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arXiv:quant-ph/9604038 · Physical Review A

Class of quantum error-correcting codes saturating the quantum Hamming bound

Daniel Gottesman

Published Sep 1, 1996• 1,227 citations
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Abstract

I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode k=n-j-2 quantum bits (qubits) in n=${2}^{\mathit{j}}$ qubits and correct t=1 error. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1862

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I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode k=n-j-2 quantum bits (qubits) in n=2j{2}^{\mathit{j}}2j qubits and correct t=1 error. \textcopyright{} 1996 The American Physical Society.
Methods of finding good quantum error-correcting codes are discussed, and many example codes are presented. The recipe ${\mathit{C}}_{2}^{\mathrm{\ensuremath{\perp}}}$\ensuremath{\subset}C1{\mathit{C}}_{1}C1​, where C1{\mathit{C}}_{1}C1​ and C2{\mathit{C}}_{2}C2​ are classical codes, is used to obtain codes for up to 16 information quantum bits (qubits) with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C1{\mathit{C}}_{1}C1​ to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error-correcting codes for 3, 4, and 5 information qubits with block lengths of 8, 10, and 11 qubits, respectively. \textcopyright{} 1996 The American Physical Society.
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2H2{\mathit{H}}_{2}H2​(2t/n) where H2{\mathit{H}}_{2}H2​(p) is the binary entropy function -plog2{\mathrm{log}}_{2}log2​p-(1-p)log2{\mathrm{log}}_{2}log2​(1-p). Upper bounds on this asymptotic rate are given. \textcopyright{} 1996 The American Physical Society.