Quantum Error Correction

01 · Purpose

Testing whether syndrome logic can recover a single noisy qubit.

The project repeats one logical value across three data qubits, applies a random Rx or Ry rotation to one qubit, and compares measurement results before and after correction.

Two ancilla qubits compute adjacent parity checks. Their syndrome identifies the corrupted position, allowing controlled-X gates to restore the repeated state before the final measurement.

01

Inject noise

Random qubit, rotation plane, and angle from 0 to π/2.

02

Read syndrome

Ancillas store q0 ⊕ q1 and q1 ⊕ q2.

03

Correct state

The syndrome selects the data qubit that receives an X correction.

02 · Code

Noise, syndrome detection, and repeatable simulation.

Rotational error model

Corrupt one statevector.

The helper applies either an X-axis or Y-axis rotation, while the randomizer selects the affected qubit and angle.

Python · Qiskit

def corrupt(angle, q, plane):
    qc_err = QuantumCircuit(1)
    if plane == 'ZX':
        qc_err.rx(angle, 0)
    else:
        qc_err.ry(angle, 0)
    return q.evolve(Operator(qc_err))

def get_noise():
    which_qubit = rng.integers(0, 3)
    plane = 'ZX' if rng.integers(0, 2) == 0 else 'ZY'
    theta = rng.uniform(0.0, np.pi * 0.5)
    return which_qubit, plane, theta

Detection + recovery

Map parity to a correction.

Syndromes 11, 10, and 01 identify the middle, first, and third data qubits.

Python · Qiskit

# Compute syndrome with two ancillas
qc.cx(0, 3); qc.cx(1, 3)
qc.cx(1, 4); qc.cx(2, 4)
qc.measure(3, 3); qc.measure(4, 4)

# 11 → q1, 10 → q0, 01 → q2
qc.append(XGate().control(2), [3, 4, 1])
qc.x(4)
qc.append(XGate().control(2), [3, 4, 0])
qc.x(4)
qc.x(3)
qc.append(XGate().control(2), [3, 4, 2])

Randomized validation

Alternate logical states.

Each trial creates new noise, runs 1,024 shots without correction, then repeats the same test with ECC.

Python · AerSimulator

for i in range(N):
    init_one = (i % 2 == 1)
    base = one if init_one else zero
    which_qubit, plane, theta = get_noise()

    qc_n = noisy_circuit(base, which_qubit, plane, theta)
    c_n = sim.run(qc_n, shots=1024).result().get_counts()

    noisy = corrupt(theta, base, plane)
    args = [base, base, base]
    args[2 - which_qubit] = noisy
    qc_e = create(args[0], args[1], args[2])
    c_e_raw = sim.run(qc_e, shots=1024).result().get_counts()

03 · Findings

The simulated correction recovered every logical measurement.

27randomized trials

1,024shots per trial

81.0%average correct without ECC

100%correct with ECC

Histogram showing noisy logical zero measurement outcomes — A large rotation reduced this example to 58.1% correct measurements.

Histogram showing corrected logical zero measurements — The corrected data register returned `000` in every shot.

Result: across all 27 trials, ECC produced 27,648 correct data-register measurements out of 27,648. The uncorrected simulations averaged 81.0%, with individual trials ranging from 49.1% to 100%.

Scope: this demonstrates correction of one simulated data-qubit error in a repeated-state model; it is not a complete fault-tolerant quantum-computing architecture.

Professional development

Quantum Professional

Black Opal certificate issued by Q-CTRL on May 28, 2026.