Category: error-correction  |  Difficulty: advanced  |  Qubits: 9  |  Gates: 9  |  Depth: 4

Shor's 9-qubit code combines a phase-flip outer code with three bit-flip inner codes to protect against any single-qubit error (bit-flip, phase-flip, or both). It was the first quantum error-correcting code discovered. The encoding circuit transforms one logical qubit q[0] into a 9-qubit codeword distributed across q[0..8]. This circuit implements only the encoding; syndrome measurement and correction require ancilla qubits and are omitted for clarity.

9-qubit encoded state (no measurement — encoding only)

The OpenQASM 2.0 circuit is in circuit.qasm.

OPENQASM 2.0;
include "qelib1.inc";
// Shor's 9-qubit code: encoding circuit
// q[0]: logical data qubit (input), q[1..8]: physical ancilla qubits
qreg q[9];
// Phase-flip outer code: spread to groups of 3
cx q[0],q[3];
cx q[0],q[6];
// Switch to X basis for phase-flip protection
h q[0]; h q[3]; h q[6];
// Bit-flip inner codes: each X-basis qubit gets 3-fold repetition
cx q[0],q[1]; cx q[0],q[2];
cx q[3],q[4]; cx q[3],q[5];
cx q[6],q[7]; cx q[6],q[8];

error-correction shor-code fault-tolerant 9-qubit

• Shor, P.W. (1995). Scheme for reducing decoherence in quantum computer memory. PRA 52(4), R2493

MIT — part of the OpenQC Algorithm Catalog.