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 $\varepsilon_0$ . 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, $\varepsilon_0 \ge 2.73\times 10^{-5}$ 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 $\varepsilon_0$. 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, $\varepsilon_0 \ge 2.73\times 10^{-5}$ 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.

Quantum Information and ComputationMar 1, 2006398 citations

Quantum accuracy threshold for concatenated distance-3 codes

Panos Aliferis, Daniel Gottesman, John P. Preskill

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 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.

arXiv:quant-ph/0504218Apr 28, 200584 citations

Accuracy threshold for concatenated error detection in one dimension

Ashley M. Stephens, Zachary W. E. Evans

Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately ${10}^{\ensuremath{-}5}$. This threshold is three orders of magnitude lower than the highest known.Estimates of the quantum accuracy threshold often tacitly assume that it is possible to interact arbitrary pairs of qubits in a quantum computer with a failure rate that is independent of the distance between them. None of the many physical systems that are candidates for quantum computing possess this property. Here we study the performance of a concatenated error-detection code in a system that permits only nearest-neighbor interactions in one dimension. We make use of a message-passing scheme that maximizes the number of errors that can be reliably corrected by the code. Our numerical results indicate that arbitrarily accurate universal quantum computation is possible if the probability of failure of each elementary physical operation is below approximately ${10}^{\ensuremath{-}5}$. This threshold is three orders of magnitude lower than the highest known.

arXiv:0902.2658Physical Review AAug 10, 200923 citations

Quantum accuracy threshold for distance-5 codes

Omar. Khalid

A quantum computer can only solve classically intractable problems like factoring large integers if it can perform quantum computations scalably. The minimum accuracy required among the components of a quantum computer to perform scalable computation is called the quantum accuracy threshold, epsilon 0. We explore the accuracy threshold for Calderbank-Shor-Steane codes of distance 5. Our accuracy thresholds are based on the threshold theorem proven by Aliferis et al. [3]. Their threshold theorem is applicable to concatenated codes and can be used to derive a rigorous lower bound on the threshold. In this thesis we consider degenerate codes with parameters [[17,1,5]] and [[19,1,5]], and a non-degenerate [[21,1,5]] code. For each code we design a set of logical operations sufficient to perform universal computation. We estimate the thresholds using simulation tools for an independent stochastic error model. Our simulations incorporate randomized sampling techniques to estimate the number of ways our largest logical operation can fail. A lower bound of epsilon0 ≥ 3.624 x 10-5 is derived for the [[17,1,5]] code.A quantum computer can only solve classically intractable problems like factoring large integers if it can perform quantum computations scalably. The minimum accuracy required among the components of a quantum computer to perform scalable computation is called the quantum accuracy threshold, epsilon 0. We explore the accuracy threshold for Calderbank-Shor-Steane codes of distance 5. Our accuracy thresholds are based on the threshold theorem proven by Aliferis et al. [3]. Their threshold theorem is applicable to concatenated codes and can be used to derive a rigorous lower bound on the threshold. In this thesis we consider degenerate codes with parameters [[17,1,5]] and [[19,1,5]], and a non-degenerate [[21,1,5]] code. For each code we design a set of logical operations sufficient to perform universal computation. We estimate the thresholds using simulation tools for an independent stochastic error model. Our simulations incorporate randomized sampling techniques to estimate the number of ways our largest logical operation can fail. A lower bound of epsilon0 ≥ 3.624 x 10-5 is derived for the [[17,1,5]] code.

Jan 1, 20060 citations

Accuracy Threshold Theorem

Frank Gaitan

In this chapter we begin to examine quantum stabilizer codes in detail. Sec tion 3.1 introduces the theoretical framework that underlies this class of quan tum error correcting codes, and some examples are given in Section 3.2. Sec tion 3.3 shows how these codes can be described in terms of a binary vector space with symplectic inner product, and finally, concatenated quantum er ror correcting codes are introduced in Section 3.4. The latter discussion will focus on concatenated codes that are put together using quantum stabilizer codes. We will see in Chapter 6 that concatenated codes play an important role in establishing the accuracy threshold theorem for fault-tolerant quantum computing.In this chapter we begin to examine quantum stabilizer codes in detail. Sec tion 3.1 introduces the theoretical framework that underlies this class of quan tum error correcting codes, and some examples are given in Section 3.2. Sec tion 3.3 shows how these codes can be described in terms of a binary vector space with symplectic inner product, and finally, concatenated quantum er ror correcting codes are introduced in Section 3.4. The latter discussion will focus on concatenated codes that are put together using quantum stabilizer codes. We will see in Chapter 6 that concatenated codes play an important role in establishing the accuracy threshold theorem for fault-tolerant quantum computing.

CRC Press eBooksFeb 7, 20080 citations

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.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/0703264Quantum Information and ComputationMar 1, 200895 citations

\varepsilon_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,

\varepsilon_0 \ge 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.

1.04\times 10^{-3}

, is well below Knill's numerical estimates.