v_{ab}^{\ell}(r_{ij})$ collects information from neighboring atoms $j$ that are sufficiently close to $i$, i.e., that satisfy $r_{ij}<r_{\text{cut}}$. $\mu_{\nu,\ell}$="" $\mu_{\nu,\ell}$="" $\sigma_{\nu,\ell}$="" $\sigma_{\nu,\ell}^{\ell}$.="" $i$,="" $v_{ab}^{\ell}(r_{ij})="\sum_{\nu}V_{\nu,ab}^{\ell}s_{\nu}^{\ell}(r_{ij}),$" $v_{ab}^{\ell}(r_{ij})$="" $z_{i,a}^{\ell}$="" $z_{i,b}^{\ell}$="" (10)="" (9)="" (9)}$.="" a="" and="" are="" as="" at="" be="" basis="" by="" c}.$="" d}.$="" distance,="" distances="" expansion="" f}$.="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$="" f}_{\text{cut}}.$
|
| 86 |
+
|
| 87 |
+
The brackets $\langle\cdot\rangle_{D}$ denote an average over molecules within a dataset $D$, $\hat{E}$ is the molecular energy predicted in Eq. (1), and $E$ is the true *ab initio* energy.
|
| 88 |
+
|
| 89 |
+
We optimize the HIP-NN model parameters to minimize both MAE and RMSE. That is, we wish to minimize a loss function,
|
| 90 |
+
|
| 91 |
+
$\mathcal{L}=\frac{1}{\sigma_{E}}(\text{MAE}+\text{RMSE})+\mathcal{L}_{L2}+\mathcal{L}_{R}.$ (14)
|
| 92 |
+
|
| 93 |
+
In this context, we select $D=D_{\text{train}}$ to be the training dataset. The natural energy scale for MAE and RMSE is the standard deviation of molecular energies,
|
| 94 |
+
|
| 95 |
+
$\sigma_{E}=\sqrt{\langle(E-\langle E\rangle)^{2}\rangle_{D}}.$ (15)
|
| 96 |
+
|
| 97 |
+
Importantly, the loss function includes two regularization terms. The first is a $L_{2}$ regularization on weight tensors appearing in the equations for energy regression (4), on-site layers (5), interaction layers (8), (9), and ResNet transformation (7):
|
| 98 |
+
|
| 99 |
+
$\mathcal{L}_{L2}=\lambda_{L2}\left(\frac{||w||_{2}^{2}}{\sigma_{E}^{2}}+||W||_{2}^{2}+||V||_{2}^{2}+||\hat{W}||_{2}^{2}+||\hat{M}||_{2}^{2}\right).$ (16)
|
| 100 |
+
|
| 101 |
+
We find that a sufficiently small hyperparameter $\lambda_{L2}$ is effective at reducing outlier HIP-NN predictions while introducing minimal bias to the model.
|
| 102 |
+
|
| 103 |
+
To encourage hierarchality of the energy terms, we include a second regularization term,
|
| 104 |
+
|
| 105 |
+
$\mathcal{L}_{R}=\lambda_{R}\langle R\rangle_{D},$ (17)
|
| 106 |
+
|
| 107 |
+
that penalizes the *non-hierarchicality* $R$ of energy contributions,
|
| 108 |
+
|
| 109 |
+
$R=\sum_{n=1}^{N_{\text{interaction}}}\sum_{i=1}^{N_{\text{atom}}}\frac{(\hat{E}_{i}^{n})^{2}}{(\hat{E}_{i}^{n})^{2}+(\hat{E}_{i}^{n-1})^{2}}.$ (18)
|
| 110 |
+
|
| 111 |
+
When HIP-NN is functioning properly, we commonly observe that $\hat{E}_{i}^{n}$ decays rapidly in $n$. A large value of $R$ thus indicates malfunction of HIP-NN for the given molecular input.
|
| 112 |
+
|
| 113 |
+
#### II.2.2 Stochastic optimization
|
| 114 |
+
|
| 115 |
+
We use the Adaptive Moment Estimation (Adam) algorithm, [45] a variant of stochastic gradient descent (SGD), to train HIP-NN. Let $U=\{w,b,W,B,V,\sigma,\mu,\hat{W},\hat{B},\hat{M}\}$ denote the full set of model parameters, Eqs. (4)–(10). The goal, then, is to evolve $U$ to minimize the loss $\mathcal{L}\big{|}_{D}$, Eq. (14), evaluated on the dataset $D=D_{\text{train}}$ of training molecules.
|
| 116 |
+
|
| 117 |
+
In SGD, one partitions the training data into random disjoint sets of *mini-batches*, $D=\hat{D}_{1}\cup\hat{D}_{2}\cdots\cup\hat{D}_{N}$. For each mini-batch $\hat{D}$, one evolves $U$ in the direction of the negative gradient $-\nabla_{U}\mathcal{L}\big{|}_{\hat{D}}$, which is a stochastic approximation to $\nabla_{U}\mathcal{L}\big{|}_{D}$, the gradient evaluated on the full dataset. Training time is measured in *epochs*. Each epoch corresponds to a pass through all mini-batches $\hat{D}_{i}$. After each epoch, the mini-batch partition is re-randomized. Compared to plain SGD, Adam speeds convergence by selecting its updates as a function of a decaying average of previous gradients. The Adam parameters are its learning rate $\eta$ and exponential decay factors $\beta_{1}$ and $\beta_{2}$.
|
| 118 |
+
|
| 119 |
+
To reduce overfitting, we use an *early stopping* procedure to terminate the learning process when the MAE on a *validation* dataset $D_{\text{validate}}$ (separate from the training data $D_{\text{train}}$) stops improving. [46] The Adam learning rate $\eta$ is initialized to $\eta_{\text{init}}$ and annealed as follows. We train the network while tracking best_score, the best validation MAE yet observed, and corresponding model parameters $U$. The learning rate is fixed to $\eta_{\text{init}}$ for the first $t_{\text{init}}$ epochs. Afterwards, if best_score plateaus (does not drop for a period of $t_{\text{patience}}$ epochs) then the learning rate $\eta$ is multiplied by $\alpha_{\text{decay}}$, causing the gradient descent procedure to take finer steps. Training is terminated if $\eta$ decreases twice without any improvement to best_score. Training is forcefully terminated if $t_{\text{max}}$ epochs elapse. The final parameter set $U$ is taken to be the one which produced the lowest validation error.
|
| 120 |
+
|
| 121 |
+
### II.6 Implementation details
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Here we discuss hyperparameters, initialization of model parameters, and our numerical implementation.
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As illustrated in Fig. (1) we use $n=0\ldots N_{\text{interaction}}$ hierarchical contributions to the energy model. We choose $N_{\text{interaction}}=2$ interaction layers, a number comparable to previous studies. [31; 32; 33; 28; 34] Each interaction layer is followed by $N_{\text{on-site}}$ on-site layers. Thus the total number of nonlinear layers is $N_{\text{interaction}}\times(1+N_{\text{on-site}})$. We fix the feature vector size to a constant $N_{\text{feature}}=|z_{i}^{\ell}|$ for all layers $\ell>0$. Recall that the input feature vector $z_{i}^{0}$ is a one-hot encoding of the atomic species. In our numerical studies, we consider models with varying $N_{\text{on-site}}$ and $N_{\text{feature}}$ hyperparameters.
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The initial network weights $w$, $W$, $V$, $\hat{W}$, and $\hat{M}$ from Eqs. (4)–(9) are drawn from a uniform distribution according to the Glorot initialization scheme. [47] We initialize the network biases $b$, $B$, and $\hat{B}$ to zero. Next, we set the zeroth-order energy model $\hat{E}^{n=0}$ to minimize the least squares error on the training data. The corresponding linear regression parameters $w_{a}^{0}$ and $b^{0}$ are held fixed for the duration of training. For subsequent orders $n>0$ we rescale the Glorot initialized weights $w_{a}^{n}$ by a factor $\sigma_{E}/10^{-2n}$ to impose the expected energy scale and hierarchical decay. During training, we factorize $w_{a}^{n}=\sigma_{E}\tilde{w}_{a}^{n}$ and treat $\tilde{w}_{a}^{n}$ as the learnable, dimensionless parameters.
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We employ $N_{\text{sensitivity}}=20$ sensitivity functions $s_{\nu}^{\ell}(r)$ as given by Eq. (10). Initially, the sensitivities are independent of layer $\ell$. We select initial inverse distances $\mu_{\nu,\ell}^{-1}$ with uniform separation between $R_{\text{low}}^{-1}$ and $R_{\text{high}}^{-1}$ for $\nu=1\ldots N_{\text{sensitivity}}$. The lower and upper distances are $R_{\text{low}}=1.7$ Bohr and $R_{\text{high}}=10$ Bohr. The
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width parameters $\sigma_{\nu,\ell}$ are initialized to the constant value $2N_{\mathrm{sensitivity}}R_{\mathrm{low}}$ , which allows moderate overlap between adjacent sensitivity functions, as shown in Fig. 2. The sensitivities are modulated by a cutoff $R_{\mathrm{cutoff}} = 15$ Bohr.
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The loss function regularization terms are weighted by $\lambda_{L2} = 10^{-6}$ and $\lambda_R = 10^{-2}$ . The training data minibatches each contain 30 molecules. An additional validation dataset of 1000 molecules (separate from both training and test data) is used to determine the early stopping time. The Adam decay hyperparameters are $\beta_{1} = 0.9$ , $\beta_{2} = 0.999$ , and the initial learning rate is $\eta_{\mathrm{init}} = 10^{-3}$ . The learning rate decay factor is $\alpha_{\mathrm{decay}} = 0.5$ . The patience is $t_{\mathrm{patience}} = 50$ epochs, the initial training time before annealing is $t_{\mathrm{init}} = 100$ , and the maximum training time is $t_{\mathrm{max}} = 2000$ .
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We implemented HIP-NN using the Theano framework $^{48}$ and Lasagne neural network library $^{49}$ with custom layers. Theano calculates gradients of the loss function using backpropagation (also known as reverse-mode automatic differentiation $^{50}$ ). Theano also compiles the model for high performance execution on GPU hardware. A single Nvidia Tesla P100 GPU requires about 1 minute to complete one training epoch for the full QM9 dataset (discussed below) with $N_{\mathrm{on-site}} = 3$ and $N_{\mathrm{feature}} = 80$ . The full training procedure typically completes in 1000 to 2000 epochs.
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# III. RESULTS
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# A. QM9 dataset
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The QM9 dataset $^{41,51}$ is comprised of about 134k organic molecules containing H and nine or fewer C, N, O, and F atoms. Properties were calculated at the B3LYP/6-31G(2df,p) level of quantum chemistry. About 3k molecules within QM9 fail a geometric consistency check $^{41}$ and are commonly removed from the dataset. $^{20,31,33}$ The authors of QM9 had difficulty in the energy-minimization procedure for 11 more molecules, $^{41}$ which we also remove. Our pruned dataset thus contains about 131k molecules. This dataset is then randomly partitioned into training, validation, and testing datasets, $D_{\mathrm{all}} = D_{\mathrm{train}} \cup D_{\mathrm{validate}} \cup D_{\mathrm{test}}$ . We benchmark on varying amounts of training data, $|D_{\mathrm{train}}| = N_{\mathrm{train}}$ . The validation dataset controls early stopping and has fixed size $|D_{\mathrm{validate}}| = 1000$ . All remaining molecules are included in the testing dataset $D_{\mathrm{test}}$ . Every HIP-NN error statistic $S$ reported below (e.g., MAE and RMSE over $D_{\mathrm{test}}$ ) is actually a sample average $\mu_S$ over $N_{\mathrm{model}} = 8$ models, each with a differently randomized split of the training/validation/testing data. We calculate error bars as $\sigma_S / \sqrt{N_{\mathrm{model}}}$ , where $\sigma_S$ is the sample standard deviation over the $N_{\mathrm{model}}$ models.
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Table I benchmarks HIP-NN against recent state-of-the-art models reported in the literature. The HIP-NN models contain $N_{\mathrm{on-site}} = 3$ on-site layers and $N_{\mathrm{feature}} = 80$ atomic features per layer. Following previous work,
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Figure 3. Larger non-hierarchicality $R$ , Eq. (18), indicates a breakdown of the energy hierarchy assumption, Eq. (2), and correlates with larger error in the HIP-NN predictions, as observed in quantile functions $Q_{\mathrm{err}}(p,R)$ from Eq. (19). The gray background shows the rescaled probability distribution of $\log R$ . The scatter of $Q_{\mathrm{err}}$ at very small and large values of $R$ is likely due to a lack of data.
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we report the mean absolute error (MAE) using training sets of three different sizes. HIP-NN achieves an MAE of 0.26 kcal/mol when trained on the largest datasets and, to our knowledge, outperforms all existing models.
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Table II shows HIP-NN performance as a function of model complexity. We fix $N_{\mathrm{on-site}} = 3$ on-site layers and allow the number of atomic features $N_{\mathrm{feature}}$ to vary between 5 and 80. The HIP-NN parameter count grows roughly as $N_{\mathrm{feature}}^2$ . For each complexity level we calculate three error statistics: (1) MAE, (2) RMSE, and (3) the percentage of molecules in the testing set whose predicted energy has an absolute error that exceeds 1 kcal/mol (a common standard of chemical accuracy). In the last two rows we report the performance of HIP-NN trained without hierarchical energy contributions [i.e., fixing $w^n = b^n = 0$ for $n = \{0,1\}$ in Eq. (4), so that only $\hat{E}^{n=2}$ contributes to $\hat{E}$ ], and without hierarchical regularization, Eq. (17). With these limitations, the MAE performance degrades by $\approx 9\%$ , the fraction of errors above 1 kcal/mol increases by $\approx 22\%$ , but the RMSE values are comparable.
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Note that with only 5 atomic features (corresponding to $1.6\mathrm{k}$ parameters) the MAE of $1.2\mathrm{kcal / mol}$ already approaches chemical accuracy. This performance is remarkable, given that the parameter count is roughly two orders of magnitude smaller than the QM9 dataset size. For reference, the standard deviation of energies in QM9 is $\sigma_{E}\approx 238~\mathrm{kcal / mol}$ . We observe that the HIP-NN error tends to decrease with increasing $N_{\mathrm{feature}}$ , but the non-hierarchical HIP-NN model with $N_{\mathrm{feature}} = 80$ performs worse than that with $N_{\mathrm{feature}} = 60$ , possibly due to overfitting.
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Even though our best MAE of $0.26\mathrm{kcal / mol}$ is well under $1\mathrm{kcal / mol}$ , approximately $2.3\%$ of the predicted molecular energies have an error that exceeds $1\mathrm{kcal / mol}$ ; there is still room for improved ML models with fewer outliers in the energy predictions.
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Table I. QM9 performance (MAE in kcal/mol) for various models reported in the literature.
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| Ntrain + Nvalidate | HIP-NN | MTM22 | SchNet33 | MPNN31 | HDAD+KRR20 | DTNN32 |
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| --- | --- | --- | --- | --- | --- | --- |
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| 110426 | 0.256 ± 0.003 | - | 0.31 | 0.42 | 0.58 | - |
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| 100000 | 0.261 ± 0.002 | - | 0.34 | - | - | 0.84 |
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| 50000 | 0.354 ± 0.004 | 0.41 | 0.49 | - | - | 0.94 |
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Table II. QM9 performance for HIP-NN models with varying complexity.
