Large-scale, fault-tolerant quantum computations will be enabled by quantum error-correcting codes (QECC). This work presents the first systematic technique to test the accuracy and effectiveness of different QECC decoding schemes by comparing a look-up table decoder to solutions generated using algorithmic decoders. Specifically, we examine the results of minimum-weight-perfect-matching and belief-propagation decoders against exhaustive look-up tables for surface codes up to distance seven and categorise where errors are accurately corrected in both decoding schemes. While our results are preliminary, we show that significant quantitative results can be generated, comparing how actual error channels are successfully or unsuccessfully decoded. We show that different decoding schemes perform very differently under the same QECC scheme and error model, and detail how decoders can be tested and classified with respect to errors that are successfully decodable. This work paves the way to the data driven tuning of decoder ensembles and will enable tailored design of hybrid decoding schemes that allow for real-time decoding, while maintaining the high theoretical thresholds allowed by specific quantum error correction codes.
Large-scale, fault-tolerant quantum computations will be enabled by quantum error-correcting codes (QECC). This work presents the first systematic technique to test the accuracy and effectiveness of different QECC decoding schemes by comparing a look-up table decoder to solutions generated using algorithmic decoders. Specifically, we examine the results of minimum-weight-perfect-matching and belief-propagation decoders against exhaustive look-up tables for surface codes up to distance seven and categorise where errors are accurately corrected in both decoding schemes. While our results are preliminary, we show that significant quantitative results can be generated, comparing how actual error channels are successfully or unsuccessfully decoded. We show that different decoding schemes perform very differently under the same QECC scheme and error model, and detail how decoders can be tested and classified with respect to errors that are successfully decodable. This work paves the way to the data driven tuning of decoder ensembles and will enable tailored design of hybrid decoding schemes that allow for real-time decoding, while maintaining the high theoretical thresholds allowed by specific quantum error correction codes.
Realizing the full potential of quantum computation requires quantum error correction (QEC), with most recent breakthrough demonstrations of QEC using the surface code.QEC codes use multiple noisy physical qubits to encode information in fewer logical qubits, enabling the identification of errors through a decoding process.This process increases the logical fidelity (or accuracy) making the computation more reliable.However, most fast (efficient run-time) decoders neglect important noise characteristics, thereby reducing their accuracy.In this work, we introduce decoders that are both fast and accurate, and can be used with a wide class of QEC codes including the surface code.Our decoders, named belief-matching and belief-find, exploit all noise information and thereby unlock higher accuracy demonstrations of QEC.Using the surface code threshold as a performance metric, we observe a threshold at 0.94% error probability for our decoders, outperforming the 0.82% threshold for a standard minimum-weight perfect matching decoder.We also test our belief-matching decoders in a theoretical case study of codes tailored to a biased noise model.We find that the decoders lead to a much higher threshold and lower qubit overhead in the tailored surface code with respect to the standard, square surface code.Surprisingly, in the wellbelow-threshold regime, the rectangular surface code becomes more resource efficient than the tailored surface code due to a previously unnoticed phenomenon that we call "fragile boundaries."Our decoders outperform all other fast decoders in terms of threshold and accuracy, enabling better results in current quantum-error-correction experiments and opening up new areas for theoretical case studies.
Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.
Decoding algorithms are essential to fault-tolerant quantum-computing architectures. In this perspective we explore decoding algorithms for the surface code; a prototypical quantum low-density parity-check code that underlies many of the leading efforts to demonstrate scalable quantum computing. Central to our discussion is the minimum-weight perfect-matching decoder. The decoder works by exploiting underlying structure that arises due to materialised symmetries among surface-code stabilizer elements. By concentrating on these symmetries, we begin to address the question of how a minimum-weight perfect-matching decoder might be generalised for other families of codes. We approach this question first by investigating examples of matching decoders for other codes. These include decoding algorithms that have been specialised to correct for noise models that demonstrate a particular structure or bias with respect to certain codes. In addition to this, we propose a systematic way of constructing a minimum-weight perfect-matching decoder for codes with certain characteristic properties. The properties we make use of are common among topological codes. We discuss the broader applicability of the proposal, and we suggest some questions we can address that may show us how to design a generalised matching decoder for arbitrary stabilizer codes.
Noise-induced errors in quantum information processing corrupt computation and cause useful algorithms to fail. Quantum ErrorCorrection (QEC) counteracts this by encoding a single logical qubit across many physical qubits and correcting errors through repeatedsyndrome measurements. The Minimum-Weight Perfect Matching (MWPM) decoder is the accuracy benchmark for surface-codedecoding, but its microsecond-scale latency can consume the entire QEC cycle time of superconducting processors, leaving almost nomargin for other control operations. Pre-decoded lookup tables reduce latency to the nanosecond scale, but their memory requirementsgrow exponentially with code size, limiting them to very small code distances. In this work, we demonstrate a single-level cache,implemented as a software prototype, that accelerates decoding by exploiting the repetitive nature of surface-code syndromes. Hitrates above 80% are achieved for code distances up to 9 at below-threshold noise rates, and effective decoding speedups of up to18.1 times over PyMatching’s standalone decoding time are observed at code distance ≥ 5. Cache lookup latency remains below 400nanoseconds in the worst case, with memory well below that of full lookup tables. We further develop a mathematical frameworkdefining the theoretical limits of any cache-based decoder.
Abstract To address qubits' high environmental sensitivity and reduce the significant error rates in current quantum devices, quantum error correction stands as one of the most dependable approaches. The topological surface code, renowned for its unique qubit lattice structure, is widely considered a pivotal tool for enabling fault-tolerant quantum computation. Through redundancy introduced across multiple qubits, the surface code safeguards quantum information and identifies errors via state changes captured by syndrome qubits. However, simultaneous errors in data and syndrome qubits substantially escalate decoding complexity. Quantum Generative Adversarial Networks (QGANs) have emerged as promising deep learning frameworks, effectively harnessing quantum advantages for practical tasks such as image processing and data optimization. Consequently, a topological code trainer for quantum-classical hybrid GANs is proposed as an auxiliary model to enhance error correction in machine learning-based decoders, demonstrating significantly improved training accuracy compared to the traditional Minimum Weight Perfect Matching (MWPM) algorithm, which achieves an accuracy of 65 . Numerical experiments reveal that the decoder achieves a fidelity threshold of P=0.1978, substantially surpassing the traditional algorithm's threshold of P=0.1024. To enhance decoding efficiency, a Transformer decoder is integrated, incorporating syndrome error outputs trained via QGANs into its framework. By leveraging its self-attention mechanism, the Transformer effectively captures long-range qubit dependencies at a global scale, enabling high-fidelity error correction over larger dimensions. Numerical validation of the surface code error threshold demonstrates an 8.5 threshold with a correction success rate exceeding 94, whereas the local MWPM decoder achieves only 55 and fails to support large-scale computation at a 4 threshold.
Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76 ± 0.02 % . This is larger than the threshold achieved by previous matching-based decoders ( 14.88 ± 0.02 % ), though still below the known upper bound of ∼ 18.9 % .