Quantum Error Correction

Small MATLAB/Octave implementations of quantum error-correction codes, syndrome decoders, and Monte Carlo simulations.

Website: sidrichardsquantum.github.io/Quantum_Error_Correction

This repository is educational and code-oriented: it uses pure state vectors, Pauli errors, stabilizer measurements, and focused scripts that generate figures under images/.

Package archives are built as Octave .tar.gz packages. Local builds write to dist/; public release archives should be attached to GitHub Releases.

Implemented Codes

Code or decoder	Physical qubits	What is implemented
3-qubit bit-flip repetition	3	Encode, syndrome, correction, Monte Carlo logical error
3-qubit phase-flip repetition	3	Encode, X-parity syndrome, Z correction
[[4,2,2]] error-detecting code	4	Two-logical-qubit encoding, stabilizer syndrome, single-qubit Pauli detection
5-qubit perfect code	5	Stabilizer encoding and single-qubit Pauli recovery
7-qubit Steane code	7	CSS stabilizer syndrome and single-qubit Pauli recovery
7-qubit triangular color code	7	Color-code-facing CSS stabilizer recovery for the smallest triangular code
9-qubit Shor code	9	Bit/phase syndrome and single-qubit Pauli recovery
3x3 Bacon-Shor subsystem code	9	Pauli-frame row/column syndrome decoder and logical residual check
1D repetition decoder	variable	Minimum-weight syndrome decoder
Surface-3 prototype	9 data qubits + 8 ancillas	X/Z/Pauli code-capacity model, noisy syndrome rounds with detector history, circuit-level schedule prototype, cached minimum-weight lookup
Generic surface-layout decoder	odd d	Variable-distance square layout, plaquette syndromes, cached small-pattern lookup, MWPM-style matching baseline, bounded graph-search baseline, peeling fallback, and configurable d=3/5/7 benchmark scripts

Quickstart

Install Octave, then load the checkout paths:

octave --no-gui --eval "qec_setup; qec_about"

Run the tests:

octave --no-gui tests/run_all_tests.m

Or use the package archive workflow:

make package
octave --no-gui --eval "pkg install dist/qec-0.3.0.tar.gz"
octave --no-gui --eval "pkg load qec; qec_about"

Published package archives, when available, are listed on GitHub Releases.

Run an example:

octave --no-gui examples/plot_logical_vs_p.m

Run a minimal recovery example:

octave --no-gui examples/minimal_bitflip_recovery.m

Run a minimal error-detection example:

octave --no-gui examples/minimal_four_two_two_detection.m

Run a Bacon-Shor Pauli-frame recovery example:

octave --no-gui examples/minimal_bacon_shor_recovery.m

Run a triangular color-code recovery example:

octave --no-gui examples/minimal_color7_recovery.m

Run a variable-distance surface-layout decoder example:

octave --no-gui examples/minimal_surface_distance_decoder.m

Run the surface-layout distance benchmark:

octave --no-gui examples/benchmark_surface_distance_decoder.m
octave --no-gui examples/plot_surface_distance_scaling.m

Override surface-layout benchmark settings:

octave --no-gui examples/benchmark_surface_distance_decoder.m -- --trials=200 --seed=7 --ps="0 0.02 0.04" --distances="3 5" --decoders=min_weight,mwpm,graph_matching

Run a longer surface-code walkthrough:

octave --no-gui examples/surface3_walkthrough.m

Run the circuit-level surface-3 schedule example:

octave --no-gui examples/minimal_surface3_circuit_level.m

Run all text examples:

octave --no-gui examples/run_text_examples.m

Run the full local validation set:

octave --no-gui tests/run_all_tests.m
octave --no-gui examples/run_text_examples.m
octave --no-gui examples/generate_simulation_report.m
octave --no-gui examples/check_generated_images.m

Generate the simulation summary:

octave --no-gui examples/generate_simulation_report.m

Regenerate all figures:

octave --no-gui examples/run_all_plots.m

Surface-3 Monte Carlo sweeps cache reusable .mat results under cache/, which is ignored by git.

Project Map

• src/: QEC implementations, stabilizer helpers, noise models, decoders, and sweeps.
• tests/: Octave-compatible correctness checks.
• examples/: Plot, figure regeneration, and report scripts.
• docs/: Longer usage, theory, setup, and results notes.
• docs/algorithms/: Per-code implementation notes for the bit-flip, phase-flip, [[4,2,2]], Shor, Steane, 5-qubit, triangular color-code, Bacon-Shor, generic surface-layout, and 1D repetition decoder helpers.
• docs/ALGORITHMS.md: Cross-reference table linking schemes, source files, examples, tests, and limits.
• tools/build_pages_site.py: GitHub Pages builder, including generated api.html source-reference output.
• tools/build_octave_package.py: Builds an Octave package archive under dist/.
• docs/SIMULATION_REPORT.md: Generated compact simulation table.
• docs/REPOSITORY_LAYOUT.md: Development layout and release package layout.
• docs/ROADMAP.md: Near-term decoder, modeling, and release priorities.
• qec_setup.m: Adds source and example paths for a cloned checkout.
• DESCRIPTION, INDEX, COPYING: Octave package metadata.
• Makefile: Shortcuts for tests, examples, reports, figures, site build, package build, and validation.
• images/: Generated PNG outputs.

Notes

• The exact recovery tests verify arbitrary logical-state recovery up to global phase.
• Depolarizing-noise examples sample independent X/Y/Z errors after an error event.
• The surface-code model is intentionally compact; it includes code-capacity helpers plus a lightweight circuit-level schedule prototype, not a calibrated hardware threshold simulator. See docs/SURFACE3.md.

See docs/USAGE.md and docs/THEORY.md for more detail.

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Support development

If this repository is useful for research, learning, or experimentation, you can support continued development through GitHub Sponsors:

https://github.com/sponsors/SidRichardsQuantum

Sponsorship supports open-source implementations of quantum algorithms, including better documentation, reproducible workflows, tests, and practical examples for quantum error correction.

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

• LinkedIn: sid-richards-21374b30b
• GitHub: SidRichardsQuantum

License

MIT. See LICENSE.