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| Nfeature | Parameter count | MAE (kcal/mol) | RMSE (kcal/mol) | Errors above 1 kcal/mol (%) |
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| --- | --- | --- | --- | --- |
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| 5 | 1.6k | 1.177 ± 0.014 | 1.851 ± 0.019 | 42.18 ± 0.58 |
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| 10 | 4.9k | 0.653 ± 0.007 | 1.077 ± 0.015 | 18.95 ± 0.37 |
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| 20 | 17k | 0.398 ± 0.005 | 0.706 ± 0.025 | 6.60 ± 0.16 |
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| 40 | 61k | 0.274 ± 0.003 | 0.539 ± 0.014 | 2.65 ± 0.06 |
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| 60 | 134k | 0.261 ± 0.004 | 0.552 ± 0.024 | 2.37 ± 0.09 |
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| 80 | 234k | 0.256 ± 0.003 | 0.527 ± 0.020 | 2.26 ± 0.07 |
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| 60 (no hierarchy) | 134k | 0.278 ± 0.008 | 0.522 ± 0.013 | 2.75 ± 0.21 |
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| 80 (no hierarchy) | 234k | 0.293 ± 0.008 | 0.539 ± 0.013 | 3.10 ± 0.21 |
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Figure 3 shows that the non-hierarchicality $R$ is an indicator of inaccurate HIP-NN energy predictions. This is reasonable because large $R$ indicates breakdown of the energy hierarchy assumption, i.e., non-decaying contributions $\hat{E}_i^n$ in Eq. (2). We quantify this correspondence by considering the distribution of absolute error $|\hat{E} - E|$ over the testing dataset $D_{\mathrm{test}}$ . In Fig. 3 we visualize the quantile function $Q_{\mathrm{err}}(p, R)$ defined to satisfy
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$$
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P \left(\left| \hat {E} - E \right| < Q _ {\text {e r r}} | R\right) = p, \tag {19}
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$$
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for various cumulative probabilities $p$ and non-hierarchicalities $R$ , combined over 8 random splits of the QM9 data using HIP-NN with $N_{\mathrm{feature}} = 80$ . In the background of the plot, we show the distribution of molecules using a histogram in $\log R$ . The bin width is $\Delta \log_{10} R = 0.066$ . The error of a random molecule, drawn from a given bin of $R$ , falls below the quantile $Q_{\mathrm{err}}(p, R)$ with probability $p$ ; thus $1 - p$ gives the empirical probability that a molecular error will exceed $Q_{\mathrm{err}}$ .
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We observe that, among the vast majority of the dataset $(R\gtrsim 3\times 10^{-2})$ , increasing $R$ corresponds to larger error quantiles. In other words, if the energy contributions $\hat{E}_i^n$ are more hierarchical in $n$ for a given molecule, then HIP-NN is more likely to be accurate. This is true both for the typical $(p = 0.5)$ and outlier $(p = 0.99)$ quantiles $Q_{\mathrm{err}}$
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# B. MD Trajectories
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Here, we demonstrate that HIP-NN also performs well when trained on energies obtained from molecular dynamics (MD) trajectories. We use datasets generated by Schütt et. al $^{32}$ consisting of MD trajectories for four molecules in vacuum: benzene, malonaldehyde, salicylic acid, and toluene. The temperature is $T = 500 \mathrm{~K}$ . Energies and forces were calculated using density-functional theory with the PBE exchange-correlation potential. $^{52}$
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Previous studies on the Gradient Domain Machine Learning (GDML) $^{53}$ and SchNet $^{33}$ models have also benchmarked on this dataset.
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We use training datasets of two sizes, $N_{\mathrm{train}} = 1\mathrm{k}$ and $50\mathrm{k}$ , randomly sampled from the full MD trajectory data. We use an additional $N_{\mathrm{validate}} = 1\mathrm{k}$ random conformations for early stopping. The remaining conformations from each MD trajectory comprise the test data. For the case with $N_{\mathrm{train}} = 1\mathrm{k}$ conformers, we use a very simple HIP-NN model with $N_{\mathrm{on-site}} = 0$ and $N_{\mathrm{feature}} = 20$ , which corresponds to about $10\mathrm{k}$ model parameters. When training on $N_{\mathrm{train}} = 50\mathrm{k}$ conformers, we instead use $N_{\mathrm{on-site}} = 3$ and $N_{\mathrm{feature}} = 40$ , which corresponds to about $59\mathrm{k}$ model parameters. The resulting MAE benchmarks are shown in Table III. When restricted to training on only energies, HIP-NN is comparable to or better than the other models included in this benchmark. However, our current implementation of HIP-NN does not train on forces. When SchNet and GDML are trained with force data, they outperform HIP-NN. Extending our model to force training is straightforward and will be reported in future work.
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Finally, we note that the energies in this dataset are only expressed with a precision of $0.1\mathrm{kcal / mol}$ ,[54] which is comparable to many MAEs in Table III. This suggests that lower MAEs may be possible with a more precise dataset, especially with training set size $N_{\mathrm{train}} = 50\mathrm{k}$ .
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# IV. DISCUSSION
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HIP-NN achieves state-of-the-art performance on both QM9 and MD trajectory datasets, with MAEs well under 1 kcal/mol. We show that HIP-NN continues to perform well even when the parameter count is drastically reduced. We attribute the success of HIP-NN to a combination of design decisions. One is the use of sensitivity functions[28,32,33] with an inverse-distance parameterization.[12,15,30,53] Thus we achieve finer sensitivity at shorter
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Table III. Accuracy of energy predictions for finite temperature molecular conformers. We report the MAE in units of kcal/mol for various training set sizes, model types, and molecule types. Including forces in the training data significantly improves the predictions. Best results for each training category are shown in bold.
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| | Ntrain = 1k | | | | Ntrain = 50k | | | |
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| --- | --- | --- | --- | --- | --- | --- | --- | --- |
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| | Training on energy | | On energy & forces | | Training on energy | | On energy & forces | |
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| | HIP-NN | SchNet33 | SchNet | GDML53 | HIP-NN | SchNet | DTNN | SchNet |
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| Benzene | 0.162 ± 0.002 | 1.19 | 0.08 | 0.07 | 0.064 ± 0.002 | 0.08 | 0.04 | 0.07 |
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| Malonaldehyde | 0.970 ± 0.019 | 2.03 | 0.13 | 0.16 | 0.094 ± 0.001 | 0.13 | 0.19 | 0.08 |
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| Salicylic acid | 1.444 ± 0.024 | 3.27 | 0.20 | 0.12 | 0.195 ± 0.002 | 0.25 | 0.41 | 0.10 |
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| Toluene | 0.880 ± 0.019 | 2.95 | 0.12 | 0.12 | 0.144 ± 0.004 | 0.16 | 0.18 | 0.09 |
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ranges and coarser sensitivity at longer ranges. Another effective design decision is the use of the ResNet transformation, Eq. (7), a now-common technique to improve deep neural networks.[26,33] A small amount of $L_{2}$ regularization, Eq. (16), is very helpful for stabilizing the root-mean-squared error, but has little effect on the MAE. Annealing the learning rate when the validation score plateaus improves optimization of the model parameters.
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The physically motivated hierarchical energy decomposition, Eq. (2), and corresponding regularization, Eq. (17), noticeably improve HIP-NN performance. Without this decomposition, the MAE increases by $9\%$ and the fraction of errors under 1 kcal/mol increases by $22\%$ . This improvement is intriguing, given that the energy decomposition negligibly increases the total parameter count. Also, the lower order energy contributions are formally redundant given that the linear pass-throughs $(\hat{M}_{el}^{I} = \delta_{ab})$ of the ResNet transformation, Eq. (7), could allow features to propagate unchanged through the network.
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We interpret the hierarchical energy terms as follows. At zeroth order, $\hat{E}_i^{n = 0}$ corresponds to the dressed atom approximation. $^{15}$ Next, $\hat{E}_i^{n = 1}$ captures information about distances between atom $i$ and its local neighbors, but goes beyond traditional pairwise-interactions by combining local pairwise information. The final term, $\hat{E}_i^{n = 2}$ , captures more detailed geometric information such as angles between atom triples. For our best performing models with fixed $N_{\mathrm{interaction}} = 3$ , we find that the truncated model energy $\hat{E}_{\mathrm{trunc}}^k = \sum_i\sum_{n = 0}^k\hat{E}_i^n$ has an MAE that decays exponentially with $k$ .
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Despite achieving state-of-the-art MAEs, we still find that the HIP-NN energy predictions on QM9 have an error exceeding 1 kcal/mol about $2.3\%$ of the time. For certain applications this error rate may not be acceptable. Future work may focus on developing models that have a lower failure rate. Another important research direction is to develop methods for inferring when the model prediction is unreliable. We provide a step in this direction by showing that large $R$ (which indicates failure of the hierarchical energy decomposition) implies that the HIP-NN energy prediction is less reliable.
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As methodology improves, the machine learning community has room to study increasingly challenging and varied datasets (e.g. Refs. 33 and 55) in pursuit of improved accuracy and transferability. Other interest-
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ing research directions include using active learning to construct diverse datasets that cover unusual corners of chemical space,[22,56-58] and using machine learning to glean chemical and physical insight.[59]
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# V. CONCLUSION
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This paper introduces and pedagogically describes HIP-NN, a machine learning technique for modeling molecular energies. By using an appropriate molecular representation, HIP-NN naturally encodes permutation, rotation and translation invariances. Inspired by the many-body expansion, HIP-NN also encodes locality and hierarchical properties that one would expect of molecular energies from physical principles. HIP-NN improves significantly upon the state-of-the-art in predicting energies on the QM9 dataset, a standard benchmark of organic molecules. HIP-NN also shows promise on datasets of finite-temperature molecular trajectories. The HIP-NN energy function is smooth, and thus can potentially drive MD simulations. In addition to enhancing performance, the hierarchical decomposition of energy yields an empirical measure of model uncertainty: If the energy hierarchy produced by HIP-NN does not decay sufficiently fast, the corresponding molecular energy prediction is less likely to be accurate.
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# ACKNOWLEDGMENTS
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The authors thank Benjamin Nebgen, Adrian Roitberg, and Sergei Tretiak for valuable discussions and feedback. Our work was supported by the Laboratory Directed Research and Development (LDRD) program, the Advanced Simulation and Computing (ASC) program, and the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory (LANL). Computations were performed using the CCS-7 Darwin cluster at LANL.
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|
| 1 |
+
# Task: Develop a Substantive Follow-up to "Hierarchical Modeling of Molecular Energies Using a Deep Neural Network"
|
| 2 |
+
|
| 3 |
+
## Objective — Here's what you're doing
|
| 4 |
+
|
| 5 |
+
Develop a substantive scientific follow-up to the Hierarchically Interacting Particle Neural Network (HIP-NN) by Lubbers, Smith, and Barros, "Hierarchical modeling of molecular energies using a deep neural network," J. Chem. Phys. 148, 241715 (2018), in the direction of **characterizing where the hierarchical decomposition breaks down**, identifying the chemical motifs responsible for those breakdowns, and building a calibrated uncertainty model from the hierarchy-failure signal.
|
| 6 |
+
|
| 7 |
+
## The Problem — Here's why it matters
|
| 8 |
+
|
| 9 |
+
A machine-learning interatomic potential (MLIP) for molecular systems maps an atomic configuration $\mathcal{C} = \{(Z_i, \mathbf{r}_i)\}$ to a scalar energy $\hat{E}(\mathcal{C})$. Modern architectures (HIP-NN, SchNet, PhysNet, MPNN families) achieve sub-1 kcal/mol mean absolute error on benchmark organic-molecule datasets, often well below "chemical accuracy" thresholds. But the headline MAE hides a tail: even at $0.26$ kcal/mol MAE on QM9, HIP-NN's original paper reports that **2.3% of molecules have prediction errors exceeding 1 kcal/mol**. For molecular dynamics, transition-state characterization, or high-throughput chemical screening, this tail matters far more than the mean — a one-in-fifty failure rate is the difference between a tool a chemist trusts and a black box that needs human spot-checking.
|
| 10 |
+
|
| 11 |
+
The deeper question is: when a HIP-NN-style model fails, *can the model itself tell you it failed?* HIP-NN's hierarchical decomposition was designed in part to make this question tractable. Energies decompose as $\hat{E} = \sum_i \sum_n \hat{E}_i^n$ over per-atom hierarchical orders $n = 0, 1, \dots, N_{\text{interaction}}$, with order $n$ capturing roughly $(n{+}1)$-body interactions. When the model functions properly, $\hat{E}_i^n$ should decay rapidly with $n$, and the non-hierarchicality
|
| 12 |
+
|
| 13 |
+
$$R = \sum_{n=1}^{N_{\text{interaction}}} \sum_{i=1}^{N_{\text{atom}}} \frac{(\hat{E}_i^n)^2}{(\hat{E}_i^n)^2 + (\hat{E}_i^{n-1})^2}$$
|
| 14 |
+
|
| 15 |
+
should be small. The original paper's Figure 3 shows that when a model is regularized using R as a component in the loss function that larger $R$ correlates with larger prediction error — the model has a built-in uncertainty signal.
|
| 16 |
+
|
| 17 |
+
But the paper does not unpack that signal further. It does not identify *which chemical motifs* drive hierarchy failure, *which structural features* correspond to high-$R$ regions of the molecular manifold, *whether the $R$ signal is calibrated* (predicted-uncertainty bins matching actual-error magnitudes), or *whether the resulting failure modes are compact* (a few dominant motifs explain most failures) versus diluted (every molecule contributes a little to the failure tail with no clear structure). These are direct follow-ups to the paper's own uncertainty claim, and they matter for any downstream use that needs calibrated prediction-confidence intervals rather than just a low mean error.
|
| 18 |
+
|
| 19 |
+
## Background — Here's what you need to know about the original paper
|
| 20 |
+
|
| 21 |
+
The Lubbers, Smith, and Barros 2018 paper introduces HIP-NN and validates it on QM9 and the original MD17 trajectory benchmarks. Read `paper.md` in full before beginning. The technical components of the original method are:
|
| 22 |
+
|
| 23 |
+
1. **Molecular representation.** Atomic positions $\mathbf{r}_i$ are encoded via pairwise distances $r_{ij} = \|\mathbf{r}_i - \mathbf{r}_j\|$ with a radial cutoff $R_{\text{cut}} = 15$ Bohr (smooth, cosine-modulated). Atomic species $Z_i$ are one-hot encoded over the elements present in the dataset (H, C, N, O, F for QM9). The representation is invariant under rigid translations, rotations, and atom-index permutations by construction.
