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Quantum accuracy threshold for distance-5 codes

Omar. Khalid

Published Jan 1, 2006• 0 citations

Abstract

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.

DOI: 10.82308/51383

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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

\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 have previously [11] shown that for quantum memories and quantum communication, a state can be transmitted over arbitrary distances with error ffl provided each gate has error at most cffl. We discuss a similar concatenation technique which can be used with fault tolerant networks to achieve any desired accuracy when computing with classical initial states, provided a minimum gate accuracy can be achieved. The technique works under realistic assumptions on operational errors. These assumptions are more general than the stochastic error heuristic used in other work. Methods are proposed to account for leakage errors, a problem not previously recognized. 1 Introduction Three recent events are promising to make extensive quantum computations as practical as classical computations. The first is the discovery by Shor [13], Steane [15] and Calderbank et al. [4, 3] of quantum error-correcting codes email: knill@lanl.gov y laflamme@lanl.gov z whz@lanl.gov which can be used to main...

I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes. First I consider only gate errors and use the depolarizing channel error model. I will follow P.Shor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code. A simple computer simulation suggests a threshold for gate errors of the order ε\approx 10^{-3} or better. I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus ε\approx 10^{-4}.

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.