Theory of quantum error-correcting codes

Emanuel Knill and Raymond Laflamme
Phys. Rev. A 55, 900 – Published 1 February 1997
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Abstract

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.

  • Received 14 June 1996

DOI:https://doi.org/10.1103/PhysRevA.55.900

©1997 American Physical Society

Authors & Affiliations

Emanuel Knill1 and Raymond Laflamme2

  • 11CIC-3, Mail Stop B265, Los Alamos National Laboratory, New Mexico 87545
  • 22T-6, Mail Stop B288, Los Alamos National Laboratory, New Mexico 87545

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Vol. 55, Iss. 2 — February 1997

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