|
| 24 |
+
|
| 25 |
+
2. **Network architecture.** HIP-NN alternates *interaction layers* (mixing information between pairs of atoms within $R_{\text{cut}}$, modulated by 20 learned inverse-distance sensitivity functions $s_\nu^\ell(r_{ij})$) and *on-site layers* (per-atom feature transformations), with residual connections and softplus activations. The baseline architecture has $N_{\text{interaction}} = 2$ interaction layers, $N_{\text{on-site}} = 3$ on-site layers per interaction block, and $N_{\text{feature}} = 80$ atomic features per layer (~234k parameters total).
|
| 26 |
+
|
| 27 |
+
3. **Hierarchical energy regression.** The total energy is $\hat{E} = \sum_i \sum_n \hat{E}_i^n$ with $\hat{E}_i^n = \sum_a w_a^n z_{i,a}^{\ell_n} + b^n$ (linear regression on the features at selected layers $\ell_n$). Order $n=0$ is the dressed-atom approximation; higher $n$ adds increasingly many-body content.
|
| 28 |
+
|
| 29 |
+
4. **Hierarchical regularization.** A loss term $\mathcal{L}_R = \lambda_R \langle R \rangle_D$ (with $\lambda_R = 10^{-2}$) penalizes molecules whose hierarchical contributions $\hat{E}_i^n$ do *not* decay with $n$. This encourages the trained model to actually use the hierarchical structure, and creates the $R$-as-uncertainty-signal correspondence shown in Figure 3.
|
| 30 |
+
|
| 31 |
+
5. **Training protocol.** Adam optimizer ($\eta_{\text{init}} = 10^{-3}$, $\beta_1 = 0.9$, $\beta_2 = 0.999$), L2 regularization at $\lambda_{L2} = 10^{-6}$, minibatch size 30, validation-based early stopping with patience $t_{\text{patience}} = 50$ and learning-rate annealing factor $\alpha_{\text{decay}} = 0.5$; training terminates with a patience of 100 epochs. Initial training period $t_{\text{init}} = 100$ epochs, max $t_{\text{max}} = 2000$ epochs. The paper trained on $\sim$8 random splits per benchmark and reported sample-mean MAE with standard-error error bars.
|
| 32 |
+
|
| 33 |
+
**Reference baseline results from the paper (Table I, QM9 MAE in kcal/mol):**
|
| 34 |
+
|
| 35 |
+
| $N_{\text{train}} + N_{\text{validate}}$ | HIP-NN |
|
| 36 |
+
|---|---|
|
| 37 |
+
| 110,426 | $0.256 \pm 0.003$ |
|
| 38 |
+
| 100,000 | $0.261 \pm 0.002$ |
|
| 39 |
+
| 50,000 | $0.354 \pm 0.004$ |
|
| 40 |
+
|
| 41 |
+
**Figure 3 (the uncertainty story):** the quantile function $Q_{\text{err}}(p, R)$ — error level at the $p$-th percentile, conditioned on non-hierarchicality $R$ — increases monotonically with $R$ across the bulk of the QM9 dataset ($R \gtrsim 3 \times 10^{-2}$). This is the empirical correspondence the paper invokes as a built-in uncertainty estimate. The paper stops at showing the correspondence — it does not characterize the chemistry driving the high-$R$ tail, does not assess calibration quantitatively, and does not measure the compactness of the failure motifs.
|
| 42 |
+
|
| 43 |
+
## Contract — Here's exactly what counts as success
|
| 44 |
+
|
| 45 |
+
The substantive contribution is to characterize where HIP-NN's hierarchical decomposition breaks down on out-of-distribution chemistry, identify the structural motifs responsible for those breakdowns, build a calibrated uncertainty model based on hierarchy-failure signals, and assess whether the failure motifs are compact (a few motifs explain most failures) or diluted (failures spread across many uncorrelated structural features). You design the motif-identification methodology; you design the uncertainty model on top of (or extending) the paper's $R$; you choose how to validate calibration. The HIP-NN architecture and training protocol are held fixed so the comparison evaluates *what the hierarchy reveals about itself*, not *what a different model would reveal*.
|
| 46 |
+
|
| 47 |
+
Your submission clears **three independent bars**:
|
| 48 |
+
|
| 49 |
+
1. **Substantive contribution** — the analysis identifies a meaningful library of failure motifs in the out-of-distribution chemistry, the proposed uncertainty model is calibrated against actual prediction errors on held-out chemistry, and the motif library is shown to be compact (a Pareto analysis of motif → failure attribution). (See "Operationalized success thresholds" below.)
|
| 50 |
+
2. **Honest framing** — motif identification reports its statistical support (how many test-set molecules each motif covers, what fraction of failures it explains); calibration is evaluated quantitatively with reported false-positive and false-negative rates; cases where the uncertainty model fails are reported as themselves.
|
| 51 |
+
3. **Code–paper consistency** — the HIP-NN architecture and training procedure described in the paper match what the code (hippynn package) implements. Hyperparameters quoted in the paper match the logs.
|
| 52 |
+
|
| 53 |
+
These three bars are checked independently by the rubric.
|
| 54 |
+
|
| 55 |
+
### Operationalized success thresholds
|
| 56 |
+
|
| 57 |
+
| Axis | Floor (must clear) | "Substantial" (the hallmark) |
|
| 58 |
+
|---|---|---|
|
| 59 |
+
| **Reimplementation of HIP-NN on the in-distribution training subset** | Reimplemented HIP-NN matches or improves on the paper-as-reported MAE at the comparable training-set size (e.g., $\leq$ 0.354 kcal/mol when comparing to the paper's 50k-training reference, with reasonable extrapolation across training-size differences between the agent's $\leq 8$-heavy-atom subset and the paper's full-QM9 training sizes) | Reimplemented HIP-NN MAE is **at or below** paper-as-reported across every training-size comparison reported — failure motifs are anchored on a faithful baseline |
|
| 60 |
+
| **Failure motif identification** | At least 3 distinct failure motifs identified, each with quantitative statistical support (test-set coverage, error-distribution skew) | At least 5 motifs identified, spanning multiple chemistry categories (structural, conformational, electronic); each motif's connection to the hierarchical decomposition failure is mechanistically explained |
|
| 61 |
+
| **Uncertainty model calibration** | Calibration curve and expected calibration error (ECE) reported on held-out 9-heavy-atom molecules; precision and recall for flagging high-error predictions stated at one or more chosen thresholds | Calibration curve close to identity (within $\pm 10\%$ across the predicted-uncertainty range); ECE $\leq 0.05$; precision and recall reported across a sweep of thresholds with a chosen operating point justified |
|
| 62 |
+
| **Motif compactness** | A Pareto-style cumulative-attribution plot (motif rank vs. cumulative fraction of failures explained) is reported | Top-$K$ motifs (with $K \leq 5$) explain $\geq 80\%$ of the identified failures, demonstrating that the failure modes are interpretable rather than diluted across the entire test set |
|
| 63 |
+
| **Conformational transfer of the motif library** | The minimized-geometry limitation of the motif library is addressed at least by substantive analytical engagement — a reasoned account of which motifs would or would not transfer to non-minimized (MD-sampled) conformations, and why. (Running the quantitative transfer test empirically on ANI-1x or rMD17 is the substantial outcome, not the floor.) | Quantitative transfer test executed on ANI-1x (or, as a smaller-scope alternative, rMD17) with per-motif transfer-rate analysis and explicit characterization of which motifs do and don't transfer from minimized to MD-sampled geometries |
|
| 64 |
+
|
| 65 |
+
### Held fixed (parity constraints)
|
| 66 |
+
|
| 67 |
+
These are not part of the design space. They must remain identical to the paper's protocol:
|
| 68 |
+
|
| 69 |
+
- **The chemistry**: HCNOF only, as in QM9. No transition metals, no halogens beyond F. (Multi-element MLIP benchmarks built around other element sets are not transferable to this comparison.)
|
| 70 |
+
- **The dataset**: QM9, as fetched into `data/qm9/`, with the standard pruning of $\sim$3k geometric-consistency-failure molecules and the 11 additional energy-minimization failures the paper removes. Use the pruned $\sim$131k-molecule dataset.
|
| 71 |
+
- **The train/test split based on heavy-atom count**: train HIP-NN on the **$\leq 8$-heavy-atom subset** of QM9 (the in-distribution set, where "heavy atom" means C/N/O/F, not H); evaluate the trained model and identify failure motifs on the **9-heavy-atom subset** (the extrapolation set). The validation split for early stopping is drawn from within the $\leq 8$-heavy-atom subset.
|
| 72 |
+
- **The HIP-NN architecture and training protocol**: $N_{\text{interaction}} = 2$ interaction layers, $N_{\text{on-site}} = 3$ on-site layers, $N_{\text{feature}} = 80$ atomic features, 20 sensitivity functions, hierarchical regularization at $\lambda_R = 10^{-2}$, L2 regularization at $\lambda_{L2} = 10^{-6}$, Adam optimizer with the paper's $\eta_{\text{init}}$, $\beta_1$, $\beta_2$, minibatch size 30, validation-based early stopping at patience 50 and decay 0.5. The motivation for holding these fixed is that the comparison evaluates the hierarchy's failure structure, not the architecture's expressive power. (If your compute budget cannot reach the paper's $t_{\text{max}} = 2000$ epochs, reduce the schedule transparently and disclose; see the matched-budget note below.)
|
| 73 |
+
- **Energy-only training**: HIP-NN in the original paper trains on molecular energies only, not forces. Match this for the QM9 experiments. (Force-only training is a different study.)
|
| 74 |
+
- **No DFT generation**: you may not generate new DFT calculations, run quantum chemistry calculations yourself, or use PySCF / Psi4 / ORCA / Gaussian / VASP to compute new reference data. The contract holds the training data fixed at QM9 as provided.
|
| 75 |
+
- **Validation anchor**: Comparison of MAE between agent's reimplementation and the paper's published number is anchored on QM9 (which has not changed since 2014). Transfer tests (on ANI-1x or rMD17) are reported as a complementary analysis, not as a substitute for the QM9 in-distribution comparison.
|
| 76 |
+
|
| 77 |
+
> **Note on matched-budget framing.** The contract evaluates your method's behaviour on the paper's published training regime ($\sim$1000–2000 epochs, $N_{\text{feature}} = 80$, $\sim$234k parameters). A reduced-regime comparison — e.g., training a smaller HIP-NN (reduced $N_{\text{feature}}$), fewer epochs, fewer seeds — does not satisfy the contract even if it produces well-defined paired comparisons within the reduced regime. If your compute budget cannot reach the published regime, run the largest model and longest schedule your budget supports, **disclose the gap to paper-as-reported in Experimental Setup**, and address whether your uncertainty-calibration and motif-compactness conclusions transfer to a full-convergence regime in your Failure Analysis. Hidden under-convergence is a contract violation; transparent under-convergence with substantive transfer analysis is acceptable.
|
| 78 |
+
|
| 79 |
+
> **Note on reimplementation accuracy as a contract-integrity requirement.** Failure motif identification depends on the reimplemented HIP-NN being a *faithful* reproduction of the paper. If your reimplementation is meaningfully weaker than paper-as-reported on every comparable training-set size, the "failure motifs" you identify may be artifacts of under-converged training (the reimplementation isn't learning some chemistry pattern) rather than properties of the hierarchical architecture (HIP-NN as designed has trouble with this chemistry). The headline contribution loses scientific validity to the extent the baseline is weak. The contract therefore enforces a strict floor: at every training-set size you report a HIP-NN run, the reimplementation's MAE must be **at or below** the paper-as-reported MAE at the comparable size (with reasonable extrapolation across training-size differences between your $\leq 8$-heavy-atom subset and the paper's full-QM9 training sizes). A reimplementation that's even slightly worse than paper-as-reported is a Tier 3 weakness; a reimplementation that's substantially worse with the gap disclosed is at the contract boundary; a reimplementation that's catastrophically worse (>2× paper MAE) or any gap silently is a Tier 2 / Tier 0 violation. This is the most important hygiene gate for the K3 direction's scientific validity.
|
| 80 |
+
|
| 81 |
+
### What you may design
|
| 82 |
+
|
| 83 |
+
- The **failure motif identification methodology**: clustering on hierarchical components $\{\hat{E}_i^n\}_{n,i}$, hand-coded structural features (e.g., functional-group counts, rotatable bonds, ring statistics — but identify your own), learned classifiers, attention-based attribution, decision trees on per-atom features, or whatever combination supports the claim. Standard cheminformatics representations (Morgan fingerprints, RDKit descriptors, etc.) are tools you may use; cite the libraries.
|
| 84 |
+
- The **uncertainty model** itself: $R$ from the paper is one signal, but you may extend it — include the per-order spread $\text{Var}_n(\hat{E}_i^n)$, the maximum hierarchical contribution $\max_n |\hat{E}_i^n|$, the total atomic energy magnitude, ensembles across multiple HIP-NN seeds, or whatever combination produces a better-calibrated uncertainty than $R$ alone. Document which signals you use and how they combine.
|
| 85 |
+
- The **calibration protocol**: how you bin predicted uncertainty, how you compute calibration error (ECE, MCE, reliability diagrams, etc.), how you choose operating thresholds for the precision/recall analysis.
|
| 86 |
+
- The **motif library structure and presentation**: whether to organize by chemistry class (e.g., conjugated systems, strained geometries, heteroatom motifs), by failure magnitude, or by some learned hierarchy.
|
| 87 |
+
- The choice between **ANI-1x** (`data/ani1x/`) and **rMD17** (`data/rmd17/`) for the conformational transfer test is part of the contribution. ANI-1x is the recommended substantial-outcome target (the canonical conformational-variance dataset used by that research group); rMD17 is acceptable as a smaller-scope alternative if ANI-1x integration is too heavy within the budget. Either way, addressing the conformational-transfer question is required (see "Conformational transfer of the motif library" below): running the empirical test on ANI-1x or rMD17 is the substantial outcome, and substantive analytical engagement with the limitation is the floor — the question is which, and how substantively.
|
| 88 |
+
|
| 89 |
+
## Evaluation — Here's how you'll be judged
|
| 90 |
+
|
| 91 |
+
Your submission will be evaluated along two axes:
|
| 92 |
+
|
| 93 |
+
1. **Performance on stated metrics** — reimplementation MAE relative to paper-as-reported, motif library coverage and statistical support, uncertainty calibration error, motif compactness ratio.
|
| 94 |
+
2. **Scientific merit** — depth of analysis, quality of the mechanistic story connecting motifs to hierarchy failure, engagement with the post-HIP-NN MLIP-uncertainty literature, soundness of the methodological choices. Scientific merit values ambition and depth without prescribing what counts as ambitious.
|
| 95 |
+
|
| 96 |
+
### Two-baseline reporting (required)
|
| 97 |
+
|
| 98 |
+
Every quantitative comparison must report two HIP-NN baselines side-by-side:
|
| 99 |
+
|
| 100 |
+
1. **Paper-as-reported HIP-NN** — values cited from Tables I/II of Lubbers, Smith, Barros 2018. Primary headline references: QM9 110k-training MAE $0.256 \pm 0.003$ kcal/mol (Table I); QM9 50k-training MAE $0.354 \pm 0.004$ kcal/mol (Table I). Cite every paper-as-reported value to its specific source location.
