## A Theory of Quantum Error-Correcting Codes (1996)

Venue: | Phys. Rev. A |

Citations: | 80 - 7 self |

### BibTeX

@ARTICLE{Knill96atheory,

author = {Emanuel Knill and Raymond Laflamme},

title = {A Theory of Quantum Error-Correcting Codes},

journal = {Phys. Rev. A},

year = {1996},

volume = {55},

pages = {900--911}

}

### Years of Citing Articles

### OpenURL

### 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 w...

### Citations

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Citation Context ...s. This has resulted not only from the development of new algorithms such as quantum factoring[1], but also as a consequence of recent experimental work on implementations of individual quantum gates =-=[2, 3, 4]-=- and of quantum cryptography [5]. Unfortunately, the quantum states required to carry out a computation are very sensitive to the imperfections of the hardware, and above all, to the decoherence[6] ca... |

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Citation Context ...ji L i, jj L i (i 6= j) and operators A a , A b in A hi L jA y a A b ji L i = hj L jA y a A b jj L i (19) and hi L jA y a A b jj L i = 0: (20) These conditions are more general than the ones given in =-=[22]-=- which are sufficient but not necessary. Since they are independent of a recovery operator, we can define an A-correcting code as one which satisfies Eq.(19) and Eq.(20) for any one (and therefore eve... |

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for example the appendix of E. Knill, Approximation by Quantum Circuits, Los Alamos National Laboratory preprint LA-UR-95-2225
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Citation Context ...L i. The existence of unitary operators satisfying the second condition follows from Eq.(19), according to which the innerproduct relationships between the A a j0 L i and the A a ji L i are identical =-=[23]-=-. Given such U i , j i r i can be made to satisfy the remaining condition by choosing the basis j 0 r i of V 0 and defining j i r i = U i j 0 r i. We show that R does indeed recover the state, i.e. fo... |