QEC Quantum Error Correction

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9-Qubit Shor Code

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9-Qubit Shor Code

The Shor code combines phase-error protection with bit-flip repetition blocks. This repository implements state-vector encoding, syndrome extraction, and single-qubit Pauli recovery.

Logical States

encode_shor(alpha_beta) builds logical states from three GHZ-like repetition blocks:

|GHZ+> = (|000> + |111>) / sqrt(2)
|GHZ-> = (|000> - |111>) / sqrt(2)

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

For an input state alpha|0> + beta|1>, the encoded state is:

alpha|0_L> + beta|1_L>

Stabilizers And Syndrome

The stabilizer generators returned by stabilizers_for_code('shor') are:

ZZIIIIIII
IZZIIIIII
IIIZZIIII
IIIIZZIII
IIIIIIZZI
IIIIIIIZZ
XXXXXXIII
IIIXXXXXX

The first six checks detect bit flips within each 3-qubit block. The final two checks compare the phase parity of neighboring blocks.

syndrome_shor(psi) returns a structured syndrome with:

phase error.

  • syn.bit: three local two-bit repetition syndromes, one per block.
  • syn.phase: a two-bit syndrome identifying the likely block affected by a

Recovery Flow

The high-level recovery helper is:

[psi_corr, syndrome] = recover_shor(psi_noisy);

correct_shor(psi, syndrome) applies X corrections inside each block based on the local bit syndromes. If the phase syndrome identifies a block, it applies a Z correction to the first qubit of that block. That block-level Z correction is equivalent, on the code space, to correcting the phase component of a single-qubit Pauli error in the block.

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

Main Files

  • src/encode_shor.m
  • src/syndrome_shor.m
  • src/correct_shor.m
  • src/recover_shor.m
  • src/stabilizers_for_code.m

Examples

The Shor 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

not a fault-tolerant syndrome-extraction circuit.

  • Recovery is designed and tested for single-qubit Pauli errors.
  • The implementation uses projective stabilizer measurements on state vectors,
  • It does not simulate encoded gates or logical-state preparation circuits.