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Quantum Error-Correcting Code

A quantum error-correcting code is a subspace of the Hilbert space of qubits that encodes quantum states so that a specified set of errors can be detected or corrected. If is the orthogonal projection onto , then a set of error operators ${E_a}$ is correctable exactly when the Knill-Laflamme conditions

hold for constants and all , (Knill and Laflamme 1997).

A quantum code encoding logical qubits into physical qubits and having distance is denoted . A stabilizer code is the common eigenspace of an abelian group contained in the -qubit Pauli group, with . This construction converts many questions about quantum error correction into questions about finite groups and linear algebra (Gottesman 1997).

Dzoba (2026) constructed binary stabilizer codes with parameters and . Their distances attain the upper bounds listed in that paper for these lengths and dimensions. Dzoba (2026) reports that Claude generated the constructions and search programs under human direction, with additional Codex-generated verification programs.

References

Dzoba, C. "New Lower Bounds for Binary Constant-Weight Codes from Prescribed-Automorphism Search, and Two Quantum Codes Meeting Their Upper Bounds." 16 Oct 2026. https://arxiv.org/abs/2610.24025.

Gottesman, D. "Stabilizer Codes and Quantum Error Correction." PhD thesis. Pasadena, CA: California Institute of Technology, 28 May 1997. https://arxiv.org/abs/quant-ph/9705052.

Knill, E. and Laflamme, R. "Theory of Quantum Error-Correcting Codes." Phys. Rev. A 55, 900-911, 1997. https://doi.org/10.1103/PhysRevA.55.900.

Nielsen, M. and Chuang, I. Quantum Computation and Quantum Information. Cambridge, England: Cambridge University Press, 2000.

Quantum Error-Correcting Code

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