1996
DOI: 10.1103/physreva.54.1098
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Good quantum error-correcting codes exist

Abstract: A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-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 − 2H 2 (2t/n) where H 2 (p) is the binary entropy functio… Show more

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Cited by 2,140 publications
(2,333 citation statements)
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References 13 publications
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“…In our language: Alice and Bob share a superposition of states | Ă f ey with ωpf q, ωpeq ď t. Then they use the fact that, roughly speaking, | Ć 0 n 0 n y is an encoding of | Ą 0 ℓ 0 ℓ y for some ℓ ă n using a random CSS code correcting t bit/phase error. (Calderbank-Shor-Steane codes [CS96,Ste96], see Appendix C.) So if Alice and Bob apply error correction and decoding to | Ă f ey, they get the state | Ą 0 ℓ 0 ℓ y. Then, if Alice and Bob measure that state, they get identical and uniformly distributed keys, and the adversary has no information.…”
Section: Revocably Hiding Tresmentioning
confidence: 99%
See 1 more Smart Citation
“…In our language: Alice and Bob share a superposition of states | Ă f ey with ωpf q, ωpeq ď t. Then they use the fact that, roughly speaking, | Ć 0 n 0 n y is an encoding of | Ą 0 ℓ 0 ℓ y for some ℓ ă n using a random CSS code correcting t bit/phase error. (Calderbank-Shor-Steane codes [CS96,Ste96], see Appendix C.) So if Alice and Bob apply error correction and decoding to | Ă f ey, they get the state | Ą 0 ℓ 0 ℓ y. Then, if Alice and Bob measure that state, they get identical and uniformly distributed keys, and the adversary has no information.…”
Section: Revocably Hiding Tresmentioning
confidence: 99%
“…For more information, see [CS96,Ste96] or the textbook [NC10, Section 10.4.2]. A CSS code with parameters n, k 1 , k 2 , t consists of two classical linear binary codes, namely an rn, k 1 s code C 1 19 and an rn, k 2 s code C 2 such that C 2 Ď C 1 and both C 1 and C K 2 can correct up to t errors.…”
Section: Css Codes -Recap and Propertiesmentioning
confidence: 99%
“…The concept of operator measurements is closely related to the Stabiliser formalism of Gottesman [39], commonly used in Quantum Error Correction (QEC) [40,41,42]. A state |Ψ is stabilised by a operator U , if U |Ψ = |Ψ .…”
Section: Generating Cluster and Stabiliser Statesmentioning
confidence: 99%
“…However, noise and imperfections still stand in our way. While the active strategies to prevent errors, such as quantum error correcting codes [17,18,19,20] may, in principle, be universal as claimed, passive prevention methods have hardware resource advantages. For example, decoherence-free subspaces (DFS) and noiseless subsystems (NS) [21,22,23,24] are based on the symmetry of the system-bath interaction, so do not require active detection and correction of errors.…”
Section: Introductionmentioning
confidence: 99%