Category: error-correction  |  Difficulty: intermediate  |  Qubits: 4  |  Gates: 4  |  Depth: 4

The [[4,2,2]] code is the smallest quantum error detecting code. It encodes 2 logical qubits into 4 physical qubits and detects (but cannot correct) any single-qubit error. The logical |00_L⟩ = (|0000⟩+|1111⟩)/√2, equivalent to a 4-qubit GHZ state. Two stabilizer generators ⟨XXXX, ZZZZ⟩ define the code space. A single-qubit error maps the state out of the code space, detectable by measuring stabilizers.

50% |0000⟩, 50% |1111⟩ (logical |00_L⟩)

The OpenQASM 2.0 circuit is in circuit.qasm.

OPENQASM 2.0; include "qelib1.inc"; // [[4,2,2]] code: prepare logical |00_L> = (|0000> + |1111>) / sqrt(2) qreg q[4]; creg c[4]; h q[0]; cx q[0],q[1]; cx q[0],q[2]; cx q[0],q[3]; measure q[0] -> c[0]; measure q[1] -> c[1]; measure q[2] -> c[2]; measure q[3] -> c[3]; 

error-detection css-code stabilizer fault-tolerant

• Vaidman, Goldenberg, Wiesner (1996). Error Prevention Scheme with Four Particles. PRA 54(3), R1745

MIT — part of the OpenQC Algorithm Catalog.