| # Fermions (Majorana Jordan-Wigner mapping) | |
| Majorana-fermion → qubit (Jordan-Wigner) mapping, one qubit per two | |
| Majorana modes: `chi_{2j-1} = (prod_{k<j} Z_k) X_j`, | |
| `chi_{2j} = (prod_{k<j} Z_k) Y_j`. Each `chi_i` is Hermitian and unit- | |
| normalized (`chi_i^2 = I`), and the anticommutation relation | |
| `{chi_a, chi_b} = 2*delta_ab*I` holds exactly — verified against the | |
| actual matrices, not assumed from the textbook formula alone. | |
| Combine the returned Pauli term with | |
| [`pauli_hamiltonian_to_matrix`](observables.md) to build any | |
| Majorana-operator Hamiltonian as a dense matrix, e.g. a sparse | |
| Sachdev-Ye-Kitaev (SYK) model: `H = sum_{ijkl} J_ijkl * chi_i*chi_j*chi_k*chi_l`. | |
| ::: dense_evolution.physics.fermions | |
| --- | |
| **See also**: [`entropy`](entropy.md) and [`trotter`](trotter.md), the | |
| other two modules promoted alongside this one from a real traversable- | |
| wormhole-inspired quantum teleportation reproduction (Gao-Jafferis-Wall | |
| theory, arXiv:2604.10090) — see the [MCP Server](../composer.md) section | |
| and [Dense-Evolution-Discovery](https://tatopenn-cell.github.io/Dense-Evolution-Discovery/wormhole_syk_teleportation/) | |
| for the real experiments built on top of these three modules. | |