QEC Quantum Error Correction

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5-Qubit Perfect Code

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5-Qubit Perfect Code

The 5-qubit perfect code is the smallest distance-3 stabilizer code that corrects an arbitrary single-qubit Pauli error. This repository implements state-vector encoding through stabilizer projection and syndrome-table recovery.

Stabilizers

fivequbit_stabilizers() returns:

XZZXI
IXZZX
XIXZZ
ZXIXZ

These four generators define a two-dimensional code space inside the five-qubit Hilbert space.

Logical States

encode_fivequbit(alpha_beta) constructs |0_L> by projecting a computational basis seed into the stabilizer +1 eigenspace and the logical-Z +1 eigenspace. It then applies logical X:

XXXXX

to construct |1_L>. The returned encoded state is:

alpha|0_L> + beta|1_L>

The implementation uses:

logical Z = ZZZZZ
logical X = XXXXX

Syndrome And Recovery

syndrome_fivequbit(psi) measures the four stabilizer generators.

correct_fivequbit(psi, syndrome) searches all single-qubit Pauli errors:

X, Y, Z on qubits 1 through 5

For each candidate error, it computes the expected syndrome with pauli_error_syndrome(...). When the expected syndrome matches the measured one, it applies the same Pauli as the correction.

The high-level recovery helper is:

[psi_corr, syndrome] = recover_fivequbit(psi_noisy);

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

Main Files

  • src/fivequbit_stabilizers.m
  • src/encode_fivequbit.m
  • src/syndrome_fivequbit.m
  • src/correct_fivequbit.m
  • src/recover_fivequbit.m
  • src/pauli_error_syndrome.m

Examples

The 5-qubit implementation is covered by the text example runner:

octave --no-gui examples/run_text_examples.m

The recovery functions can also be called directly after adding src/ to the Octave path.

Tests

The main checks live in:

  • tests/test_stabilizer_codes.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

projection, not an encoding circuit.

  • Recovery is designed and tested for single-qubit Pauli errors.
  • The implementation uses exact state-vector operations and stabilizer
  • Logical gates and fault-tolerant measurement circuits are not implemented.