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arXiv:quant-ph/9512032 · Physical Review A

Good quantum error-correcting codes exist

A.R. Calderbank, Peter W. Shor

Published Aug 1, 1996• 2,517 citations
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Abstract

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=1-2${\mathit{H}}_{2}$(2t/n) where ${\mathit{H}}_{2}$(p) is the binary entropy function -p${\mathrm{log}}_{2}$p-(1-p)${\mathrm{log}}_{2}$(1-p). Upper bounds on this asymptotic rate are given. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1098

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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.
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.
We present a quantum error correction code which protects a qubit of information against general one qubit errors. To accomplish this, we encode the original state by distributing quantum information over five qubits, the minimal number required for this task. We describe a circuit which takes the initial state with four extra qubits in the state ∣0〉|0〉∣0〉 to the encoded state. It can also be converted into a decoder by running it backward. The original state of the encoded qubit can then be restored by a simple unitary transformation.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
We first present a useful characterization of additive (stabilizer) quantum error-correcting codes. Then we present several examples of We first present a useful characterization of additive (stabilizer) quantum error--correcting codes. Then we present several examples of nonadditive codes. We show that there exist infinitely many non-trivial nonadditive codes with different minimum distances, and high rates. In fact, we show that nonadditive codes that correct t errors can reach the asymptotic rate R=1-2H(2t/n), where H(x) is the binary entropy function. Finally, we introduce the notion of strongly nonadditive codes (i.e., quantum codes with the following property: the trivial code consisting of the entire Hilbert space is the only additive code that is equivalent to any code containing the given code), and provide a construction for an ((11,2,3)) strongly nonadditive code.
It is shown that encoding the state of a quantum computer (or other quantum information-processing device) for error correction has the effect of making its operating states macroscopically indistinguishable; all the more, the more ``stable'' the code is, that is, the more errors it can correct in each pass. It is also shown that for nondegenerate codes the expectation values of macroscopic observables in such encoded states are identical to those that would be obtained for a totally mixed state of maximum entropy, where again the degree of indistinguishability increases with the number of errors corrected by the code. The results can be used to estimate the decoherence rates for systems of qubits under certain forms of interaction with the environment: it is found that when the qubits are encoded all coherent superpositions decay, to lowest order, at the same rate, which scales more slowly with the number of physical qubits than the decoherence rate for macroscopic coherent superpositions (``Schr\"odinger cats'') of the bare qubits.