|
| 101 |
+
2. **Reimplemented HIP-NN** — your own run via the hippynn package on the $\leq 8$-heavy-atom QM9 subset at the held-fixed architecture and protocol. This is the paired-comparison baseline against which your motif identification and uncertainty calibration are evaluated.
|
| 102 |
+
|
| 103 |
+
Your motif analysis and uncertainty model build on the reimplemented HIP-NN's hierarchical outputs. The paper-as-reported baseline serves as an external anchor — if your reimplementation diverges substantially from paper-as-reported on the comparable training size, the gap must be documented and addressed in your failure analysis (under-convergence may itself contribute to motif identification artifacts).
|
| 104 |
+
|
| 105 |
+
### Training / evaluation protocol
|
| 106 |
+
|
| 107 |
+
Reproduce the paper's per-split methodology with adjustments for the heavy-atom-extrapolation framing:
|
| 108 |
+
|
| 109 |
+
- **Train set**: QM9 molecules with $\leq 8$ heavy atoms. Use $\sim$1k molecules as the validation split (for early stopping); the remainder is training data.
|
| 110 |
+
- **Test set**: QM9 molecules with $= 9$ heavy atoms. This is the out-of-distribution extrapolation set on which failure motifs are identified and the uncertainty model is calibrated.
|
| 111 |
+
- **Number of independent splits / seeds**: report at least **3 independent random shuffles** of the train/validation partition within the $\leq 8$-heavy-atom set, with mean ± standard error across the seeds for every reported number. (The paper uses 8; if compute permits, more is better. If compute requires fewer, document the reduction.)
|
| 112 |
+
- **Energy metric**: mean absolute error (MAE) on the test set, in kcal/mol, computed per-molecule, then averaged.
|
| 113 |
+
|
| 114 |
+
### Failure motif identification protocol
|
| 115 |
+
|
| 116 |
+
For each trained HIP-NN model on the $\leq 8$-heavy-atom training set:
|
| 117 |
+
|
| 118 |
+
- Evaluate on the 9-heavy-atom test set.
|
| 119 |
+
- Compute non-hierarchicality $R$ (Eq. 18 of the paper) and per-order contributions $\hat{E}_i^n$ for every test molecule.
|
| 120 |
+
- Identify a library of failure motifs: structural / conformational / chemical features that correlate with high $R$ *and* high prediction error.
|
| 121 |
+
- For each motif, report: definition (precise enough to be applied to a new molecule), test-set coverage (fraction of 9-heavy-atom molecules matching the motif), conditional error distribution (mean and percentiles of $|\hat{E} - E|$ for molecules matching the motif), and statistical support (compared to non-matching molecules, with a stated significance test).
|
| 122 |
+
|
| 123 |
+
### Uncertainty model calibration protocol
|
| 124 |
+
|
| 125 |
+
For your uncertainty model $\hat{u}(\mathcal{C})$ on test molecules:
|
| 126 |
+
|
| 127 |
+
- Compute $\hat{u}$ on every 9-heavy-atom test molecule.
|
| 128 |
+
- Bin molecules by predicted uncertainty (e.g., deciles or quartiles).
|
| 129 |
+
- For each bin, compute the actual error distribution (mean, std, quantiles of $|\hat{E} - E|$).
|
| 130 |
+
- Report a **calibration curve**: x-axis = predicted-uncertainty bin midpoint, y-axis = actual mean error in bin. A perfectly calibrated model has the curve on the identity line (up to a multiplicative scale).
|
| 131 |
+
- Report **expected calibration error (ECE)** or an equivalent scalar calibration metric.
|
| 132 |
+
- Choose one or more thresholds for flagging "high-uncertainty" predictions and report **precision** (of flagged predictions, what fraction are actually high-error?) and **recall** (of high-error predictions, what fraction are flagged?). Justify the threshold choice in the prose.
|
| 133 |
+
|
| 134 |
+
### Motif compactness protocol
|
| 135 |
+
|
| 136 |
+
Quantify whether the failure motifs are compact (interpretable) or diluted:
|
| 137 |
+
|
| 138 |
+
- For each identified motif, compute its **failure attribution**: the fraction of total test-set failures (e.g., predictions with $|\hat{E} - E| > 1$ kcal/mol) that match the motif.
|
| 139 |
+
- Sort motifs by attribution; report a **Pareto curve** (rank vs. cumulative attribution).
|
| 140 |
+
- State the **compactness ratio**: the smallest $K$ such that the top $K$ motifs cover $\geq 80\%$ of failures. A small $K$ (say, $\leq 5$) indicates compact, interpretable failure modes; a large $K$ indicates the failures are diluted across the motif library.
|
| 141 |
+
|
| 142 |
+
### Per-motif justification
|
| 143 |
+
|
| 144 |
+
For any motif identified as a failure mode, the paper must contain a methodological justification connecting the motif's structural / chemical properties to the hierarchical decomposition itself — for instance, why does this particular structural class produce non-decaying $\hat{E}_i^n$? Generic statements ("this motif is harder for the model") do not satisfy the principle. The form of the justification is up to you; what matters is that it ties a property of the motif to a property of the hierarchy.
|
| 145 |
+
|
| 146 |
+
### Conformational transfer of the motif library
|
| 147 |
+
|
| 148 |
+
The motif library you identify is conditioned on **minimized geometries** — every QM9 configuration was the energy-minimized structure under B3LYP/6-31G(2df,p). This is a load-bearing assumption: models trained on minimized geometries are typically not robust to other conformations of the same molecules, and a motif library identified on minimized chemistry may or may not predict failures on the conformations a molecule would actually see in molecular dynamics, transition-state characterization, or high-throughput screening. The contract therefore requires the submission to address the conformational transfer of the motif library — empirically on ANI-1x or rMD17 for the substantial outcome, or at minimum through substantive analytical engagement with the limitation for the floor.
|
| 149 |
+
|
| 150 |
+
**Protocol.** Take your motif library (identified on the 9-heavy-atom QM9 subset of minimized geometries) and apply it to a conformational-variance dataset of a subset chemistry (HCNO). Two pre-staged options:
|
| 151 |
+
|
| 152 |
+
- **ANI-1x** (`data/ani1x/`) — the canonical conformationally-varying training set from that research group, used for active-learning work. ~5M off-equilibrium conformations at the ωB97X/6-31G(d) DFT level. HCNO chemistry, a subset of QM9's HCNOF, but covers conformational space rather than just minimized geometries. The DFT functional differs (ωB97X/6-31G(d) vs. QM9's B3LYP/6-31G(2df,p)), so absolute energies are on different scales — for the transfer test, the relevant question is whether motif-flagged molecules show systematically larger *relative* errors, not whether absolute energies match. Recommended for the substantial transfer test.
|
| 153 |
+
|
| 154 |
+
- **rMD17** (`data/rmd17/`) — revised MD17, 100k MD-sampled conformations each of 10 small organic molecules at PBE/def2-SVP. Smaller and more focused than ANI-1x; useful for a smaller-scope transfer test if ANI-1x integration is too heavy within the budget. Note that the 4 HIP-NN-relevant rMD17 molecules (benzene, malonaldehyde, toluene, uracil) are all $\leq 8$ heavy atoms, so they're in your HIP-NN's training distribution by heavy-atom count — the test is whether the motif library still flags failures on their MD-sampled conformations.
|
| 155 |
+
|
| 156 |
+
**What to report.** For each motif in your library, evaluate its **transfer rate**: on the conformational-variance dataset, do molecules matching the motif show systematically larger HIP-NN prediction errors than molecules not matching the motif? Report per-motif transfer rates with statistical support, and characterize which motifs transfer cleanly (the failure mode is a property of the chemistry, not the geometry distribution) versus which do not transfer (the motif's failure signature was specific to minimized geometries and doesn't predict MD-conformation failures). Discuss the implication for the motif library's downstream usability in non-minimized-geometry contexts.
|
| 157 |
+
|
| 158 |
+
The transfer test executed substantively on ANI-1x is the substantial outcome; the same test on rMD17 (smaller scope) is the modest outcome; the test omitted with only an analytical discussion of the limitation is the floor; the limitation not addressed at all is a contract failure.
|
| 159 |
+
|
| 160 |
+
## Rules — Here are the rules
|
| 161 |
+
|
| 162 |
+
### Available resources
|
| 163 |
+
|
| 164 |
+
The following are provided in the working directory:
|
| 165 |
+
|
| 166 |
+
- `paper.md`, `paper.pdf` — Lubbers, Smith, Barros 2018, the source paper.
|
| 167 |
+
- `images/` — figures extracted from the source paper.
|
| 168 |
+
- `code/hippynn/` — the LANL hippynn package, the modern PyTorch implementation of HIP-NN maintained by the Lubbers group. **Primary working dependency.** Contains reference implementations of the HIP-NN architecture, hierarchical regularization, training loops, and the QM9 data loaders. The original 2018 paper used Theano + Lasagne; hippynn is the PyTorch successor that the authors recommend for current work. Inspect this before writing training code.
|
| 169 |
+
- `data/qm9/` — QM9 dataset. The standard $\sim$134k-molecule organic-chemistry benchmark, pre-staged from its figshare release. Apply the paper's standard pruning ($\sim$3k geometric-consistency failures + 11 minimization failures) to reach the $\sim$131k-molecule clean dataset before train/test partitioning.
|
| 170 |
+
- `data/rmd17/` — revised MD17 (Christensen & von Lilienfeld 2020), the duplicate-pruned and re-DFT'd successor to the original MD17 trajectory dataset. PBE/def2-SVP precision via ORCA. Available as a conformational-extrapolation supplement for the motif library's transfer test (smaller-scope alternative to ANI-1x); the original MD17 has a known 0.1 kcal/mol energy rounding issue that makes some published MAE comparisons unreliable. Any MAE comparisons with the paper's published numbers should be anchored on QM9, not rMD17. rMD17 results count as supporting evidence for the conformational transfer test, not as a substitute for the QM9 in-distribution comparison.
|
| 171 |
+
- `data/ani1x/` — ANI-1x dataset (Smith, Zubatyuk, Nebgen, Lubbers, Barros, Roitberg, Isayev, Tretiak — Sci. Data 7, 134, 2020). ~5M off-equilibrium conformations of organic molecules at the ωB97X/6-31G(d) DFT level, sampled via active learning from chemical space. HCNO chemistry, a subset of that in QM9, but covers conformational space rather than just minimized geometries. ANI-1x is the canonical conformationally-varying training set from that research group, and is the recommended dataset for the paper-1 follow-up's conformational transfer test. Extraction requires the `ani1x_datasets` package (github.com/aiqm/ANI1x_datasets) — the file is HDF5 with a specific schema; the package provides the loaders. DFT functional differs from QM9 (ωB97X/6-31G(d) vs. B3LYP/6-31G(2df,p)), so absolute energies are on different scales; for the transfer test, the relevant signal is whether motif-flagged molecules show systematically larger *relative* errors than non-flagged ones.
|
| 172 |
+
- `proposal/` — your output goes here, organized as `proposal/code/`, `proposal/logs/`, `proposal/codebase_notes.md`, `proposal/attempts_log.md`, with the final paper at `proposal/report.tex` (and rendered `proposal/report.pdf`).
|
| 173 |
+
|
| 174 |
+
### Compute budget
|
| 175 |
+
|
| 176 |
+
- **Hardware**: 1× NVIDIA H100 80GB, 16 vCPU, 128 GB RAM, 500 GB disk.
|
| 177 |
+
|
| 178 |
+
If compute pressure forces tradeoffs:
|
| 179 |
+
- **Priority 1**: train at least one reimplemented HIP-NN at the held-fixed architecture on the $\leq 8$-heavy-atom subset (the reimplementation baseline + the basis for motif analysis).
|
| 180 |
+
- **Priority 2**: motif identification + uncertainty calibration on that single trained model (the headline scientific content).
|
| 181 |
+
- **Priority 3**: additional seeds to demonstrate that the motif library and calibration are stable across random initializations.
|
| 182 |
+
- **Priority 4**: motif compactness analysis at finer granularity.
|
| 183 |
+
|
| 184 |
+
Document any compute-driven choices transparently in Methodology. Submissions that finish early without exhausting the budget typically have unrealized improvements.
|
| 185 |
+
|
| 186 |
+
### Autonomy and external resources
|
| 187 |
+
|
| 188 |
+
**Internet search is permitted and encouraged.** Use it for literature lookup, library documentation, citations of post-HIP-NN MLIP uncertainty literature (e.g., ensemble-based UQ, Bayesian neural-network MLIPs, dropout-based UQ, conformal-prediction MLIPs), and identification of post-2018 developments in molecular-property prediction. Cite anything you use.
|
| 189 |
+
|
| 190 |
+
**Workspace constraint.** You may not read local files outside the provided working directory.
|
| 191 |
+
|
| 192 |
+
**No DFT generation.** No quantum chemistry engine is provided in the workspace, and you may not run new DFT calculations or quantum chemistry of any kind. The training and evaluation data are restricted to the pre-staged QM9, rMD17, and ANI-1x datasets. Adding new molecules via DFT-of-your-choice would change the study from "characterizing the hierarchy's failure on existing chemistry" to "designing a custom probe set" — out of scope under this contract.
|
| 193 |
+
|
| 194 |
+
**Anti-replication.** Independent researchers working on MLIP uncertainty will naturally converge on some shared methodologies — that kind of convergence is fine and expected. What is not allowed is doing exactly what a published method already did: re-implementing the same uncertainty methodology, the same motif library, the same calibration protocol, and presenting the result as a new contribution. The contract evaluates your scientific work, not your literature search. If the evaluator finds that your submission is substantively a re-implementation of an existing published uncertainty-for-MLIP study, your submission will be severely degraded. Acknowledged convergence with prior work is fine; silent or misrepresented substantive overlap is not.
|
| 195 |
+
|
| 196 |
+
**No questions.** Do not query the user for clarification. Make your own design decisions and justify them in your write-up.
|
| 197 |
+
|
| 198 |
+
### No reward hacking
|
| 199 |
+
|
| 200 |
+
The following checks will be applied to your submission. They are disclosed up front so the contract is clear.
|
| 201 |
+
|
| 202 |
+
1. **Number traceability.** Every quantitative claim in the paper (MAE values, motif coverages, calibration errors, precision/recall numbers, attribution percentages) must trace to a log entry or to a specific citation in the source paper. Numbers without a verifiable source will be flagged.
|
| 203 |
+
|
| 204 |
+
2. **Two-baseline integrity.** Paper-as-reported HIP-NN values must trace to specific tables/figures in Lubbers, Smith, Barros 2018 (e.g., "Table I, row $N_{\text{train+val}} = 110{,}426$"). Reimplemented HIP-NN values must trace to your own hippynn training logs. Mixing the two — citing your reimplementation as if it were paper-as-reported, or vice versa — will be flagged.
|
| 205 |
+
|
| 206 |
+
3. **Pre-stated method.** The motif-identification methodology, the uncertainty model definition, and the calibration protocol must be stated in Methodology before any empirical results are described.
|
| 207 |
+
|
| 208 |
+
4. **Held-fixed disclosure.** Any deviation from the strictly-held-fixed items (chemistry, dataset pruning, train/test split by heavy-atom count, HIP-NN architecture, training protocol, energy-only training) must be transparently disclosed in Methodology. Silent or misrepresented deviations will be flagged.
