3-Qubit Bit-Flip Repetition Code

The bit-flip repetition code protects one logical qubit against a single Pauli-X error by encoding the logical state redundantly across three physical qubits.

Logical States

|0_L> = |000>
|1_L> = |111>

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

alpha|000> + beta|111>

Stabilizers And Syndrome

The implemented stabilizer checks are:

Z1 Z2
Z2 Z3

In code, these are represented as:

ZZI
IZZ

syndrome_bitflip(psi) measures the two parity checks. A nonzero syndrome identifies the most likely single-qubit X error.

The correction convention used by correct_bitflip(psi, syndrome) is:

[1 0] -> X on qubit 1
[1 1] -> X on qubit 2
[0 1] -> X on qubit 3
[0 0] -> no correction

Recovery Flow

The high-level recovery helper is:

[psi_corr, syndrome] = recover_bitflip(psi_noisy);

It measures the syndrome and applies the corresponding X correction. Recovery preserves the encoded logical state up to global phase for zero or one X error. Two or three X errors can produce a logical failure.

Noise And Simulation

The bit-flip Monte Carlo helpers sample independent X errors on the three data qubits. simulate_bitflip_once(p) runs one trial at physical error probability p, and sweep_logical_error(...) estimates logical failure rates over a range of probabilities.

Noisy syndrome examples repeat syndrome extraction and majority-vote each syndrome bit. This models classical readout flips on the reported syndrome, not an ancilla-level circuit.

Main Files

• src/encode_bitflip.m
• src/syndrome_bitflip.m
• src/correct_bitflip.m
• src/recover_bitflip.m
• src/simulate_bitflip_once.m
• src/simulate_noisy_syndrome_bitflip_once.m
• src/sweep_logical_error.m
• src/sweep_noisy_syndrome_bitflip.m

Examples

octave --no-gui examples/minimal_bitflip_recovery.m
octave --no-gui examples/plot_logical_vs_p.m
octave --no-gui examples/plot_noisy_syndrome_rounds.m

Tests

The main checks live in:

• tests/test_bitflip.m
• tests/test_decoders_and_simulations.m

Run them through:

octave --no-gui tests/run_all_tests.m

Current Limits

full measurement circuit.

• Corrects X-type bit flips only.
• Does not protect against phase errors.
• Noisy syndrome mode flips classical syndrome bits rather than simulating a