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arXiv:quant-ph/0703264 · Quantum Information and Computation

Accuracy threshold for postselected quantum computation

Panos Aliferis, Daniel Gottesman, John P. Preskill

Published Mar 1, 2008• 95 citations
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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, 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.

DOI
10.26421/qic8.3-4-1

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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.
We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold epsilon_0. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, epsilon_0 > 2.73 \times 10^{-5} for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.
We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold ε0\varepsilon_0ε0​. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, ε0≥2.73×10−5\varepsilon_0 \ge 2.73\times 10^{-5}ε0​≥2.73×10−5 for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.
We rigorously analyze Knill's Fibonacci scheme for fault-tolerant quantum computation, which is based on the recursive preparation of Bell states protected by a concatenated error-detecting code. We prove lower bounds on the threshold fault rate of $0.67\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for adversarial local stochastic noise, and $1.25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for independent depolarizing noise. In contrast to other schemes with comparable proved accuracy thresholds, the Fibonacci scheme has a significantly reduced overhead cost because it uses postselection far more sparingly.
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 this paper, we place bounds on when it is impossible to purify a noisy two-qubit state if all the gates used in the purification protocol are subject to adversarial, local, independent noise. It is found that the gate operations must be subject to less than 5.3% error. An existing proof that purification is equivalent to error correction is used to show that this bound can also be applied to concatenated error correcting codes in the presence of noisy gates, and hence gives a limit to the tolerable error rate for a fault-tolerant quantum computer formed by concatenation. This is shown to apply also to the case where error detection and postselection, as proposed by Knill, is used to enhance the threshold. We demonstrate the trade-off between gate and environmentally induced faulty rotations and qubit loss errors.