QEC Quantum Error Correction

Documentation

Results

All docs Raw markdown View on GitHub

Results

This document summarizes the generated outputs in images/ and the behavior each script is intended to show. The exact recovery checks live in tests/ and can be run with:

octave --no-gui tests/run_all_tests.m

The compact generated table in docs/SIMULATION_REPORT.md can be refreshed with:

octave --no-gui examples/generate_simulation_report.m

All figures can be regenerated with:

octave --no-gui examples/run_all_plots.m

Exact Recovery Coverage

The test suite checks that the implemented distance-3 codes recover arbitrary logical states after correctable single-qubit errors:

Code Error model checked Expected behavior
3-qubit bit-flip repetition Single X errors Recovers the encoded logical state
3-qubit phase-flip repetition Single Z errors Recovers the encoded logical state
[[4,2,2]] error-detecting code Single-qubit X, Y, or Z errors Produces a nonzero detection syndrome
5-qubit perfect code Single-qubit X, Y, or Z errors Recovers the encoded logical state
7-qubit Steane code Single-qubit X, Y, or Z errors Recovers the encoded logical state
9-qubit Shor code Single-qubit X, Y, or Z errors Recovers the encoded logical state
3x3 Bacon-Shor subsystem code Single-qubit X, Y, or Z errors Produces a non-logical Pauli-frame residual
Surface-3 prototype Single X, Z, or Y errors on 9 data qubits Returns to the zero-syndrome class without a logical residual

Bit-Flip Repetition Code

The 3-qubit repetition examples are the smallest end-to-end demonstrations of syndrome extraction, correction, and Monte Carlo logical-error estimation.

Logical Error Probability

images/bitflip_logical_error_vs_physical_error.png

Bit-flip logical error probability vs physical error probability

This sweep shows quadratic suppression at small physical error probability. The analytic logical failure probability for independent bit flips is:

P_fail = 3p^2(1-p) + p^3

Syndrome Histogram

images/bitflip_syndrome_distribution.png

Bit-flip syndrome distribution

The histogram shows how syndrome outcomes identify the most likely single-qubit flip without measuring the logical value directly.

Decoder Confusion Matrix

images/bitflip_decoder_confusion_matrix.png

Bit-flip syndrome decoder confusion matrix

Rows are true error patterns and columns are inferred corrections. Diagonal entries represent correct single-error identification; off-diagonal entries come from ambiguous multi-error patterns.

Error-Weight Distribution

images/bitflip_error_weight_distribution.png

Bit-flip error-weight distribution

This plot explains why the repetition code works well at small p: weight-0 and weight-1 events dominate. Weight-2 and weight-3 events become the logical-failure contribution as p grows.

Monte Carlo Demo

images/bitflip_monte_carlo_demo.png

Bit-flip Monte Carlo demo

This example provides a compact visual run of the bit-flip simulation workflow.

Multi-Code Recovery Comparison

images/qec_recovery_failure_by_error_weight.png

Recovery failure by error weight across QEC codes

This figure compares exact Pauli-recovery behavior by error weight across the implemented codes. The distance-3 codes are expected to handle all single-qubit Pauli errors and to fail on some higher-weight patterns.

Depolarizing Noise Sweep

images/qec_depolarizing_logical_error_comparison.png

Depolarizing logical error comparison across QEC codes

This Monte Carlo sweep applies independent depolarizing noise and estimates logical failure after recovery. It is a simple state-vector noise model, not a circuit-level threshold simulation.

Noisy Syndrome Rounds

images/bitflip_noisy_syndrome_rounds.png

Bit-flip noisy syndrome rounds

This plot models classical readout errors in the reported bit-flip syndrome. Repeating syndrome extraction and aggregating the result reduces readout-induced decoder mistakes. The noisy-syndrome trial output includes both raw syndrome history and detector-history differences between adjacent rounds.

Surface-3 Prototype

images/surface3_logical_error_vs_x_error.png

Surface-3 prototype logical error vs X error probability

The surface-code example is a compact 9-data-qubit code-capacity model using Z-check syndromes for X errors, X-check syndromes for Z errors, cached minimum-weight lookups, and repeated noisy syndrome readout. It also includes a lightweight circuit-level schedule prototype with 8 ancillas, data errors between rounds, and hook-like gate faults for studying how measurement order can affect residual errors.

images/surface3_channel_logical_error_comparison.png

Surface-3 logical error comparison by error channel

This comparison shows X-only, Z-only, and independent Pauli X/Y/Z logical failure estimates for the same compact surface-3 layout.

images/surface3_noisy_syndrome_rounds.png

Surface-3 noisy syndrome rounds

This plot models classical readout errors on surface-3 syndrome bits. Each syndrome bit is measured repeatedly and majority-voted before decoding. Surface-3 noisy-syndrome and circuit-level trials expose detector-history matrices alongside the raw measurement history.

Generic Surface-Layout Decoder Scaling

images/surface_distance_logical_error_scaling.png

Generic surface-layout decoder logical error scaling

This benchmark compares the variable-distance square-layout decoder for d = 3, d = 5, and d = 7 under the same code-capacity Pauli noise model. It now plots the default cached lookup plus peeling decoder alongside an MWPM-style matching baseline and a bounded syndrome-graph baseline. The matching and graph baselines are exact only within their searched candidate-fragment weights; the peeling branch is a compact heuristic for unresolved larger-distance syndromes.

Generated Report

docs/SIMULATION_REPORT.md contains a small, reproducible Markdown table generated by examples/generate_simulation_report.m. It uses intentionally low sample counts so it can run quickly in CI.

Key Takeaways

  • The repetition-code examples demonstrate the basic QEC loop: encode, apply noise, measure syndrome, correct, and estimate logical failure.
  • The [[4,2,2]] implementation verifies single-qubit Pauli detection without presenting it as correction.
  • The 5-qubit, Steane, and Shor implementations verify exact recovery for every single-qubit Pauli error.
  • The depolarizing and noisy-syndrome examples are lightweight educational simulations, not hardware-calibrated noise models.
  • The surface-3 code now supports X-only, Z-only, combined Pauli code-capacity simulations, repeated noisy syndrome readout, and a compact circuit-level schedule prototype.
  • The generic surface-layout path now includes d=3/5/7 benchmark scripts with configurable trial count, seed, physical error range, and decoder list.

---

Citation

If you use this repository in a project, cite it as:

Sid Richards (2026). Quantum_Error_Correction: MATLAB/Octave implementations of quantum error-correction codes and simulations.

Author

Sid Richards

License

MIT. See LICENSE.