Coherent-parity-check (CPC) code

Coherent-parity-check (CPC) code[1–3]

Description

A qubit stabilizer code for which two binary linear codes are used to directly construct encoding and decoding circuits against \(X\)- and \(Z\)-type errors, respectively, via ZX calculus [4,5]. CPC codes can be obtained from numerical search [2].

Cousins

Linear binary code— The CPC Construction uses two binary linear codes.
\([2^r-1,2^r-r-1,3]\) Hamming code— Tripartite CPC codes are constructed from Hamming codes via the CPC construction [1; Thm. 4].
\([[4,2,2]]\) Four-qubit code— CPC gadgets for the \([[4,2,2]]\) code have been implemented on the IBM 5Q superconducting device [2].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Qubit stabilizer code

CPC codes are a type of stabilizer code. A teleported version of the CPC construction, the Clifford noise reduction (CliNR) scheme, can reduce noise in Clifford circuits with Pauli measurements with at most a three-fold overhead in the number of qubits and gates [6,7]. There is a simple formula for the probability that a Clifford circuit contains a logical error [3].

Coherent-parity-check (CPC) code

Children

\([[2^r+r, 2^r-r-2, 3]]\) Ring CPC code

Qubit CSS codeQuantum Reed-Muller Color BB Homological Kitaev surface

CSS codes are a subset of CPC codes [1], with the latter not requiring the two classical codes to be related.

References

[1]: N. Chancellor, A. Kissinger, S. Zohren, J. Roffe, and D. Horsman, “Graphical structures for design and verification of quantum error correction”, Quantum Science and Technology 8, 045028 (2023) arXiv:1611.08012 DOI
[2]: J. Roffe, D. Headley, N. Chancellor, D. Horsman, and V. Kendon, “Protecting quantum memories using coherent parity check codes”, Quantum Science and Technology 3, 035010 (2018) arXiv:1709.01866 DOI
[3]: D. M. Debroy and K. R. Brown, “Extended flag gadgets for low-overhead circuit verification”, Physical Review A 102, (2020) arXiv:2009.07752 DOI
[4]: B. Coecke and R. Duncan, “Interacting Quantum Observables”, Automata, Languages and Programming 298 DOI
[5]: B. Coecke and R. Duncan, “Interacting quantum observables: categorical algebra and diagrammatics”, New Journal of Physics 13, 043016 (2011) arXiv:0906.4725 DOI
[6]: N. Delfosse and E. Tham, “Low-cost noise reduction for Clifford circuits”, (2024) arXiv:2407.06583
[7]: E. Tham and N. Delfosse, “Optimized Clifford Noise Reduction: Theory, Simulations and Experiments”, (2025) arXiv:2504.13356

Page edit log

Victor V. Albert (2023-11-14) — most recent

Cite as:

“Coherent-parity-check (CPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/cpc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/cpc.yml.