Shor Code

The Shor code, published by Peter Shor in 1995, was the first quantum error-correcting code. It proves that quantum information can be protected against decoherence, resolving a fundamental question about whether fault-tolerant quantum computing is even possible.

Encoding

The Shor code encodes one logical qubit into nine physical qubits. The encoding works in two stages:

Phase correction (outer code). The logical state is encoded into three blocks of three qubits each, using a repetition code in the X basis:

|0⟩_L → |+⟩|+⟩|+⟩  (each |+⟩ = (|000⟩ + |111⟩)/√2)
|1⟩_L → |-⟩|-⟩|-⟩  (each |-⟩ = (|000⟩ - |111⟩)/√2)

Bit-flip correction (inner code). Within each block of three qubits, the state is further encoded using a bit-flip repetition code:

|0⟩ → |000⟩
|1⟩ → |111⟩

The combination of the two repetition codes corrects any single-qubit error, including:

• Bit flips (X errors)
• Phase flips (Z errors)
• Combined Y errors
• Any single-qubit unitary error (by linearity)

Error Correction

The Shor code uses six stabilizer measurements (four-qubit parity checks) to identify which qubit suffered an error without measuring the encoded logical state:

• Bit-flip syndromes: compare adjacent qubits within each group of three
• Phase-flip syndromes: compare the phases between the three blocks

After measuring the six syndrome bits, the correction circuit applies X and/or Z gates to restore the logical qubit.

Why the Surface Code Replaced It

The Shor code uses 9 physical qubits per logical qubit and requires long-range gates between non-neighbouring qubits. The surface code achieves comparable error correction with:

• Only nearest-neighbor interactions (crucial for physical hardware)
• Higher error thresholds (~1% vs ~0.1% for Shor)
• Better efficiency at large code distances

The Shor code remains historically important and conceptually illuminating, but is not used in modern fault-tolerant architectures.

See also

• Surface Code
• Steane Code
• Stabilizer Code
• Quantum Error Correction