CalderbankShor

The Calderbank-Shor code is a quantum error-correcting code introduced by Andrew Calderbank and Peter Shor in 1996 as part of the early development of CSS (Calderbank-Shor-Steane) codes. It provides a method to protect quantum information from errors due to decoherence and imperfect operations by encoding logical qubits into a larger set of physical qubits and by using measurements that reveal error syndromes without collapsing the encoded quantum information.

The construction of a Calderbank-Shor code relies on two classical binary linear codes, C1 and C2, of the same length with C2 contained in C1. The quantum code encodes k = dim(C1) − dim(C2) logical qubits into n physical qubits. The stabilizer group of the code comprises two sets of operators: Z-type generators derived from the parity checks of C2 and X-type generators derived from the parity checks of C1. Bit-flip (X) errors are detected by the Z-type stabilizers, while phase-flip (Z) errors are detected by the X-type stabilizers, allowing X and Z errors to be corrected independently.

The performance of a Calderbank-Shor code, like other CSS codes, depends on the properties of the underlying classical codes. If C1 has a larger distance d1 and C2 has distance d2, the resulting quantum code can correct up to ⌊(d1−1)/2⌋ X-errors and ⌊(d2−1)/2⌋ Z-errors. This framework highlights a key advantage: quantum error correction can be reduced to manipulating two classical codes with well-understood properties.

In practice, Calderbank-Shor codes are viewed as an early instance of CSS codes, a family that includes notable examples such as the Steane code. They remain foundational in discussions of stabilizer formalism and fault-tolerant quantum computation.