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arXiv:quant-ph/9605021 · Physical Review A

Simple quantum error-correcting codes

Andrew M. Steane

Published Dec 1, 1996• 523 citations
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Abstract

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}${\mathit{C}}_{1}$, where ${\mathit{C}}_{1}$ and ${\mathit{C}}_{2}$ 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 ${\mathit{C}}_{1}$ 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.

DOI
10.1103/physreva.54.4741

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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.
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.
The hypergraph product (HGP) construction of quantum error-correcting codes (QECC) offers a general and explicit method for building a QECC from two classical codes, thereby paving the way for the discovery of good quantum low-density parity-check codes. In this letter, we propose a general and explicit construction recipe for QECCs from a total of D classical codes for arbitrary D. Following this recipe guarantees the obtainment of a QECC within the stabilizer formalism and nearly exhausts all possible constructions. As examples, we demonstrate that our construction recovers the HGP construction when D = 2 and leads to four distinct types of constructions for D = 3, including a previously studied case as one of them. When the input classical codes are repetition codes, our D = 3 constructions unify various three-dimensional lattice models into a single framework, encompassing the three-dimensional toric code model, a fracton model, and two other intriguing models not previously investigated. Among these, two types of constructions exhibit a trade-off between code distance and code dimension for a fixed number of qubits by adjusting the lengths of the different classical codes, and the optimal choice can simultaneously achieve relatively large values for both code distance and code dimension. Our general construction protocol provides another perspective for enriching the structure of QECCs and enables the exploration of richer possibilities for good codes.
The theory of quantum error correction was established more than a decade ago as the primary tool for fighting decoherence in quantum information processing. Although great progress has already been made in this field, limited methods are available in constructing new quantum error correction codes from old codes. Here we exhibit a simple and general method to construct new quantum error correction codes by nesting certain quantum codes together. The problem of finding long quantum error correction codes is reduced to that of searching several short length quantum codes with certain properties. Our method works for all length and all distance codes, and is quite efficient to construct optimal or near optimal codes. Two main known methods in constructing new codes from old codes in quantum error-correction theory, the concatenating and pasting, can be understood in the framework of nested quantum error correction codes.