QEC Quantum Error Correction

Documentation

3x3 Bacon-Shor Subsystem Code

All docs Raw markdown View on GitHub

3x3 Bacon-Shor Subsystem Code

The Bacon-Shor implementation is a compact Pauli-frame model of the 3x3 subsystem code. It focuses on syndrome algebra, representative corrections, and logical residual checks rather than full state-vector gauge dynamics.

Layout

Data qubits use a 3x3 row-major grid:

1 2 3
4 5 6
7 8 9

bacon_shor_layout() returns this layout as a matrix.

Stabilizers

The compact stabilizer representatives are:

ZZIZZIZZI
IZZIZZIZZ
XXXXXXIII
IIIXXXXXX

The Z-type stabilizers compare neighboring column parities and detect X-error components. The X-type stabilizers compare neighboring row parities and detect Z-error components.

Syndrome Convention

bacon_shor_x_syndrome(x_error) returns two bits:

[col1 xor col2, col2 xor col3]

where each column value is the parity of X-error components in that column.

bacon_shor_z_syndrome(z_error) returns:

[row1 xor row2, row2 xor row3]

where each row value is the parity of Z-error components in that row.

The syndrome-to-index convention is the same as the 3-bit repetition decoder:

[1 0] -> first row or column
[1 1] -> second row or column
[0 1] -> third row or column
[0 0] -> no correction

Recovery Flow

The high-level Pauli-frame recovery helper is:

out = recover_bacon_shor_pauli(error_string);

For an input Pauli string such as IIIIYIIII, it separates the X and Z components, computes row/column syndromes, applies representative X/Z corrections, and reports whether the residual has logical parity.

The X decoder, decode_bacon_shor_x(s), applies a representative X correction on the first row of the identified column. The Z decoder, decode_bacon_shor_z(s), applies a representative Z correction on the first column of the identified row. Different representatives can differ by gauge operators; the logical residual check is the invariant quantity used here.

Logical Failure Check

bacon_shor_logical_failure(x_residual, z_residual) reports failure when:

  • an X residual has odd parity in any column, or
  • a Z residual has odd parity in any row.

This matches the compact subsystem-code Pauli-frame model used by the decoder.

Noise And Simulation

simulate_bacon_shor_pauli_once(p) samples an independent 9-qubit Pauli error string with sample_pauli_error_string(9, p) and runs recover_bacon_shor_pauli(...).

Main Files

  • src/bacon_shor_layout.m
  • src/bacon_shor_stabilizers.m
  • src/bacon_shor_x_syndrome.m
  • src/bacon_shor_z_syndrome.m
  • src/decode_bacon_shor_x.m
  • src/decode_bacon_shor_z.m
  • src/bacon_shor_logical_failure.m
  • src/recover_bacon_shor_pauli.m
  • src/simulate_bacon_shor_pauli_once.m

Examples

octave --no-gui examples/minimal_bacon_shor_recovery.m

The example is also included in:

octave --no-gui examples/run_text_examples.m

Tests

The main checks live in:

  • tests/test_bacon_shor.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

simulation of gauge-qubit dynamics.

space-time detector decoder.

  • This is a Pauli-frame subsystem-code model, not a full state-vector
  • Recovery is tested for every single-qubit X, Y, and Z error.
  • It does not include noisy gauge-measurement circuits, hook errors, or a