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arXiv:2004.07247 · Scientific Reports

Cellular automaton decoders for topological quantum codes with noisy measurements and beyond

Michael Vasmer, Dan E. Browne, Aleksander Kubica

Published Jan 21, 2021• 36 citations
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Abstract

We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is also essentially insensitive to the details of the lattice and noise model. Our work constitutes a step towards finding simple and high-performance decoding strategies for a wide range of quantum low-density parity-check codes.

DOI
10.1038/s41598-021-81138-2

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We propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is also essentially insensitive to the details of the lattice and noise model. Our work constitutes a step towards finding simple and high-performance decoding strategies for a wide range of quantum low-density parity-check codes.
Execution of quantum algorithms on large-scale quantum computers will require extremely low logical error rates, which necessitates the development of scalable decoding architectures. Local decoders are promising candidates for this task, as they avoid the communication and data processing bottlenecks inherent in global decoding strategies. Cellular automaton (CA) decoders represent a distinct class of local decoders, offering a path toward the low-latency, real-time decoding required for practical applications. In this work, we present SCALA (Signaling CA with Local Attraction), a novel non-hierarchical cellular automaton decoder for quantum repetition and toric codes. By evaluating SCALA alongside the hierarchical CA decoder proposed by Harrington, we provide a direct comparison between non-hierarchical and renormalization-group-style local decoding strategies. We characterize SCALA across three key metrics: Performance, scalability, and robustness. Our results show that SCALA achieves a code-capacity threshold of approximately pc≈7.5%p_c\approx 7.5\%pc​≈7.5% and provides strong sub-threshold scaling of about pL∝pd/4p_L\propto p^{d/4}pL​∝pd/4 on the toric code. In terms of scalability, our non-hierarchical design ensures that the local computational resources remain independent of system size, yielding a modular local architecture suitable for hardware implementation. Finally, SCALA demonstrates strong robustness to qubit measurement errors and noise within the decoder itself, a critical advantage for real-time decoding on noisy hardware. Our results establish SCALA as a high-performance, scalable, and robust local decoder for scalable quantum error correction.
We propose a new cellular automaton (CA), the sweep rule, which generalizes Toom's rule to any locally Euclidean lattice. We use the sweep rule to design a local decoder for the toric code in d≥3 dimensions, the sweep decoder, and rigorously establish a lower bound on its performance. We also numerically estimate the sweep decoder threshold for the three-dimensional toric code on the cubic and body-centered cubic lattices for phenomenological phase-flip noise. Our results lead to new CA decoders with provable error-correction thresholds for other topological quantum codes including the color code.
Abstract Local decoders, also known as cellular-automaton decoders, offer a promising path toward real-time quantum error-correction by replacing centralized classical decoding, with inherent hardware constraints, with a natively parallel and streamlined architecture from a simple local transition rule. We propose two new types of local decoders for the quantum repetition code in one dimension. The signal-rule decoders interpret odd parities between neighboring qubits as defects, attracted to each other via the exchange of binary-signals. We prove the existence of a threshold in the code-capacity model and present numerical evidence of exponential logical error suppression under a phenomenological noise model, with data and measurement errors at each error correction cycle. Compared to previously known local decoders that suffer from sub-optimal threshold and scaling, our construction significantly narrows the gap with global decoders for practical system sizes and error rates. Implementation requirements can be further reduced by eliminating the need for local classical memories, with a new rule defined on two rows of qubits. This shearing-rule works well at relevant system sizes making it an appealing short-term solution. When combined with biased-noise qubits, such as cat qubits, these decoders enable a fully local quantum memory in one dimension.
We analyze a time-continuous version of a cellular automaton decoder for the toric code in the form of a Lindblad master equation. In this setting, a self-correcting quantum memory becomes a thermodynamical phase of the steady state, which manifests itself through the steady state being topologically ordered. We compute the steady state phase diagram, finding a competition between the error correction rate and the update rate for the classical field of the cellular automaton. Strikingly, we find that self-correction of errors is possible even in situations where conventional quantum error correction does not have a finite 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.
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d′)(d,d')(d,d′)-Toric code with d′,d−d′≥2d',d-d'\geq 2d′,d−d′≥2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays. Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results: 1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d′)(d,d')(d,d′)-Toric complexes, resolving the local-existence question of Kubica and Preskill. 2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead. 3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate ppp below threshold, nd+1n^{d+1}nd+1 noiseless sweep cycles recover the encoded state on a side-length-nnn (d,d′)(d,d')(d,d′)-Toric code with probability at least 1−O(poly⁡(n))pΩ(n)1-O(\operatorname{poly}(n))p^{Ω(n)}1−O(poly(n))pΩ(n), extending earlier guarantees for independent errors.