|
| 209 |
+
|
| 210 |
+
5. **Train/test split integrity.** The $\leq 8$-heavy-atom training set must not include any 9-heavy-atom molecules; the 9-heavy-atom test set must not include any $\leq 8$-heavy-atom molecules; the validation split used for early stopping must come from within the training set ($\leq 8$ heavy atoms), not from the test set. Logs must demonstrate the partition.
|
| 211 |
+
|
| 212 |
+
6. **No selective reporting.** Motif library entries must include all motifs identified in the analysis, including ones that turn out to have low statistical support — not just the ones that look clean. Calibration curves must show the full range of predicted-uncertainty bins, not just the favorable middle.
|
| 213 |
+
|
| 214 |
+
7. **Calibration honesty.** ECE and precision/recall numbers must be computed on **held-out** chemistry. If you use any of the 9-heavy-atom test molecules to tune the uncertainty model's hyperparameters (e.g., choosing how to combine $R$ with other signals), that subset must not be used for calibration evaluation. The calibration protocol must include a clean held-out split.
|
| 215 |
+
|
| 216 |
+
8. **Motif statistical support.** Each reported motif must have stated test-set coverage and a statistical-significance test comparing its conditional error distribution against the non-matching molecules. Motifs identified from $\leq 5$ test-set molecules will be flagged as not having statistical support unless explicitly disclosed as illustrative cases.
|
| 217 |
+
|
| 218 |
+
9. **Code–paper alignment.** The HIP-NN architecture and training procedure described in Methodology must match what hippynn / your code actually runs. Hyperparameters quoted in the paper must match the logs.
|
| 219 |
+
|
| 220 |
+
10. **No hardcoded reference values.** Code must not contain hardcoded DFT energies, hardcoded molecular descriptors, hardcoded motif assignments derived from external sources, or any quantity that bypasses actual computation.
|
| 221 |
+
|
| 222 |
+
11. **No DFT generation.** Your submission must not contain quantum chemistry calculations you ran yourself. Training data is QM9 as provided; supplementary evidence and the conformational transfer test may come from rMD17 or ANI-1x (also as provided). The held-fixed training data clause is checked against your data-loading code paths.
|
| 223 |
+
|
| 224 |
+
12. **Codebase inspection.** Document in `codebase_notes.md` what you inspected in the hippynn package before writing your own training pipeline. Reimplementing from scratch what hippynn already provides (HIP-NN architecture, hierarchical regularization, QM9 data loaders) is wasted effort and a comparison-quality risk; if you did reimplement, document why.
|
| 225 |
+
|
| 226 |
+
13. **No replication of published uncertainty-for-MLIP methods.** Your uncertainty model and motif-identification methodology must not be a re-implementation of a published uncertainty-for-MLIP study (e.g., named ensemble-based UQ, dropout-based UQ, conformal prediction for molecular energies, learned uncertainty heads on existing MLIPs). Acknowledged convergence with prior work is fine; silent substantive overlap is not (see "Anti-replication" above).
|
| 227 |
+
|
| 228 |
+
14. **Effort evidence.** If you describe any work as partial, unresolved, or attempted-but-failed (e.g., a motif you tried to define but couldn't operationalize, an uncertainty signal you experimented with but didn't include), `proposal/attempts_log.md` must document the approaches tried and why each was set aside.
|
| 229 |
+
|
| 230 |
+
15. **Consistent calibration reporting.** Calibration metrics (ECE, precision, recall) must use consistent definitions and consistent error-threshold conventions across the paper. If you switch between "high-error" $> 1$ kcal/mol and "high-error" $> 2$ kcal/mol in different sections, this must be disclosed and the rationale stated.
|
| 231 |
+
|
| 232 |
+
16. **Reimplementation floor across training sizes.** At every training-set size you report a HIP-NN run, the reimplemented MAE must be at or below the paper-as-reported MAE at the comparable size (with reasonable extrapolation across training-size differences between your $\leq 8$-heavy-atom subset and the paper's full-QM9 training sizes). Reimplementations that miss the floor at some training sizes must be disclosed; reimplementations that miss the floor at every reported size without disclosure are flagged as silent under-convergence and weaken the motif analysis's anchoring. Active misrepresentation (citing paper-as-reported numbers as if they were the agent's reimplementation) is a more severe violation.
|
| 233 |
+
|
| 234 |
+
17. **Motif identification protocol integrity.** Motif definitions must be derived from structural / chemical features applied independently of test-set per-molecule error magnitudes. If you derived motif candidates by inspecting which test molecules have high prediction errors and matching their structural features (post-hoc identification), the post-hoc nature must be disclosed *and* you must either (a) validate motif predictions on a separate held-out subset with the discovery and validation splits clearly separated, or (b) apply post-selection inference statistics (conditional p-values, sample-splitting estimators) with stated methodology. Silent post-hoc identification — where the motif library was fit to test-set residuals and presented as predictive — is the K3-direction analog of test-set leakage at the methodology-design level and will be flagged as a serious integrity violation.
|
| 235 |
+
|
| 236 |
+
18. **Multiple-testing correction.** Motif-significance claims must account for testing and selecting among candidate motif definitions, including candidates recorded in `attempts_log.md`. When multiple hypotheses support inferential claims, control the family-wise error rate over the relevant hypothesis family using Holm, Bonferroni, or another procedure that provides family-wise error control under the applicable assumptions. State the hypothesis family, significance level, correction method, and treatment of outcome-dependent hypothesis selection. Benjamini–Hochberg FDR alone does not satisfy this requirement. Preregistration identifies the planned hypotheses but does not, by itself, waive multiplicity control. A genuinely single prespecified test does not require a multiple-testing adjustment. Exploratory effect-size summaries may be reported descriptively, but they do not substitute for multiplicity-controlled inferential evidence where such evidence is required. Reporting selected uncorrected p-values as though no selection or multiplicity occurred, while concealing the actual procedure, is an integrity concern; a disclosed inadequate correction is a scientific shortfall, not by itself deception.
|
| 237 |
+
|
| 238 |
+
19. **Conformational transfer sampling honesty.** For the conformational transfer test (on ANI-1x or rMD17), the evaluation subset must be selected representatively rather than hand-picked. Acceptable selections: the full available dataset, a representative random sample with stated RNG seed and sample size, or a stratified sample with stated stratification rule and rationale. A hand-picked subset where the motif library happens to transfer well — without disclosure of the selection — will be flagged.
|
| 239 |
+
|
| 240 |
+
## Deliverables — Here's what to hand in
|
| 241 |
+
|
| 242 |
+
1. **Codebase and logs.** Code organized under `proposal/code/`; logs under `proposal/logs/`. Logs must include:
|
| 243 |
+
- Per-seed HIP-NN training run records: hyperparameters, RNG seeds, per-epoch training/validation MAE, intermediate-epoch checkpoints if used, final-epoch metrics, timestamps.
|
| 244 |
+
- The trained model checkpoints (one per seed) — small enough that they can be saved alongside the logs.
|
| 245 |
+
- Per-test-molecule outputs at evaluation: predicted energy, non-hierarchicality $R$, per-atom per-order hierarchical contributions $\hat{E}_i^n$, any additional uncertainty signals you compute. This per-molecule output is the raw material for both motif identification and calibration.
|
| 246 |
+
- Motif-assignment logs: for each test molecule, which motifs it matches (under your operationalized motif definitions).
|
| 247 |
+
- Calibration analysis logs: binned predicted-uncertainty vs. actual-error data, computed ECE and precision/recall numbers with the seed-resolved breakdown.
|
| 248 |
+
- A `proposal/logs/paper_reference.md` (or equivalent) listing every paper-as-reported value cited in your paper with its specific table or figure source in Lubbers, Smith, Barros 2018.
|
| 249 |
+
|
| 250 |
+
2. **Follow-up paper.** A LaTeX paper at `proposal/report.tex` rendered to `proposal/report.pdf`, using the NeurIPS style file (`neurips.sty`, provided in the workspace). The paper should read as a self-contained journal-style contribution and must include:
|
| 251 |
+
- **Motivation** for the failure-characterization study — why the long-tail-of-errors and uncertainty-calibration questions matter for HIP-NN usage downstream.
|
| 252 |
+
- **A description of the experimental setup**: the train ($\leq 8$ heavy atoms) / test ($= 9$ heavy atoms) QM9 partition, the held-fixed HIP-NN architecture, the number of independent seeds, the two-baseline rows, the motif-identification methodology, the uncertainty model definition, the calibration protocol.
|
| 253 |
+
- **A description of the motif-identification methodology**: precise enough that a reader can reproduce it. Whether you use clustering on hierarchical components, hand-coded structural descriptors, learned classifiers, or some combination — and which cheminformatics libraries (RDKit, etc.) you use.
|
| 254 |
+
- **A description of the uncertainty model**: precise definition (including any extensions of $R$), with the rationale for each signal included.
|
| 255 |
+
- **A primary results table** labeled `\label{tab:main_results}` with two-baseline rows. Suggested structure (you may adapt column names and add rows for variants of your method):
|
| 256 |
+
|
| 257 |
+
| Method | Training subset | $N_{\text{train}}$ | MAE on test (kcal/mol) | ECE on test | Compactness $K$ (≥80% attribution) |
|
| 258 |
+
|---|---|---|---|---|---|
|
| 259 |
+
| HIP-NN (paper as reported) | full QM9, 110k | 110,426 | $0.256 \pm 0.003$ (Table I) | — | — |
|
| 260 |
+
| HIP-NN (paper as reported) | full QM9, 50k | 50,000 | $0.354 \pm 0.004$ (Table I) | — | — |
|
| 261 |
+
| HIP-NN (reimplemented) | $\leq 8$ heavy atoms | [agent's count] | [agent's MAE] | — | — |
|
| 262 |
+
| This work | $\leq 8$ heavy atoms | [agent's count] | [same as reimpl.] | [agent's ECE] | [agent's $K$] |
|
| 263 |
+
|
| 264 |
+
Numbers in `[brackets]` are placeholders; report actual values. The "This work" row uses the same trained HIP-NN as the reimplemented row — the difference is that "This work" adds the uncertainty-model and motif-analysis outputs.
|
| 265 |
+
|
| 266 |
+
- **A motif library table** with at least 3 motifs, each with definition, coverage, conditional error distribution, and significance support.
|
| 267 |
+
- **A calibration figure**: predicted-uncertainty bin vs. actual mean error, with the identity line overlaid; the ECE value annotated.
|
| 268 |
+
- **A Pareto figure for motif compactness**: cumulative attribution vs. motif rank, with the compactness $K$ annotated.
|
| 269 |
+
- **A mechanism discussion**: why does each identified motif drive hierarchical decomposition failure? Connect motif structure to the architecture's behaviour (which $n$ orders fail to decay, what aspects of the structural environment produce the failure).
|
| 270 |
+
- **A failure analysis**: which 9-heavy-atom molecules are *not* captured by your motif library, and what does that suggest? Where does your uncertainty model fail (high uncertainty / low actual error, or vice versa) and what does the failure reveal about the model's assumptions?
|
| 271 |
+
- **Computational-cost discussion**: per-seed wall-clock for HIP-NN training, per-test-molecule wall-clock for uncertainty + motif evaluation. State the cost-quality tradeoff if your method extends $R$ substantially.
|
| 272 |
+
- **Engagement with related work**: position your uncertainty model and motif identification relative to the published MLIP-uncertainty literature (ensemble UQ, dropout UQ, Bayesian neural-network MLIPs, conformal prediction for molecules, etc., including post-2018 developments). Cite the families of approaches your method is closest to and explicitly differentiate your contribution.
|
| 273 |
+
- **Limitations and conclusions**.
|
| 274 |
+
|
| 275 |
+
Appendix material is for supplementary tables and figures that did not fit in the main body. The substantive contract — methodology, reimplemented MAE, motif library, calibration analysis, compactness analysis, mechanism, failure — must live in the main paper, not the appendix.
|
| 276 |
+
|
| 277 |
+
3. **Supporting documents.**
|
| 278 |
+
- `proposal/codebase_notes.md` — a record of what you found and reused in the hippynn package: which components are usable, which you adapted, which you bypassed. The notes don't need to be exhaustive — a few paragraphs is enough — but they must demonstrate that you inspected what was available before writing code.
|
| 279 |
+
- `proposal/attempts_log.md` — distinct approaches tried during the work, including motifs you considered but discarded, uncertainty signals you experimented with but didn't include, calibration protocols you tried but rejected. Required whenever the submission describes any work as partial or unresolved.
|
| 280 |
+
|
| 281 |
+
### Paper quality
|
| 282 |
+
|
| 283 |
+
Your submission is an academic paper, not a class project report. The writing should resemble what would be reviewed at a top scientific journal in structure, integrity, depth, and language: a clear thesis stated up front, claims supported by quantitative evidence, prose rather than bullet enumeration in main body sections, careful citation, and explicit honest discussion of limitations.
|
| 284 |
+
|
| 285 |
+
**Independence from this task spec.** Your paper should read as a self-contained journal contribution to a reader who has not seen this task spec. Do not transliterate spec terminology, do not adopt the spec's section structure verbatim, do not use placeholder names like "the proposed motif library" or "the proposed uncertainty model" when the contribution has a name. The section content described under "Deliverables" specifies what the paper must *contain*, not the titles or order in which the content must appear. Organize the paper however best serves the contribution.
|
| 286 |
+
|
| 287 |
+
**Rendered output verification.** Before submitting, render your LaTeX to PDF (e.g., `pdflatex report.tex`) and visually inspect the resulting PDF. Equations must not overflow page margins; tables must fit within the text width without column wrapping that breaks readability; figures must not be cut off. A submission with formatting issues in the rendered PDF will be flagged as not having been visually verified.
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| 288 |
+
|
| 289 |
+
Good luck.
