7-Qubit Steane Code

The Steane code is a CSS stabilizer code built from the classical Hamming parity-check matrix. This repository implements state-vector encoding, separate X/Z syndrome extraction, and single-qubit Pauli recovery.

Parity-Check Matrix

steane_parity_check() returns:

1 0 1 0 1 0 1
0 1 1 0 0 1 1
0 0 0 1 1 1 1

The same matrix is used for Z-type and X-type stabilizer checks.

Logical States

encode_steane(alpha_beta) constructs |0_L> as the equal superposition over the eight codewords generated by the rows of the parity-check matrix. |1_L> is the bitwise complement superposition. The returned encoded state is:

alpha|0_L> + beta|1_L>

Stabilizers And Syndrome

stabilizers_for_code('steane') builds six generators:

three Z-type checks from rows of the Hamming matrix,
three X-type checks from the same rows.

syndrome_steane(psi) returns:

error.

syn.x: the syndrome from Z checks, used to locate the X component of an
syn.z: the syndrome from X checks, used to locate the Z component of an

For a Y error, both syndrome parts are nonzero because Y has both X and Z components.

Recovery Flow

The high-level recovery helper is:

[psi_corr, syndrome] = recover_steane(psi_noisy);

correct_steane(psi, syndrome) converts each three-bit syndrome into a qubit index with binary_syndrome_index(...). It applies an X correction for syn.x and a Z correction for syn.z.

The tests verify recovery from every single-qubit X, Y, and Z error on an arbitrary encoded logical state.

Main Files

src/steane_parity_check.m
src/encode_steane.m
src/syndrome_steane.m
src/correct_steane.m
src/recover_steane.m
src/stabilizers_for_code.m

Examples

octave --no-gui examples/minimal_steane_recovery.m

Tests

The main checks live in:

tests/test_stabilizer_codes.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

syndrome extraction model.

Recovery is designed and tested for single-qubit Pauli errors.
The code uses direct projective stabilizer measurements, not a circuit-level
Transversal logical gates are not implemented.