docs/00_Mathematical_Modeling_Index.md

00 — End-to-End Mathematical Modeling Index (AVID_EPJ_Patches)

Overview

This documentation suite presents the complete, end-to-end mathematical modeling of the Adaptive Variational Inference Decoder (AVID) baseline architecture (AVID_EPJ_Patches, implementing Policy Items A01–A09 and Checklist Q01–Q21). Each chapter formulates one mathematical layer of the closed-loop quantum error correction pipeline—from the algebraic topology of the rotated CSS planar surface code through the exact $2^m$-state correlated dynamic programming likelihood, structural identifiability theory, quasi-Newton optimization, signed Q7.8 fixed-point quantization, cycle-exact Normalized Min-Sum Belief Propagation (BP), and dual-timescale causal hardware control.

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Document Index

1. 01_CSS_Surface_Code_Algebra_and_Tanner_Geometry.md Formulates the symplectic commutativity invariant $H_X H_Z^\top \equiv \mathbf{0} \pmod 2$, $\mathbb{F}2$-rank code dimension $k = n - \operatorname{rank}{\mathbb{F}2}(H_X) - \operatorname{rank}{\mathbb{F}_2}(H_Z)$, canonical rotated $d=3$ ([[9, 1, 3]]) and $d=5$ ([[25, 1, 5]]) stabilizer matrices, logical operator coset distance $d_X = d_Z = d$, and degree-padded hardware Tanner graph masks.
2. 02_Correlated_Fault_Channel_and_Observation_Algebra.md Defines the physical fault mechanisms—single-qubit phase/bit flips ($g_i = H(:, i)$), nearest-neighbor dispersive $Z_i Z_j$ crosstalk pairs ($g_{ij} = g_i \oplus g_j$), and syndrome readout bit-flips ($d_t = s_t \oplus s_{t-1}$)—and proves algebraically why folding correlated pair probabilities into independent single-qubit rates fails on shared checks.
3. 03_Exact_Joint_DP_Likelihood_and_Coset_Gradients.md Derives the exact $2^m$-state group-algebra $\mathbb{R}[\mathbb{F}_2^m]$ XOR-convolution dynamic programming recurrence $P^{(k)}(s)$, its dual Walsh–Hadamard spectral representation, the proof of the Exact Single-Pass Coset-Inversion Gradient Theorem, and the prior-tethered MAP cross-entropy objective.
4. 04_Structural_Identifiability_Gauge_Orbits_and_Fisher_Rank.md Proves the structural rank theorem for the $2^m \times E$ Jacobian $J(\theta)$ and Fisher Information Matrix $\mathcal{I}(\theta)$, partitions the 21 physical fault mechanisms into 9 identifiable non-zero syndrome signature orbits and 1 zero-signature boundary orbit, and establishes full-rank ($\operatorname{rank} = 21$, $\kappa = 234.05$) convergence under Gaussian prior regularization.
5. 05_LBFGS_Optimization_XOR_Marginalization_and_Q78_Numerics.md Details the zero-allocation two-loop L-BFGS quasi-Newton recursion with projected Armijo backtracking line search, exact associative XOR marginalization $p_{\text{eff}}(v) = p_v \boxplus \bigoplus_{e \ni v} p_e$ to variable-node priors, and signed 16-bit Q7.8 fixed-point quantization.
6. 06_Hardware_MinSum_BP_Decoder_and_Cycle_Latency_Contract.md Formulates the bit-exact Q7.8 Normalized Min-Sum BP equations, active-edge masking for mixed weight-2/weight-4 checks, DSP-free shift-subtract scaling $\mathcal{S}_{0.75}(x) = x - (x \gg 2)$, weighted OSD-0 residual syndrome tie-breaking, and the exact $L(K) = 2 + 5K = 17$-cycle ($68.0,\text{ns}$ @ $250,\text{MHz}$) hardware latency contract.
7. 07_Dual_Timescale_Causal_Control_DMA_and_Baselines_B0_B4.md Models the dual-timescale causal epoch commit law $t_{\text{commit}}$, bp_ready_in-gated atomic shadow-to-active register swap, 4-state DMA ring buffer ownership FSM, 4-state safety fallback FSM, and the mathematical equations of the 5 shared-input baselines (B0–B4).
8. 08_Weighted_Bellman_R2D2_and_World_Model_Math.md Formulates the Non-Vision Bipartite Graph Perception layer, R2D2-Graph-RSSM with burn-in warmup, Kapturowski invertible value rescaling $h(u)$, Uncertainty-Weighted $n$-Step Bellman TD Loss ($\mathcal{L}_{\text{WB-R2D2}}$), Pair-Aware Logical-Triangle LLR Projection, and Rank-4 Regime LoRA Adaptation.
9. mathematical_modeling_complete.md & identifiability_proof.md Consolidated single-file mathematical reference and structural identifiability proof.

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References

• [R1] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, "Surface codes: Towards practical large-scale quantum computation," Physical Review A, vol. 86, no. 3, p. 032324, 2012.
• [R2] N. Liyanage et al., "Scalable quantum error correction for surface codes using FPGA," arXiv:2301.08419, 2023.
• [R3] J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, "The XZZX surface code," Nature Communications, vol. 12, p. 2172, 2021.
• [R4] E. van den Berg, Z. K. Minev, A. Kandala, and K. Temme, "Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors," Nature Physics, vol. 19, pp. 1116–1121, 2023.
• [R5] S. T. Flammia and J. J. Wallman, "Efficient estimation of Pauli channels," ACM Transactions on Quantum Computing, vol. 1, no. 1, pp. 1–32, 2020.
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• [R8] Y. Tomita and K. M. Svore, "Low-distance surface codes under realistic quantum noise," Physical Review A, vol. 90, no. 6, p. 062320, 2014.
• [R9] Xilinx Inc., Zynq UltraScale+ Device Technical Reference Manual (UG1085), v2.4, 2023.