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| 1 |
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# Task: Develop a Substantive Follow-up to "Efficient Berry Phase Calculation via Adaptive Variational Quantum Computing Approach"
|
| 2 |
+
|
| 3 |
+
## Objective — Here's what you're doing
|
| 4 |
+
|
| 5 |
+
Develop a substantive scientific follow-up to the AVQDS-based Berry-phase algorithm by Mootz and Yao, "Efficient Berry phase calculation via adaptive variational quantum computing approach," 2025. Your contribution is to **characterize the resilience of this algorithm to realistic quantum hardware noise — establishing, with rigorous evidence, the noise levels below which the computed Berry phase stays accurate, and explaining why the method behaves the way it does under noise.** The original paper reports only noiseless state-vector results; your follow-up asks the question the paper leaves open — *how much noise can this method tolerate before its topological output degrades* — and answers it as a self-contained academic paper.
|
| 6 |
+
|
| 7 |
+
## The Problem — Here's why it matters
|
| 8 |
+
|
| 9 |
+
The **Berry phase** is a geometric property of quantum states that underlies the quantum Hall effect, topological insulators, and topological superconductors; for a time-reversal-invariant 1D crystalline system it is quantized, taking the value $0$ in a topologically trivial phase and $\pi$ in a topologically nontrivial phase. Computing it for interacting many-body systems on a classical computer is hard, which motivates quantum algorithms that evolve the system under a cyclic Hamiltonian and read the accumulated phase off a quantum circuit.
|
| 10 |
+
|
| 11 |
+
The original paper's advance is efficiency: by combining adaptive variational quantum dynamics simulation (AVQDS) with a time-reversed adiabatic loop, it computes the Berry phase in circuits of a few tens to a few hundred layers — orders of magnitude shallower than a Trotterized adiabatic evolution, which for the same 4-site chain would need on the order of $10^5$–$10^6$ CNOTs ($\gtrsim 10^3\times$ more). Shallow circuits are what make the method a candidate for near-term hardware.
|
| 12 |
+
|
| 13 |
+
But the paper's results are **entirely noiseless**: all Berry phases are obtained from exact state-vector simulation, "neglecting errors from quantum gates and finite sampling." The scientific question this leaves open is the one that decides whether the shallow-circuit advantage actually matters in practice: **under realistic device noise, how accurate does the Berry-phase estimate remain, and at what noise level does it break?** Two facts make this non-trivial rather than a routine sweep. First, the Berry phase is read out from a **measured probability**, so **finite-shot sampling noise is intrinsic and irreducible** — it persists even for a fault-tolerant device, because the final logical measurement is still a projective sample. Second, the method's noiseless robustness (it returns the correct phase even under strong non-adiabaticity) comes from two phase contributions that **co-evolve along the same variational trajectory so their non-adiabatic errors cancel** ($\varphi_B=\varphi_{\text{qc}}+\varphi_G$). That cancellation is a property of the co-evolving, adaptively grown ansatz — and it does not automatically carry over to *stochastic* device noise. In the required scope of this task (below) the ansatz and the variational trajectory $\theta(t)$ are frozen at their noiseless values (so the McLachlan integration is not re-run under noise), but the quantities the device must measure to evaluate $\varphi_G$ — the energy expectation $\langle\hat{\mathcal{H}}\rangle$ that feeds $\varphi_{G,1}$ and the geometric-connection overlaps $\langle\Psi|\partial_{\theta_\mu}\Psi\rangle$ that feed $\varphi_{G,2}$ — are re-evaluated under the same noise as the readout, so **both $\varphi_{\text{qc}}$ and $\varphi_G$ carry stochastic noise** and the extracted threshold is a genuine whole-method number. The natural question is then how the two contributions' errors combine — the *deterministic* bias they share through the frozen trajectory can partially cancel, but their *independent* stochastic (shot/incoherent) fluctuations add rather than cancel, and $\varphi_{G,1}=-\int\langle\hat{\mathcal{H}}\rangle\,dt'$ is a dynamical-phase leg whose measured-$\langle\hat{\mathcal{H}}\rangle$ noise integrates over the loop into a non-cancelling drift distinct from the $\varphi_{\text{qc}}$/$\varphi_{G,2}$ overlaps; feeding the sampling noise all the way back into the McLachlan integration (noisy $M,V$, so the trajectory itself becomes stochastic) is the harder, optional study.
|
| 14 |
+
|
| 15 |
+
## Background — Here's what you need to know about the original paper
|
| 16 |
+
|
| 17 |
+
Read `paper.md` in full before beginning. The algorithm you will characterize (and hold fixed) has these components:
|
| 18 |
+
|
| 19 |
+
1. **Cyclic adiabatic evolution.** A Hamiltonian $\hat{\mathcal{H}}(\lambda)$ depends on a cyclic parameter $\lambda(t)\in[0,2\pi]$ with $\hat{\mathcal{H}}(2\pi)=\hat{\mathcal{H}}(0)$. The ground state adiabatically follows $|G(\lambda(t))\rangle$, accumulating $|G(2\pi)\rangle = e^{-i\varphi_D}e^{i\varphi_B}|G(0)\rangle$ with dynamical phase $\varphi_D$ and Berry phase $\varphi_B = -i\oint d\lambda\,\langle G(\lambda)|\partial_\lambda G(\lambda)\rangle$.
|
| 20 |
+
|
| 21 |
+
2. **Time-reversal-symmetric loop** (Eq. 13): $U_{\text{loop}} = U_\circlearrowleft(0,T/2)\,U_\circlearrowleft(T/2,0)$ — forward evolution $\lambda:0\to\pi$ then time-reversed propagation $\lambda:\pi\to 2\pi$. For a time-reversal-invariant Hamiltonian this cancels $\varphi_D$ exactly while $\varphi_B$ adds.
|
| 22 |
+
|
| 23 |
+
3. **AVQITE ground-state preparation.** An adaptive ansatz $|\Psi[\theta]\rangle=\prod_\mu e^{-i\theta_\mu\hat{\mathcal{A}}_\mu}|\Psi_0\rangle$ is grown by appending pool operators to keep the McLachlan distance $L^2$ below a cutoff. Used to prepare $|G(\lambda=0)\rangle$.
|
| 24 |
+
|
| 25 |
+
4. **AVQDS adaptive time evolution.** The same adaptive structure applied to real-time cyclic dynamics: variational parameters evolve via the McLachlan equation of motion $\sum_\nu M_{\mu\nu}\dot\theta_\nu = V_\mu$ (Eq. 6), integrated with RK4 at adaptive step size ($\delta\theta_{\max}=0.01$); when $L^2\ge L^2_{\text{cut}}$ the ansatz is expanded (in the disjoint-support "TETRIS" form that keeps circuits shallow).
|
| 26 |
+
|
| 27 |
+
5. **Berry-phase readout (Hadamard test, Fig. 1).** An ancilla measures the overlap phase: the ancilla's $|0\rangle$ probability is $P_{|0\rangle}=\tfrac12(1+\cos\varphi_{\text{qc}})$, from which $\varphi_{\text{qc}}$ is inferred. Because AVQDS does not track the global phase, a separately integrated variational phase $\varphi_G$ (Eq. 5) is added: $\varphi_B = \varphi_{\text{qc}} + \varphi_G$. **The final Berry phase is the sum of two separately obtained quantities** — one from a measured probability ($\varphi_{\text{qc}}$), one from a classical-side integration (Eq. 5) of device-measured quantities along the trajectory: the energy expectation $\langle\hat{\mathcal{H}}\rangle$ (giving $\varphi_{G,1}$) and the geometric-connection overlaps $\langle\Psi|\partial_{\theta_\mu}\Psi\rangle$ (giving $\varphi_{G,2}$). ($M,V$ enter only the McLachlan equation of motion, Eq. 6, not $\varphi_G$.)
|
| 28 |
+
|
| 29 |
+
6. **Hyperparameters.** $L^2_{\text{cut}}=10^{-4}$ for AVQDS, $\delta\theta_{\max}=0.01$ for the adaptive RK4 step, and the UCCSD-like operator pool (products of Paulis with an odd number of $\hat\sigma^y$); for the AVQDS evolution the pool is the set of Hamiltonian Pauli strings ($N_H=18$ for $U=0$, $N_H=22$ for $U=10$).
|
| 30 |
+
|
| 31 |
+
7. **SSHH test bed** (Eq. 19). A dimerized Fermi-Hubbard (Su–Schrieffer–Heeger–Hubbard) chain, staggered hopping $t_{j,j+1}=t(1+(-1)^j\delta)$, on-site $U$, particle-hole symmetric. $N=4$ sites, $t=1$, mapped to **8 qubits** by Jordan–Wigner; the twist $\lambda$ is applied to spin-up fermions only. Two interaction regimes: $U=0$ (noninteracting SSH) and $U=10$ (strongly correlated). The topological character is set by the dimerization $\delta$, with a phase transition at $\delta=0$.
|
| 32 |
+
|
| 33 |
+
**Reference baseline behavior (noiseless, from the paper):**
|
| 34 |
+
|
| 35 |
+
- **Berry phase.** Reproduces the exact-diagonalization value to $\sim 10^{-3}$ or better in both regimes: $\varphi_B=0$ in the topologically trivial regime, $\varphi_B=\pi$ in the nontrivial regime. All loop times $T$ resolve the $\delta=0$ transition.
|
| 36 |
+
- **Circuit resources.** Peak circuit depth up to **106 layers** ($U=0$) and **279 layers** ($U=10$); the nontrivial region costs several times more CNOTs than the trivial region. The AVQITE ground-state preparation adds $\sim 20$ CNOTs ($U=0$) / $\sim 92$ CNOTs ($U=10$).
|
| 37 |
+
- **Trotter comparison ($U=0$).** A first-order Trotterized adiabatic evolution at matched accuracy needs $\sim 10^3\times$ more CNOTs (e.g. $\sim 4.8\times 10^5$ at $T=20$), which is the sense in which AVQDS is "shallow."
|
| 38 |
+
- **Noiseless robustness the paper emphasizes.** The time-reversed loop cancels the dynamical phase; the method returns the correct $\varphi_B$ even for short loops with strong non-adiabaticity, because $\varphi_B$ is a gauge-invariant closed-loop quantity and the fast dynamical phases of excited-state admixtures average out over the loop. **This robustness is to non-adiabaticity, established under exact arithmetic — it is not a claim about stochastic hardware noise, and the two are physically different.**
|
| 39 |
+
|
| 40 |
+
**Reference data.** `data/figshare_data/AVQS-Berry.zip` contains the authors' per-$\lambda$ traces: `results{run}p0.h5` (AVQDS phase-component, fidelity, per-layer CNOT and depth traces) and `exact{run}p0.h5` (exact-diagonalization reference). These are the ground truth for reproducing the noiseless numbers and validating any re-implementation. Note the phase traces are stored **unfolded / wrapped**, not as a clean $0$-or-$\pi$ column: the reported Berry phase is reconstructed from the stored per-$\lambda$ phase components and folded to the principal value (the authors' figure scripts evaluate it at the loop endpoint and take the magnitude, yielding $0$ in the trivial and $\pi$ in the nontrivial regime). Reproduction against the archive must be judged **after** you apply and state a branch/fold/sign convention; do not read a trace column directly as the Berry phase. The archive does **not** contain the AVQITE/AVQDS engine (the authors' driver only invokes an external code base). It does ship the authors' figure-generation scripts, and — for the $U=10$ runs — per-run `incar` files holding the Jordan–Wigner-mapped Hamiltonian and operator pools, plus per-run `ansatz.h5` files holding an already-grown, already-optimized ansatz (operator selections and parameters). You must implement the engine, construct the Hamiltonian and Jordan–Wigner mapping yourself, and grow your own ansatz: these shipped artifacts may be **inspected and cross-checked against, but not loaded in place of your own construction** (see the held-fixed and reward-hacking rules).
|
| 41 |
+
|
| 42 |
+
## Contract — Here's exactly what counts as success
|
| 43 |
+
|
| 44 |
+
You will produce a **noise-resilience characterization** of the AVQDS Berry-phase algorithm on the SSHH test bed. Concretely:
|
| 45 |
+
|
| 46 |
+
1. **Inspect the provided reference data.** Document, in `proposal/codebase_notes.md`, what you inspected in `data/figshare_data/` (the trace schema, the noiseless Berry-phase and resource numbers) and how you used it to validate your re-implementation.
|
| 47 |
+
|
| 48 |
+
2. **Reproduce the noiseless anchor.** Re-implement the AVQITE + AVQDS Berry-phase pipeline and reproduce, under noiseless simulation, the paper's Berry-phase values ($0$ and $\pi$) in both regimes, together with circuit resources consistent with the paper for your stated $(\delta, T)$: peak depths of order tens of layers for $U=0$ and up to a few hundred for $U=10$ (the paper spans roughly 47–106 layers for $U=0$ and 82–279 for $U=10$ across its $(\delta, T)$), with the $U=10$ circuits several times deeper than $U=0$ and the nontrivial region costing several times more CNOTs than the trivial region — reproduced for the specific $(\delta, T)$ you choose and cross-checked against the figshare traces at that $(\delta, T)$. Exact numerical match to the paper's peak figures is not expected. This anchors every noisy comparison to a correct noiseless result.
|
| 49 |
+
|
| 50 |
+
3. **State your noise model and accuracy criterion in the Methodology before any results.** A domain expert should be able to reproduce your noise study from the Methodology alone: the channels and rates, how each maps onto the circuit, the phase estimator, the shot model, the accuracy tolerance, and the rule by which the threshold is extracted must all be stated up front (see §"Held fixed" and §"Where the contribution lives").
|
| 51 |
+
|
| 52 |
+
4. **Model, at minimum, the required realistic-noise sources** (see the Evaluation Protocol): finite-shot sampling of the readout, two-qubit gate error, ancilla readout error, and $T_1/T_2$ decoherence. Injected noise must act on the actual circuit in a genuine density-matrix (or quantum-trajectory) simulation — not be emulated by perturbing the final phase. In the required scope this noise acts on **both** the Hadamard-test readout that yields $\varphi_{\text{qc}}$ **and** the circuits that measure the quantities feeding $\varphi_G$ — the energy expectation $\langle\hat{\mathcal{H}}\rangle$ (for $\varphi_{G,1}$) and the geometric-connection overlaps $\langle\Psi|\partial_{\theta_\mu}\Psi\rangle$ (for $\varphi_{G,2}$) — evaluated along the frozen noiseless trajectory, so both contributions to $\varphi_B=\varphi_{\text{qc}}+\varphi_G$ carry noise. In $\varphi_{G,2}$ the frozen noiseless $\dot\theta_\mu(t)$ multiplies the noisy connection overlaps; $\dot\theta$ (fixed by the noiseless $M,V$) is not itself re-derived under noise — re-deriving it from noisy $M,V$ is the optional noisy-construction study.
|
| 53 |
+
|
| 54 |
+
5. **Establish the noise threshold.** For each required configuration ($U=0$ and $U=10$; both the topologically trivial and nontrivial regimes), characterize how the Berry-phase estimate degrades as noise increases, and extract the **noise threshold** — the largest incoherent-noise strength at which the estimate stays within your pre-stated accuracy tolerance — by your pre-stated extraction rule, with uncertainty from multiple seeds. Because in the required scope both the readout and the $\varphi_G$-integrand measurements carry noise while the ansatz and trajectory are frozen (see "Held fixed" and "Noisy ansatz construction"), this threshold characterizes the resilience of the whole measured Berry phase $\varphi_B=\varphi_{\text{qc}}+\varphi_G$ at a fixed variational trajectory — not the noise-driven evolution of that trajectory, which is the optional noisy-construction study; state the scope as such. Report the threshold in **device-referenced units** at a **stated reference shot count $S_{\text{ref}}$** (an equivalent two-qubit error rate, readout error, and $T_1/T_2$), so it is interpretable as "accurate below roughly $X$ two-qubit error at $S_{\text{ref}}$ shots," and report **separately** the minimum-shots floor $S^\ast$ (the fewest shots that keep the estimate within tolerance at fixed low incoherent noise), since finite-shot sampling is a distinct, irreducible axis. Report thresholds separately for the shallow ($U=0$) and deep ($U=10$) circuits so the depth dependence is visible; where the trivial and nontrivial regimes of the same $U$ give different thresholds, the per-$U$ threshold below which the two topological phases stay distinguishable is the more conservative (smaller) of the two.
