Channel-Level Diagnostic for Symmetry Breaking in Noisy Equivariant Quantum Neural Networks

Reference implementation, data, and reproducibility package for

A Channel-Level Diagnostic for Symmetry Breaking in Noisy Equivariant Quantum Neural Networks Hassan Ugail and Newton Howard

This repository provides a training-free, channel-level audit that tests whether a noisy quantum neural network (QNN) still respects an intended symmetry once the ideal parameterised circuit is composed with the device's noise channel. The audit reports a normalised commutator-defect ratio $\Delta_G \in [0, 1]$ and a bounded compliance score $C_G = \exp(-\gamma \Delta_G) \in [e^{-\gamma}, 1]$, where $C_G = 1$ exactly when the realised channel commutes with the target group action on the tested representation.

The repository contains everything needed to reproduce every figure, every verdict table, and every numerical value reported in the paper from a CPU runtime, with no proprietary hardware or non-public software dependencies.

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What's in this repository

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├── notebooks/
│   ├── Channel-Level-Diagnostic_pilot_notebook_.ipynb            # main experiments
│   ├── Channel-Level-Diagnostic_figures_.ipynb                   # figure generation
│   └── Channel-Level-Diagnostic_ibm_hardware_bloch_ordering_probe.ipynb
│                                                                 # appendix hardware probe
├── results/                  # CSV outputs from the pilot notebook
└── README.md

The three notebooks are the heart of the repository. They are intentionally separated so that the simulation step, the figure step, and the optional hardware step can each be run, re-run, or audited independently of the others.

Channel-Level-Diagnostic_pilot_notebook_.ipynb — main experiments

Computes $\Delta_G$ and $C_G$ for the two symmetry settings of the paper: $U(1)$ phase symmetry on two qubits, and collective $SU(2)$ rotation symmetry on two qubits. Sweeps three noise families (depolarising, amplitude damping, coherent $X$ over-rotation) plus a mixed-noise scenario, and runs the full pre-specified validation battery: ten correctness checks (V1–V10) and nine robustness checks (R1–R9). Writes nine CSV files to results/ and prints a pass/fail verdict for every check. Runs end-to-end on a single CPU core.

Channel-Level-Diagnostic_figures_.ipynb — figure generation

Channel-Level-Diagnostic_ibm_hardware_bloch_ordering_probe.ipynb —

Runs the optional one-qubit ordering probe on real IBM Quantum hardware via Qiskit Runtime. Compares two orderings of non-commuting single-qubit rotations $G = R_z(\pi/2)$ and $E = R_x(\pi/2)$, measures empirical Bloch vectors in the $X$, $Y$, and $Z$ bases, and reports the Bloch-vector distance with a 5000-resample bootstrap confidence interval. Outputs are saved to hardware_results/ named by <backend>_<job_id>. This probe is a hardware-execution sanity check, not a hardware estimate of $C_G$ or $\Delta_G$.

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License

This repository is released under the MIT License.