QEC Quantum Error Correction

Documentation

3-Qubit Phase-Flip Repetition Code

All docs Raw markdown View on GitHub

3-Qubit Phase-Flip Repetition Code

The phase-flip repetition code is the X-basis counterpart of the bit-flip repetition code. It protects one logical qubit against a single Pauli-Z error.

Logical States

The implementation uses:

|+> = (|0> + |1>) / sqrt(2)
|-> = (|0> - |1>) / sqrt(2)

|0_L> = |+++>
|1_L> = |--->

For an input state alpha|0> + beta|1>, encode_phaseflip(alpha_beta) returns:

alpha|+++> + beta|--->

Stabilizers And Syndrome

The implemented stabilizer checks are X-parity checks:

X1 X2
X2 X3

In code, these are represented as:

XXI
IXX

syndrome_phaseflip(psi) measures these checks. A nonzero syndrome identifies the most likely single-qubit Z error.

Recovery Flow

The high-level recovery helper is:

[psi_corr, syndrome] = recover_phaseflip(psi_noisy);

It measures the X-parity syndrome and applies the corresponding Z correction. Recovery preserves the encoded logical state up to global phase for zero or one Z error.

Main Files

  • src/encode_phaseflip.m
  • src/syndrome_phaseflip.m
  • src/correct_phaseflip.m
  • src/recover_phaseflip.m
  • src/apply_phase_error_pattern.m

Examples

The phase-flip implementation is covered by the text example runner:

octave --no-gui examples/run_text_examples.m

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

Tests

The main checks live in:

  • tests/test_phaseflip.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

  • Corrects Z-type phase flips only.
  • Does not correct arbitrary Pauli errors by itself.
  • No dedicated plotting script is currently provided for this code.