|
| 55 |
+
|
| 56 |
+
6. **Distinguish topological phases under noise.** Below the threshold, the method must still correctly separate the trivial ($\varphi_B=0$) from the nontrivial ($\varphi_B=\pi$) regime. A characterization that only tracks one regime does not satisfy the contract.
|
| 57 |
+
|
| 58 |
+
7. **Explain the mechanism.** Analyze *why* the threshold sits where it does. Engage specifically with the fate of the method's noiseless phase-cancellation robustness under noise: the Berry phase is $\varphi_B=\varphi_{\text{qc}}+\varphi_G$, and in the required scope both contributions are sampled under noise. Reason carefully about how their errors combine, distinguishing two effects: (i) the *deterministic* bias the contributions inherit from the shared frozen trajectory (e.g. a coherent over-rotation, or the finite-$T$ non-adiabatic residual whose noiseless cancellation between $\varphi_{G,2}$ and $\varphi_{\text{qc}}$ the paper derives in Appendix A) *can* partially cancel; (ii) the *stochastic* shot/incoherent fluctuations on the separate $\varphi_{\text{qc}}$, $\langle\hat{\mathcal{H}}\rangle$, and overlap circuits are drawn independently, so their variances *add* (sharing the trajectory does not make them cancel). Note also that $\varphi_{G,1}=-\int\langle\hat{\mathcal{H}}\rangle\,dt'$ is a dynamical-phase leg whose measured-$\langle\hat{\mathcal{H}}\rangle$ noise integrates over the loop into a non-cancelling bias/random-walk, physically distinct from the co-evolving $\varphi_{\text{qc}}$/$\varphi_{G,2}$ overlaps — so a rigorous analysis decomposes $\varphi_G$ into its $\varphi_{G,1}$ and $\varphi_{G,2}$ legs rather than treating it as one lumped quantity, and supports the picture with evidence (a $\varphi_{\text{qc}}$-only vs $\varphi_{G,1}$-vs-$\varphi_{G,2}$ decomposition, or per-channel isolation). If you also make the construction noisy (optional), analyze instead how re-integrating the McLachlan equation under noisy $M,V$ — a stochastic trajectory — changes that picture. Also identify which noise source and which circuit property (depth, CNOT count, the single-ancilla readout) dominates the degradation.
|
| 59 |
+
|
| 60 |
+
**Held fixed (parity constraints).** These are not part of the design space; they must remain identical to the original paper so the object you are characterizing is the paper's method:
|
| 61 |
+
|
| 62 |
+
- **The algorithm.** The AVQITE + AVQDS Berry-phase pipeline as defined in the paper: McLachlan variational principle and equation of motion (Eq. 6), the adaptive disjoint-support ansatz growth, the RK4 integrator at $\delta\theta_{\max}=0.01$, the UCCSD-like operator pool, the time-reversed adiabatic loop (Eq. 13), and the Hadamard-test phase readout with $\varphi_B=\varphi_{\text{qc}}+\varphi_G$. You are characterizing this algorithm's noise resilience, not designing a new algorithm; do not substitute a different ground-state-preparation or dynamics method.
|
| 63 |
+
- **The hyperparameter floor.** $L^2_{\text{cut}}=10^{-4}$ for AVQDS (you may use a smaller or equal cutoff; a larger one degrades variational accuracy and is not allowed). $\delta\theta_{\max}=0.01$. Loosening the cutoff to trim circuit depth — and thereby reduce apparent noise sensitivity — is a held-fixed violation, not a contribution.
|
| 64 |
+
- **The test bed.** The SSHH chain at $N=4$, $t=1$, both $U=0$ and $U=10$, mapped to 8 qubits by Jordan–Wigner, with the twist applied to spin-up fermions only, and with the trivial and nontrivial dimerization regimes the paper uses. Exact-diagonalization Berry phases ($0$ and $\pi$) are the accuracy references.
|
| 65 |
+
- **The noiseless result being anchored to.** Your noiseless re-implementation must reproduce the paper's Berry phases and approximate resources before you attach noise; a noise threshold measured against a wrong noiseless result is meaningless.
|
| 66 |
+
- **Variational-simulation integrity.** The variational state must be evolved through the parametrized circuit, and ansatz-growth decisions must use only quantities a device could measure (the McLachlan distance $L^2$ via $M,V$ and the energy variance) — never fidelity to a classically computed exact ground state. Circuit resource counts (CNOTs, depth) must come from the logged operator selections, not a closed-form estimate. The Hamiltonian and its Jordan–Wigner mapping must be constructed by your own pipeline (do not read the archive's `incar` Hamiltonian strings or operator pools as your model), and the ansatz must be grown by your own AVQITE/AVQDS run (do not load the archive's `ansatz.h5` operator selections or parameters in place of your own growth); inspecting these artifacts to cross-check is fine.
|
| 67 |
+
|
| 68 |
+
**Where the contribution lives.** Your design choices — and where the scientific merit is judged — are: the composition of the noise model beyond the required sources; the phase estimator (see below); the noise-strength parameterization and sweep grid; the accuracy tolerance and threshold-extraction rule; the depth of the mechanism analysis; and any error-mitigation or breadth extensions. If you find a reason to deviate from a held-fixed item, justify it explicitly and run a comparison that isolates the effect of the deviation; a silent deviation is a violation regardless of merit.
|
| 69 |
+
|
| 70 |
+
**The readout estimator.** The variational Hadamard test measures $\varphi_{\text{qc}}$ (with $\varphi_B=\varphi_{\text{qc}}+\varphi_G$), not $\varphi_B$ directly, via $P_{|0\rangle}=\tfrac12(1+\cos\varphi_{\text{qc}})$. Wherever $\varphi_{\text{qc}}$ sits near $0$ or $\pi$ — as it does in the near-adiabatic limit where $\varphi_G\to 0$, and generically wherever the accumulated $\varphi_G$ leaves $\varphi_{\text{qc}}$ close to a quantized value — the single real-part readout has vanishing slope and is ill-conditioned to the small noise-induced deviation. Because $\varphi_{\text{qc}}$'s location depends on your $(\delta, T)$ and the residual $\varphi_G$, use a phase estimator that stays well-conditioned across the loop — for instance one that also measures the imaginary part of the overlap, giving nonzero slope at $0$ and $\pi$ — **or** show, at your operating point, that $\varphi_{\text{qc}}$ sits far enough from $0/\pi$ that the single real-part readout remains reliable. State your estimator in the Methodology.
|
| 71 |
+
|
| 72 |
+
**Branch convention.** The Berry phase is defined modulo $2\pi$, and for $U=10$ the raw $\varphi_B=\varphi_{\text{qc}}+\varphi_G$ comes out as an odd multiple of $\pi$ (the paper reports raw values such as $-3\pi$ and $-5\pi$) that only equals the reference $\pi$ after a principal-value fold — whose branch boundary sits exactly at the $U=10$ nontrivial target, where an infinitesimal noise perturbation can flip the folded value between $+\pi$ and $-\pi$. State your branch policy (continuous phase accumulation / unwrapping vs. a discontinuous principal-value fold), and compute **all** Berry-phase deviations $|\varphi_B-\varphi_B^{\text{ED}}|$ and the trivial-vs-nontrivial discrimination margin **on the circle** (map estimate and reference to the same principal value and take the shorter angular distance), so estimates straddling the $\pm\pi$ branch boundary are compared correctly.
|
| 73 |
+
|
| 74 |
+
**A comparison to a deep-circuit baseline (optional, rewarded).** The practical case for this method is that its shallow circuits should tolerate more noise than a deep Trotterized adiabatic evolution at matched noiseless accuracy. Demonstrating this — that the AVQDS noise threshold is higher than a Trotter baseline's at the same task — is a strong, well-motivated extension. It is not required; it is rewarded.
|
| 75 |
+
|
| 76 |
+
**Noisy ansatz construction (optional, rewarded).** The required scope freezes the paper's noiselessly constructed ansatz and the variational trajectory $\theta(t)$, and injects noise into the execution of the readout **and** the $\varphi_G$-integrand measurements ($\langle\hat{\mathcal{H}}\rangle$ and the geometric-connection overlaps) evaluated along that frozen trajectory. Additionally feeding sampling noise back into the classical construction loop — noisy $M,V$ estimation, hence a noisy McLachlan integration that makes the *trajectory itself* stochastic and re-grows the ansatz under noise — is a harder and separately interesting study; it is optional and rewarded, not required.
|
| 77 |
+
|
| 78 |
+
**Effort evidence is mandatory.** Any contract item you attempt but cannot bring to completion must be documented in `proposal/attempts_log.md` with what you tried and why it did not work. An honest, well-reasoned partial characterization is scored above an incomplete or overclaimed one.
|
| 79 |
+
|
| 80 |
+
**Main paper vs. appendix.** The main paper carries the complete, deep characterization; the appendix is a fallback for work you attempted but could not bring to journal quality, with matching `attempts_log.md` entries. The expected outcome is an empty or near-empty appendix, achieved by doing the work — not by relegating contract content.
|
| 81 |
+
|
| 82 |
+
**Four principles your contribution must satisfy.**
|
| 83 |
+
1. **Originality.** The noise study should be your own scientific work; convergence with independently published noise studies is expected and fine when acknowledged, but substantive replication of an existing follow-up's structure is not (see §"Autonomy").
|
| 84 |
+
2. **Predictive sharpness.** The noise model, estimator, accuracy tolerance, and threshold-extraction rule must be fully stated in the Methodology before any results are shown. Qualitative gestures ("we add realistic noise") do not satisfy this.
|
| 85 |
+
3. **Diagnostic depth.** The paper must contain substantive mechanism, failure, and computational-cost analyses — not just threshold numbers.
|
| 86 |
+
4. **Statistical sharpness.** Report all noise levels, seeds, and regimes you ran (or use principled aggregation with error bars). Reporting only the configurations where the method looks most robust, while hiding others, is selective reporting.
|
| 87 |
+
|
| 88 |
+
## Evaluation — Here's how you'll be judged
|
| 89 |
+
|
| 90 |
+
Your submission is graded on a leaf-level rubric that combines **gates** (failing one precludes a high score) with **differentiators** (which separate a solid submission from an excellent one).
|
| 91 |
+
|
| 92 |
+
**Gates (all must pass):**
|
| 93 |
+
1. **Integrity.** Quantitative claims trace to logs; the paper describes what the code actually does; noise is injected in a genuine circuit simulation and shot noise comes from actual finite sampling; the held-fixed algorithm, test bed, and hyperparameter floor are respected (or any deviation is disclosed). Integrity violations are disqualifying.
|
| 94 |
+
2. **Noiseless anchor reproduced.** The noiseless Berry phases ($0$ and $\pi$) and approximate resources are reproduced in both regimes and cross-checked against the reference data.
|
| 95 |
+
3. **Required noise sources modeled, on the whole method.** Finite-shot sampling, two-qubit gate error, readout error, and $T_1/T_2$ decoherence are all present and faithfully modeled, acting on both the $\varphi_{\text{qc}}$ readout and the $\varphi_G$-integrand measurements ($\langle\hat{\mathcal{H}}\rangle$ and the geometric-connection overlaps) at the frozen trajectory; a $\varphi_G$-exact, readout-only scope must be disclosed as narrower.
|
| 96 |
+
4. **Threshold established on the required configurations.** A noise threshold with uncertainty is extracted, by the pre-stated rule, for $U=0$ and $U=10$ in both topological regimes, and the trivial/nontrivial phases are distinguished below threshold.
|
| 97 |
+
5. **Honest accounting.** All swept noise levels and seeds are reported; the threshold is not cherry-picked; the accuracy tolerance and extraction rule were fixed before the results.
|
| 98 |
+
|
| 99 |
+
**Differentiators (among gate-passing submissions):**
|
| 100 |
+
- Depth and rigor of the **mechanism analysis**, especially whether the noiseless phase-cancellation robustness survives stochastic noise, and which noise source / circuit property dominates the degradation.
|
| 101 |
+
- **Device-referenced, interpretable thresholds** (equivalent two-qubit error, readout error, $T_1/T_2$, shots) and a clear account of the depth dependence between the $U=0$ and $U=10$ circuits.
|
| 102 |
+
- A **well-conditioned phase estimator** at the quantized targets, or a rigorous justification of the chosen estimator.
|
| 103 |
+
- **Breadth** of the noise model beyond the required sources (single-qubit error, coherent over-rotation, crosstalk), where scientifically motivated.
|
| 104 |
+
- A **shallow-vs-deep (AVQDS-vs-Trotter) noise-tolerance comparison** at matched noiseless accuracy.
|
| 105 |
+
- An **error-mitigation** demonstration that measurably raises the threshold.
|
| 106 |
+
- Quality of the **failure analysis**, **computational-cost analysis**, **originality / literature engagement**, and **writing**.
|
| 107 |
+
|
| 108 |
+
A submission that overclaims robustness — reporting a favorable threshold without the uncertainty, the honest per-configuration picture, or the reproduced noiseless anchor — is worse than one that honestly reports a modest threshold and explains what limits it.
|
| 109 |
+
|
| 110 |
+
## Rules — Here are the rules
|
| 111 |
+
|
| 112 |
+
### Available resources
|
| 113 |
+
|
| 114 |
+
Everything you need is in the working directory.
|
| 115 |
+
|
| 116 |
+
- `paper.md` — markdown version of the original paper (Mootz & Yao, 2025): the AVQITE+AVQDS framework, the time-reversed-loop derivation, the SSHH test bed, the resource analysis, and the noiseless numerical results.
|
| 117 |
+
- `paper.pdf` — original paper PDF.
|
| 118 |
+
- `images/` — figures from the original paper (Hadamard-test circuits, loop diagrams, dimerization-vs-Berry-phase curves, depth-scaling plots).
|
| 119 |
+
- `code/` — empty; the paper's core AVQITE/AVQDS engine (the McLachlan solver, growth loop, integrator) is not provided, so implement the framework yourself from Sec. II–III of the paper. A classical density-matrix (or state-vector + noise) simulator of the 8-system-qubit-plus-ancilla circuit is sufficient; full fault-tolerant emulation is not required. (The figshare archive ships the authors' figure scripts, the $U=10$ Jordan–Wigner Hamiltonian and pools in `incar`, and pre-grown `ansatz.h5` files; these are reference artifacts to cross-check against, not to load in place of your own Hamiltonian construction and ansatz growth.)
|
| 120 |
+
- `data/figshare_data/AVQS-Berry.zip` — the authors' figshare archive. Unzip to access per-$\lambda$ traces: `results*p0.h5` (AVQDS Berry phase, fidelity, CNOT and depth traces) and `exact*p0.h5` (exact-diagonalization reference). Use it to validate your re-implementation and to source the noiseless anchor numbers. (Reading the HDF5 files may require matching your `numpy`/`h5py` build; the `h5dump`/`h5ls` command-line tools are a reliable fallback.)
|
| 121 |
+
|
| 122 |
+
For state-vector and density-matrix simulation, noise modeling, exact diagonalization, and analysis, standard scientific Python tooling is available (NumPy, SciPy, and quantum-simulation libraries such as Qiskit / qiskit-aer, Cirq, or QuTiP). For Pauli-string algebra and the Jordan–Wigner mapping, use OpenFermion or your own implementation.
