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arXiv:2207.00217 · Advanced Quantum Technologies

On the Fault‐Tolerance Threshold for Surface Codes with General Noise

Jing Hao Chai, Hui Khoon Ng

Published Aug 15, 2022• 1 citation
arXivHTMLPDF

Abstract

Abstract Fault‐tolerant quantum computing based on surface codes has emerged as a popular route to large‐scale quantum computers capable of accurate computation even in the presence of noise. Its popularity is, in part, because the fault‐tolerance or accuracy threshold for surface codes is believed to be less stringent than competing schemes. This threshold is the noise level below which computational accuracy can be increased by increasing physical resources for noise removal, and is an important engineering target for realizing quantum devices. The current conclusions about surface code thresholds are, however, drawn largely from studies of probabilistic noise. While probabilistic noise is a natural assumption, current devices experience noise beyond such a model, raising the question of whether conventional statements about the thresholds apply. This work attempts to extend past proof techniques to derive the fault‐tolerance threshold for surface codes subjected to general noise with no particular structure. Surprisingly, no nontrivial threshold is found, i.e., there is no guarantee the surface code prescription works for general noise. While this is not a proof that the scheme fails, it appears that current proof techniques are likely unable to provide an answer. A genuinely new idea is needed to reaffirm the feasibility of surface code quantum computing.

DOI
10.1002/qute.202200008

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Abstract Fault‐tolerant quantum computing based on surface codes has emerged as a popular route to large‐scale quantum computers capable of accurate computation even in the presence of noise. Its popularity is, in part, because the fault‐tolerance or accuracy threshold for surface codes is believed to be less stringent than competing schemes. This threshold is the noise level below which computational accuracy can be increased by increasing physical resources for noise removal, and is an important engineering target for realizing quantum devices. The current conclusions about surface code thresholds are, however, drawn largely from studies of probabilistic noise. While probabilistic noise is a natural assumption, current devices experience noise beyond such a model, raising the question of whether conventional statements about the thresholds apply. This work attempts to extend past proof techniques to derive the fault‐tolerance threshold for surface codes subjected to general noise with no particular structure. Surprisingly, no nontrivial threshold is found, i.e., there is no guarantee the surface code prescription works for general noise. While this is not a proof that the scheme fails, it appears that current proof techniques are likely unable to provide an answer. A genuinely new idea is needed to reaffirm the feasibility of surface code quantum computing.
Abstract Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.
With the intense interest in small, noisy quantum computing devices comes the push for larger, more accurate---and hence more useful---quantum computers. While fully fault-tolerant quantum computers are, in principle, capable of achieving arbitrarily accurate calculations using devices subjected to general noise, they require immense resources far beyond our current reach. An intermediate step would be to construct quantum computers of limited accuracy enhanced by lower-level, and hence lower-cost, noise-removal techniques. This is the motivation for our paper, which looks into fault-tolerant encoded quantum computation targeted at the dominant noise afflicting the quantum device. Specifically, we develop a protocol for fault-tolerant encoded quantum computing components in the presence of amplitude-damping noise, using a 4-qubit code and a recovery procedure tailored to such noise. We describe a universal set of fault-tolerant encoded gadgets and compute the pseudothreshold for the noise, below which our scheme leads to more accurate computation. Our paper demonstrates the possibility of applying the ideas of quantum fault tolerance to targeted noise models, generalizing the recent pursuit of biased-noise fault tolerance beyond the usual Pauli noise models. We also illustrate how certain aspects of the standard fault tolerance intuition, largely acquired through Pauli-noise considerations, can fail in the face of more general noise.
Abstract Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.
Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code-the XZZX code-offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.
This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, ηηη, is smaller than a constant threshold, ηcη_cηc​. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.