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Single-Shot Error Correction of Three-Dimensional Homological Product Codes

Armanda O. Quintavalle, Michael Vasmer, Joschka Roffe, Earl T. Campbell

Published Jan 1, 2021• 6 citations
Publisher

Abstract

Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their X components and are therefore single shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of 3.08(4)% and 2.90(2)% for three-dimensional (3D) surface and toric codes, respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed family of 3D homological product codes.

DOI
10.17169/refubium-32119

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Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their X components and are therefore single shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of 3.08(4)% and 2.90(2)% for three-dimensional (3D) surface and toric codes, respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed family of 3D homological product codes.
Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code's measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.
In this work, we study the single-shot performance of higher-dimensional hypergraph product codes decoded using belief propagation and ordered-statistics decoding [P.Panteleev and G. Kalachev, Quantum 5, 585 (2021)].We find that decoding data-qubit and syndrome-measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes.For the three-dimensional toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for Z errors, compared to the threshold of 2.90% previously found using a two-stage decoder [Quintavalle et al., 2021].For the four-dimensional (4D) toric code, for which both X and Z error correction is single shot, our results are consistent with a sustainable single-shot threshold of 4.3%, which is even higher than the threshold of 2.93% for the two-dimensional toric code for the same noise model but using L rounds of stabilizer measurement.We also explore the performance of balanced-product and 4D hypergraph-product codes, which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.
Abstract Single-shot error correction is a technique for correcting physical errors using only a single round of noisy check measurements, such that any residual noise affects a small number of qubits. We propose a general theory of single-shot error correction and establish a sufficient condition called good soundness of the code’s measurement checks. Good code soundness in topological (or low-density parity check, LDPC) codes is shown to entail a macroscopic energy barrier for the associated Hamiltonian. Consequently, 2D topological codes with local checks can not have good soundness. In tension with this, we also show that for any code a specific choice of measurement checks does exist that provides good soundness. In other words, every code can perform single-shot error correction but the required checks may be nonlocal and act on many qubits. If we desire codes with both good soundness and simple measurement checks (the LDPC property) then careful constructions are needed. Finally, we use a double application of the homological product to construct quantum LDPC codes with single-shot error correcting capabilities. Our double homological product codes exploit redundancy in measurements checks through a process we call metachecking.
Preparing logical codewords with a single round of noisy syndrome measurements can reduce the time overhead of fault-tolerant quantum computation. It remains unclear which structural properties of the code suffice to guarantee a threshold for single-shot state preparation. Here we prove that linear confinement suffices for a nonzero single-shot state-preparation threshold for CSS quantum low-density parity-check codes, in contrast to the common belief that soundness is necessary. When the distance is at least logarithmic, we prove that below nonzero local stochastic readout and data error thresholds, the logical error probability after a formal ideal final recovery is exponentially small and approaches zero when the code size approaches infinity. Under the stronger assumption of linear soundness, we show that the residual data error of the preparation protocol is local stochastic, directly allowing composition with other fault-tolerant gadgets. To investigate single-shot state-preparation threshold beyond the above assumptions, we further develop a statistical mechanical mapping that expresses the final logical error rate in terms of a classical partition function. We perform Monte Carlo simulations of this model for the three-dimensional toric code. The numerical results suggest that sublinear soundness can also support a single-shot state-preparation threshold.
Abstract Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to bepSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.
For quantum error correction codes the required number of measurement rounds typically increases with the code distance when measurements are faulty. Single-shot error correction allows for an error threshold with only one round of noisy syndrome measurements regardless of the code size. Here we implement single-shot check operators for toric codes. The single-shot checks are constructed by Gaussian elimination following Campbell [Campbell, Quantum Sci. Technol. 4, 025006 (2019)]. The single-shot check operators result in a sustainable threshold at 5.62% for an error model with noisy measurements, outperforming the conventional toric code-check operators with multiple rounds of noisy measurement. The cost of the transformation is nonlocal high-weight stabilizer generators. We then consider a gate-based error model that leads to increased measurement error with stabilizer weight. Here we find no single-shot threshold behavior and instead find the code family will have an optimal code size for a fixed error rate. For this error model the conventional check operators with multiple measurements yields a lower logical error rate.