|
| 123 |
+
|
| 124 |
+
### Evaluation Protocol
|
| 125 |
+
|
| 126 |
+
The following are held fixed so all submissions are comparable.
|
| 127 |
+
|
| 128 |
+
#### Test bed
|
| 129 |
+
|
| 130 |
+
The SSHH chain of Background §7: $N=4$ sites, $t=1$, 8 qubits after Jordan–Wigner, twist on spin-up fermions only. Both interaction regimes, $U=0$ and $U=10$. In each regime, both a topologically **trivial** dimerization (exact $\varphi_B=0$) and a topologically **nontrivial** dimerization (exact $\varphi_B=\pi$) — four required configurations in total. You choose the specific dimerization values and loop time within the paper's accurate range and state them; the exact-diagonalization Berry phase fixes the reference for each.
|
| 131 |
+
|
| 132 |
+
#### Required noise sources
|
| 133 |
+
|
| 134 |
+
At minimum, model all four, injected on the actual circuit:
|
| 135 |
+
- **Finite-shot sampling** of the readout measurement (draw outcomes from the exact probability; report the number of shots).
|
| 136 |
+
- **Two-qubit gate error** on entangling gates (e.g. a depolarizing channel per two-qubit gate).
|
| 137 |
+
- **Readout / measurement error** on the ancilla (and any measured system qubits).
|
| 138 |
+
- **$T_1/T_2$ decoherence** (amplitude + phase damping / thermal relaxation) keyed to gate times and circuit depth.
|
| 139 |
+
|
| 140 |
+
These channels act on **every** circuit the method executes under noise — both the Hadamard-test readout that yields $\varphi_{\text{qc}}$ and the circuits that estimate the $\varphi_G$-integrand quantities (the energy expectation $\langle\hat{\mathcal{H}}\rangle$ for $\varphi_{G,1}$ and the geometric-connection overlaps $\langle\Psi|\partial_{\theta_\mu}\Psi\rangle$ for $\varphi_{G,2}$), evaluated at the frozen noiseless trajectory, so the extracted threshold is a whole-method number. A required-scope threshold in which $\varphi_G$ is held at its exact noiseless value (only $\varphi_{\text{qc}}$ sampled) is a narrower, readout-only characterization: it must be disclosed as such, and a threshold presented as whole-method while $\varphi_G$ is noise-free does not meet the required scope.
|
| 141 |
+
|
| 142 |
+
Model these jointly, and parameterize noise strength so it can be swept (for instance a single scalar that scales the incoherent rates against a fixed realistic device profile), with the number of shots reported as its own axis. State the concrete base device profile you scale against — the base two-qubit error, ancilla readout error, $T_1/T_2$, and per-gate-type gate times — and report and interpret every threshold relative to it, noting that the four channels co-vary along the scalar in the ratio this profile fixes. (The paper's resource counts assume all-to-all connectivity, so a base profile in that setting is the most faithful match to its gate counts.) Single-qubit gate error, coherent over/under-rotation, and crosstalk are encouraged additions but not required.
|
| 143 |
+
|
| 144 |
+
#### Metrics (logged for every run)
|
| 145 |
+
|
| 146 |
+
The computed Berry phase and its deviation from the exact-diagonalization reference; the output-state infidelity under noise; the noise-model parameters (rates, $T_1/T_2$, readout error) and shot count; the circuit depth and CNOT count from the logged operator selections; the extracted threshold with its uncertainty.
|
| 147 |
+
|
| 148 |
+
#### Reference computation
|
| 149 |
+
|
| 150 |
+
Exact-diagonalization Berry phases (the $0$ and $\pi$ references) must be computed at runtime from the model Hamiltonian by your own independent pipeline, not hardcoded. The figshare `exact*p0.h5` traces may be used to cross-check.
|
| 151 |
+
|
| 152 |
+
#### Comparison methods
|
| 153 |
+
|
| 154 |
+
The **noiseless AVQDS Berry phase** (the reproduced anchor); the **AVQDS Berry phase under noise** at each swept level (the object of study); and, optionally, a **Trotterized adiabatic baseline under the same noise** at matched noiseless accuracy. The original paper's reported numbers are a verification reference, not the contract baseline.
|
| 155 |
+
|
| 156 |
+
### Autonomy and external resources
|
| 157 |
+
|
| 158 |
+
**Internet search is permitted and encouraged.** Use it for literature on Berry-phase and topological-invariant quantum algorithms, AVQITE/AVQDS, NISQ noise modeling, quantum error mitigation, and shot-allocation; for documentation of the simulation libraries; and to read work published since the original paper so you can position and differentiate your contribution. Cite anything you use.
|
| 159 |
+
|
| 160 |
+
**On replication.** Independent researchers will converge on some shared choices — that convergence is fine and expected. What is not allowed is substantive replication of an existing follow-up: the same noise model, the same threshold definition, the same conclusions, in a structure that mirrors prior work rather than developing its own. If the evaluator finds your submission is substantively a reproduction of an existing follow-up, it will be severely degraded. The contract evaluates your scientific work, not your literature search.
|
| 161 |
+
|
| 162 |
+
**Working-directory constraint.** Do not read local files outside the provided working directory.
|
| 163 |
+
|
| 164 |
+
**No questions.** Make your own decisions and justify them in your write-up; do not ask for clarification mid-task.
|
| 165 |
+
|
| 166 |
+
### Compute budget
|
| 167 |
+
|
| 168 |
+
- **Hardware.** A CPU machine with 32 vCPU cores, ~60 GB RAM (no swap), and ~470 GB working disk. No GPU. Treat ~50 GB as a hard memory ceiling (the node has ~60 GB RAM, no swap, and the OS + agent harness need headroom): size grids, dense matrices, ensembles, batch sizes, and parallel workers to keep peak resident memory under it; estimate/monitor peak memory before scaling up and prefer chunked / streaming / sparse computation. A process that exhausts RAM is killed by the OS. The per-configuration density matrix is small (a 9-qubit density matrix is a few megabytes), but under the whole-method scope each swept (noise level, seed, configuration) point requires, along the frozen trajectory, a noisy evaluation of the $\varphi_{\text{qc}}$ readout **and** the $\varphi_G$ integrand — $\langle\hat{\mathcal{H}}\rangle$ for $\varphi_{G,1}$ and one geometric-connection-overlap circuit per variational parameter ($N_\theta = O(100)$) at each integration node — so per-point cost scales as (number of integration nodes) $\times N_\theta$ noisy circuit evaluations, not a single readout. This is still tractable on 32 vCPU (each sim is a few MB / milliseconds, the frozen trajectory has no inner McLachlan loop, and a coarse $\lambda$ grid suffices because $\varphi_G(\lambda)$ is smooth), but budget the $\varphi_G$-integrand sampling explicitly and parallelize across cores.
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- **Where compute goes.** Favor a noise sweep dense enough to resolve the threshold with error bars over pushing to larger systems; the contract is about characterization quality, not system size.
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### No reward hacking
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Your submission and logs will be inspected for the following. These are disclosed up front so the contract is clear.
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1. **Number traceability.** Every quantitative claim (Berry-phase values, deviations, noise rates, thresholds, shot counts, CNOT and depth numbers, fidelities) must trace to a log entry with the same value. Numbers with no logged source will be flagged.
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2. **Pre-stated method and consistent reporting.** The noise model, phase estimator, accuracy tolerance, and threshold-extraction rule must appear in the Methodology before any results, and the rates and settings quoted in the paper must match the logged run configurations. Refining a choice after seeing results is allowed only if disclosed; silent restatement is not.
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| 177 |
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3. **Baseline honesty.** The noiseless-anchor numbers and any under-noise or Trotter-baseline numbers must come from your own logged runs. The original paper's reported values are a verification reference, not to be transcribed and presented as your own measurements.
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| 178 |
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4. **Held-fixed disclosure.** Any deviation from the §"Held fixed" clause — the algorithm, the hyperparameter floor ($L^2_{\text{cut}}$, $\delta\theta_{\max}$), the test bed, the variational-simulation integrity — must be disclosed transparently in the Methodology or Experimental Setup. Silent deviation, or a paper claim of compliance contradicted by the logs, is a deception.
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5. **Noise-simulation integrity.** Injected gate and decoherence noise must act on the actual circuit in a genuine density-matrix or quantum-trajectory simulation, and finite-shot noise must come from actual finite sampling (e.g. binomial draws) of the measured probability — for the $\varphi_{\text{qc}}$ readout and, in the required whole-method scope, the $\varphi_G$-integrand measurements ($\langle\hat{\mathcal{H}}\rangle$ and the geometric-connection overlaps) — not from adding a Gaussian to the final phase or substituting an analytic variance formula for a real sampling procedure.
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| 180 |
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6. **No selective reporting.** All swept noise levels, seeds, and both topological regimes must be reported (or aggregated with error bars). Reporting only the configurations where the method looks most robust, or picking the threshold from the most favorable seed, is selective reporting.
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| 181 |
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7. **Shot-sampling integrity.** The reported shot-noise behavior must come from real finite-sample estimation at the logged shot counts, for the $\varphi_{\text{qc}}$ readout and, in the required whole-method scope, the $\varphi_G$-integrand measurements. Claiming a shots axis while computing phases (or the $\varphi_G$-integrand expectation values) from exact probabilities is a misrepresentation. Holding $\varphi_G$ at its exact noiseless value is a narrower readout-only scope: when disclosed as such it is a narrower-scope limitation (graded as a weakness), not a misrepresentation; presenting a $\varphi_G$-exact threshold as whole-method is the violation.
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8. **Code-paper alignment.** The Methodology must accurately describe what the code does — the noise channels and rates, the estimator, the shot model, the ansatz, the threshold rule — and these must match the implementation.
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9. **Variational-simulation integrity.** The variational state must be evolved through the parametrized circuit; ansatz-growth decisions must use only device-measurable quantities (the McLachlan distance $L^2$), never fidelity to a classically computed exact ground state; and CNOT/depth counts must come from your own logged operator selections, not a closed-form formula and not the archive's pre-grown `ansatz.h5`/`ngates`. Loading the authors' `ansatz.h5` selections or parameters in place of your own AVQITE/AVQDS growth is a violation.
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| 184 |
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10. **No hardcoded reference values.** The code must not contain hardcoded exact-diagonalization states, target Berry-phase values, or threshold literals that bypass the actual simulation and analysis. References are computed at runtime, and the model Hamiltonian is constructed by your own Jordan–Wigner pipeline — reading the archive's `incar` Hamiltonian strings or operator pools as your model counts as a hardcoded input.
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11. **Threshold-extraction integrity.** The noise threshold must be extracted by the pre-stated criterion applied uniformly across configurations, not chosen post hoc to maximize the reported robustness.
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12. **Experiment completeness.** Every experiment described in the paper must have corresponding logs (noiseless-anchor runs, per-noise-level runs, seed repeats, threshold analysis, reference computations). Suspicious gaps will be flagged.
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13. **Codebase inspection evidence.** `proposal/codebase_notes.md` must document what you inspected in `data/figshare_data/` and reused or cross-checked, and any external simulation code you built on.
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14. **No substantive replication of published follow-ups.** If your noise model, threshold definition, and conclusions substantively replicate a published follow-up without acknowledgment, that will be flagged. Acknowledged convergence is fine; silent structural replication is not.
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15. **No undisclosed external code.** Non-trivial methodology code (the AVQDS engine, the noise model, the phase estimator) must be authored by you during this run or be from a standard scientific library with attribution in `codebase_notes.md`.
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16. **Effort evidence when partial work is claimed.** If any contract item is described as attempted-but-incomplete, `proposal/attempts_log.md` must contain at least two documented attempts with reasoning.
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## Deliverables — Here's what to hand in
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Organize all outputs under `proposal/`. Submit:
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**Codebase, logs, codebase notes, and attempts log.**
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- `proposal/code/` — your AVQITE + AVQDS implementation, the noise model, the phase estimator, the threshold-extraction analysis, and the exact-diagonalization reference pipeline.
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| 198 |
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- `proposal/codebase_notes.md` — what you inspected in the reference data and any external simulation code you reused.
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| 199 |
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- `proposal/logs/` — including: the noiseless-anchor reproduction (Berry phase, fidelity, depth/CNOT traces) cross-checked against the figshare data; per-(configuration, noise-level, seed) runs logging the noise parameters, shot count, computed Berry phase, deviation from the exact reference, and infidelity; the threshold-extraction analysis; and the runtime exact-diagonalization references.
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- `proposal/attempts_log.md` — distinct documented attempts for any unresolved or partial item.
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| 202 |
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Every quantitative claim in the paper must be reproducible from a log entry.
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**Follow-up paper.** A LaTeX paper at `proposal/report.tex`, compiled to `proposal/report.pdf`, using the provided `neurips.sty`. The paper must contain: motivation and the scientific question; a description of the method being characterized and the held-fixed setup; the noise model, phase estimator, accuracy criterion, and threshold-extraction rule; the noiseless-anchor reproduction; the noise-resilience results; a mechanism analysis (why the threshold sits where it does, and whether the noiseless cancellations survive noise); a failure analysis; a computational-cost discussion; engagement with related work; and limitations and conclusions. Choose section names that fit the contribution — do not organize the paper as a transcription of this spec.
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It must include a primary results table labeled `\label{tab:main_results}` reporting, for each configuration, the noiseless and under-noise Berry phase, the deviation from the exact reference, the extracted threshold with uncertainty (in device-referenced units), and the circuit resources. A skeleton is below; the row and method labels are placeholders — name your configurations and method as fits the paper.
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| Configuration | Noise setting | Berry phase | $|\varphi_B - \varphi_B^{\text{ED}}|$ | Threshold (device units) | Depth | $N_{\text{CX}}$ |
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| 209 |
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|---|---|---|---|---|---|---|
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| (U, regime) | noiseless | | | | | |
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| (U, regime) | at threshold | | | | | |
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| (additional rows for each configuration and swept level) | | | | | | |
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| 213 |
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Beyond the table, the results should include: Berry-phase-error-versus-noise-strength curves with the threshold crossing and seed error bars, for both $U$ regimes; a shots-versus-phase-uncertainty relationship; and the infidelity-versus-noise trend alongside the phase error. Additional analyses are encouraged.
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### Paper quality
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| 217 |
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This is an academic paper, not a class-project report; it should read as it would in a strong journal. Do not frame it as "a follow-up to Mootz and Yao"; frame it as a self-contained study of the noise resilience of adaptive-variational Berry-phase computation, citing the original where appropriate.
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| 219 |
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| 220 |
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**Independence from this task spec.** A reader who has not seen this spec should not be able to reverse-engineer it from your paper's structure or terminology. Do not reuse spec phrases ("required noise sources," "the contract," "differentiators") or transliterate its section order. The title should describe the scientific contribution, not the structure of the investigation.
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**Rendered output verification.** Before submitting, compile the LaTeX to PDF and inspect it: equations must not overflow the margins, tables must fit, and figures must not be cut off.
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Good luck.
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