✨ Add error correction module with Shor and Steane logical circuit transpilers

[Felix-Gundlach]

Summary

This PR introduces a dedicated Error Correction Module for MQT Bench. It adds two high-level transpilers that encode standard benchmark circuits into fault-tolerant logical circuits using the 7-qubit Steane code and the 9-qubit Shor code, respectively. The module is designed to integrate cleanly with the existing benchmark_generation API.

Changes

New files

• src/mqt/bench/error_correction/steane_transpiler.py — SteaneTranspiler class: encodes logical qubits into 7-qubit Steane blocks, applies transversal Clifford gates (H, X, Z, S, CX, CZ), and handles T-gates via magic state injection with ancilla teleportation.
• src/mqt/bench/error_correction/shor_transpiler.py — ShorTranspiler class: encodes logical qubits into 9-qubit Shor blocks, implements the full logical Clifford gate set (including the non-transversal H via bit-swap), and supports S- and T-gates via magic state teleportation gadgets.
• tests/test_error_correction.py — pytest suite covering gate-level equivalence (via MQT QCEC), correctness under injected bit- and phase-flip errors (via Hellinger fidelity with Aer), and circuit structure validation against pre-computed gate counts for GHZ, Bernstein–Vazirani, graph state, and QFT benchmarks.

How it works

Both transpilers follow the same three-phase pipeline:

1. Encode — each logical qubit is replaced by a physical register (7 or 9 qubits) and initialized using the respective encoding circuit.
2. Replace gates — the original circuit is scanned instruction by instruction and each gate is replaced with its logical equivalent. High-level gates (e.g. QFTGate) are first decomposed into the supported basis set {H, X, Z, S, T, CX, CZ}.
3. Insert syndromes — after each logical operation, syndrome extraction and conditional correction cycles are appended using Qiskit's dynamic circuit if_test feature.

T-gates, which are non-transversal, are handled via a teleportation gadget: a magic state ancilla block is prepared, a logical CNOT is applied from the data qubit to the ancilla, the ancilla is decoded and measured, and a conditional logical S correction is applied on outcome 1.

Testing

• Gate equivalence is verified using mqt.qcec for all single- and two-qubit Clifford gates.
• Correctness under noise is verified by injecting a physical X or Z error after encoding and checking that the Hellinger fidelity between the corrected and uncorrected logical distributions is ≥ 0.99.
• Circuit structure is validated against a reference gate_counts.json for qubit counts ranging from 3 to 9 across all supported algorithms.

Notes

• QFT is currently excluded from the correctness simulation tests due to circuit depth making simulation infeasible.
• Syndrome insertion can be toggled off via add_syndromes=False for lightweight structural or equivalence checks.

Checklist

• The pull request only contains commits that are focused and relevant to this change.
• We have added appropriate tests that cover the new/changed functionality.
• We have updated the documentation to reflect these changes.
• We have added entries to the changelog for any noteworthy additions, changes, fixes, or removals.
• We have added migration instructions to the upgrade guide (if needed).
• The changes follow the project's style guidelines and introduce no new warnings.
• The changes are fully tested and pass the CI checks.
• We have reviewed our own code changes.

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