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Quantum Information and Computation

Accuracy threshold for postselected quantum computation

AliferisPanos, GottesmanDaniel, PreskillJohn

Published Mar 1, 2008• 1 citation
Publisher

Abstract

We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill,...
DOI
10.5555/2011763.2011764

Related papers

Accuracy threshold for postselected quantum computation

Panos Aliferis, Daniel Gottesman, John P. Preskill

We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated error-detecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated error-correcting codes) not to strongly-correlated adversarial noise. Our rigorously established lower bound on the accuracy threshold, $1.04\times 10^{-3}$, is well below Knill's numerical estimates.

arXiv:quant-ph/0703264•Quantum Information and Computation•Mar 1, 2008•95 citations

Fault-Tolerant Postselected Quantum Computation: Threshold Analysis

Emanuel H. Knill

The schemes for fault-tolerant postselected quantum computation given in [Knill, Fault-Tolerant Postselected Quantum Computation: Schemes, http://arxiv.org/abs/quant-ph/0402171] are analyzed to determine their error-tolerance. The analysis is based on computer-assisted heuristics. It indicates that if classical and quantum communication delays are negligible, then scalable qubit-based quantum computation is possible with errors above 1% per elementary quantum gate.

arXiv:quant-ph/0404104•Apr 19, 2004•44 citations

Postselection threshold against biased noise

Ben W. Reichardt

The highest current estimates for the amount of noise a quantum computer can tolerate are based on fault-tolerance schemes relying heavily on postselecting on no detected errors. However, there has been no proof that these schemes give even a positive tolerable noise threshold. A technique to prove a positive threshold, for probabilistic noise models, is presented. The main idea is to maintain strong control over the distribution of errors in the quantum state at all times. This distribution has correlations which conceivably could grow out of control with postselection. But in fact, the error distribution can be written as a mixture of nearby distributions each satisfying strong independence properties, so there are no correlations for postselection to amplify

arXiv:quant-ph/0608018•Annual Symposium on Foundations of Computer Science•Jan 1, 2006•2 citations

Quantum Mechanics

Frank Gaitan

In Chapters 2-5 we gave a detailed presentation of the theory of quantum error correction and fault-tolerant quantum computing. The stage is finally set for a proof of the accuracy threshold theorem [2-9]. As pointed out in Chapter 1, this theorem spells out the conditions under which quantum computing can be done reliably using imperfect quantum gates and in the presence of noise. The proof presented in this chapter follows Ref. [6]. Section 6.1 addresses a number of preliminary topics. It explains why a concatenated quantum error correcting code (QECC) should be used to protect the computational data when fault-tolerance is an issue; it states the principal assumptions underlying the threshold calculations presented in Section 6.2; and finally, it closes with a statement of the accuracy threshold theorem. The proof of this theorem is taken up in Section 6.2. For the error model introduced in Section 6.1, the accuracy threshold is calculated for gates in the Clifford group N(Qn), storage registers, and the Toffoli gate. The calculations make explicit use of the recursive structure of a concatenated QECC and the fault-tolerant procedures introduced in Chapter 5. The theorem is proved by showing that all three cases yield non-zero threshold values.

CRC Press eBooks•Feb 7, 2008•0 citations

Quantum Error Correction and Fault Tolerant Quantum Computing

Frank Gaitan

It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists. Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to date—quantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code. Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.

CRC Press eBooks•Dec 19, 2013•99 citations

Fault-Tolerant Postselected Quantum Computation: Schemes

Emanuel H. Knill

Postselected quantum computation is distinguished from regular quantum computation by accepting the output only if measurement outcomes satisfy predetermined conditions. The output must be accepted with nonzero probability. Methods for implementing postselected quantum computation with noisy gates are proposed. These methods are based on error-detecting codes. Conditionally on detecting no errors, it is expected that the encoded computation can be made to be arbitrarily accurate. Although the probability of success of the encoded computation decreases dramatically with accuracy, it is possible to apply the proposed methods to the problem of preparing arbitrary stabilizer states in large error-correcting codes with local residual errors. Together with teleported error-correction, this may improve the error tolerance of non-postselected quantum computation.

arXiv:quant-ph/0402171•Feb 23, 2004•75 citations
We prove an accuracy threshold theorem for fault-tolerant quantum computation based on error detection and postselection. Our proof provides a rigorous foundation for the scheme suggested by Knill, in which preparation circuits for ancilla states are protected by a concatenated error-detecting code and the preparation is aborted if an error is detected. The proof applies to independent stochastic noise but (in contrast to proofs of the quantum accuracy threshold theorem based on concatenated error-correcting codes) not to strongly-correlated adversarial noise. Our rigorously established lower bound on the accuracy threshold, 1.04×10−31.04\times 10^{-3}1.04×10−3, is well below Knill's numerical estimates.
The highest current estimates for the amount of noise a quantum computer can tolerate are based on fault-tolerance schemes relying heavily on postselecting on no detected errors. However, there has been no proof that these schemes give even a positive tolerable noise threshold. A technique to prove a positive threshold, for probabilistic noise models, is presented. The main idea is to maintain strong control over the distribution of errors in the quantum state at all times. This distribution has correlations which conceivably could grow out of control with postselection. But in fact, the error distribution can be written as a mixture of nearby distributions each satisfying strong independence properties, so there are no correlations for postselection to amplify
In Chapters 2-5 we gave a detailed presentation of the theory of quantum error correction and fault-tolerant quantum computing. The stage is finally set for a proof of the accuracy threshold theorem [2-9]. As pointed out in Chapter 1, this theorem spells out the conditions under which quantum computing can be done reliably using imperfect quantum gates and in the presence of noise. The proof presented in this chapter follows Ref. [6]. Section 6.1 addresses a number of preliminary topics. It explains why a concatenated quantum error correcting code (QECC) should be used to protect the computational data when fault-tolerance is an issue; it states the principal assumptions underlying the threshold calculations presented in Section 6.2; and finally, it closes with a statement of the accuracy threshold theorem. The proof of this theorem is taken up in Section 6.2. For the error model introduced in Section 6.1, the accuracy threshold is calculated for gates in the Clifford group N(Qn), storage registers, and the Toffoli gate. The calculations make explicit use of the recursive structure of a concatenated QECC and the fault-tolerant procedures introduced in Chapter 5. The theorem is proved by showing that all three cases yield non-zero threshold values.
It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists. Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to date—quantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code. Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.
Postselected quantum computation is distinguished from regular quantum computation by accepting the output only if measurement outcomes satisfy predetermined conditions. The output must be accepted with nonzero probability. Methods for implementing postselected quantum computation with noisy gates are proposed. These methods are based on error-detecting codes. Conditionally on detecting no errors, it is expected that the encoded computation can be made to be arbitrarily accurate. Although the probability of success of the encoded computation decreases dramatically with accuracy, it is possible to apply the proposed methods to the problem of preparing arbitrary stabilizer states in large error-correcting codes with local residual errors. Together with teleported error-correction, this may improve the error tolerance of non-postselected quantum computation.