mqt.bench.benchmarks.shors_nine_qubit_code

Shor’s 9 Qubit Code benchmark definition.

Module Contents

create_circuit(num_qubits: int) → QuantumCircuit[source]

Returns a quantum circuit implementing Shor’s 9 Qubit Code.

Shor’s code is a quantum error-correcting code, capable of correcting arbitrary single-qubit errors. It encodes 1 logical qubit into 9 physical qubits using a concatenation of the 3-qubit bit-flip and phase-flip repetition codes.

Encoding:

  1. Phase-flip encoding: The logical qubit is encoded across three blocks using |0> → |+++> and |1> → |—> on qubits 0, 3, and 6.

  2. Bit-flip encoding: Each of the three qubits is then encoded using the 3-qubit repetition code (|0> → |000>, |1> → |111>), giving three blocks of three qubits each (0-2, 3-5, 6-8).

  3. So the overall encoding is - |0> -> (|000> + |111>) ⊗ (|000> + |111>) ⊗ (|000> + |111>) - |1> -> (|000> - |111>) ⊗ (|000> - |111>) ⊗ (|000> - |111>)

Syndrome Extraction:

  • Bit-flip syndrome: For each block, 2 ancilla qubits measure the parity of qubit pairs to detect which qubit (if any) experienced a bit flip. Syndrome 01 → qubit 0, syndrome 10 → qubit 1, syndrome 11 → qubit 2.

  • Phase-flip syndrome: 2 ancilla qubits detect phase differences between the three blocks. Syndrome 01 → block 1 (qubits 0-2), syndrome 10 → block 2 (qubits 3-5), syndrome 11 → block 3 (qubits 6-8).

Error Correction:

  • Bit-flip correction: Based on the 6-bit syndrome measurement, X gates are conditionally applied to correct bit flips on any of the 9 data qubits.

  • Phase-flip correction: Based on the 2-bit syndrome measurement, Z gates are conditionally applied to the first qubit of the affected block.

Circuit Structure (per logical qubit):
  • 17 qubits:

    • 9 data qubits (q): The encoded logical qubit

    • 6 bit-flip syndrome qubits (bs): 2 per block for bit-flip detection

    • 2 phase-flip syndrome qubits (ps): For phase-flip detection between blocks

  • 9 classical bits:

    • 6 bit-flip syndrome measurement bits (bsm)

    • 2 phase-flip syndrome measurement bits (psm)

    • 1 logical qubit measurement bit (m)

Parameters:

num_qubits – number of qubits of the returned quantum circuit (must be divisible by 17)

Returns:

QuantumCircuit – a quantum circuit implementing Shor’s 9 Qubit Code