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arXiv:quant-ph/0703230

Level Reduction and the Quantum Threshold Theorem

Panos Aliferis

Published Mar 25, 2007• 35 citations
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Abstract

The quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.

DOI
10.48550/arxiv.quant-ph/0703230

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The quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.
Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow. Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology. The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise. One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.
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.
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.
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.