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Proceedings - International Symposium on Computer Architecture/Proceedings

Astrea: Accurate Quantum Error-Decoding via Practical Minimum-Weight Perfect-Matching

Suhas Vittal, Poulami Das, Moinuddin K. Qureshi

Published Jun 16, 2023• 48 citations
Publisher

Abstract

Quantum devices suffer from high error rates, which makes them ineffective for running practical applications. Quantum computers can be made fault tolerant using Quantum Error Correction (QEC), which protects quantum information by encoding logical qubits using data qubits and parity qubits. The data qubits collectively store the quantum information and the parity qubits are measured periodically to produce a syndrome, which is decoded by a classical decoder to identify the location and type of errors. To prevent errors from accumulating and causing a logical error, decoders must accurately identify errors in real-time, necessitating the use of hardware solutions because software decoders are slow. Ideally, a real-time decoder must match the performance of the Minimum-Weight Perfect Matching (MWPM) decoder. However, due to the complexity of the underlying Blossom algorithm, state-of-the-art real-time decoders either use lookup tables, which are not scalable, or use approximate decoding, which significantly increases logical error rates.

DOI
10.1145/3579371.3589037

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Quantum devices suffer from high error rates, which makes them ineffective for running practical applications. Quantum computers can be made fault tolerant using Quantum Error Correction (QEC), which protects quantum information by encoding logical qubits using data qubits and parity qubits. The data qubits collectively store the quantum information and the parity qubits are measured periodically to produce a syndrome, which is decoded by a classical decoder to identify the location and type of errors. To prevent errors from accumulating and causing a logical error, decoders must accurately identify errors in real-time, necessitating the use of hardware solutions because software decoders are slow. Ideally, a real-time decoder must match the performance of the Minimum-Weight Perfect Matching (MWPM) decoder. However, due to the complexity of the underlying Blossom algorithm, state-of-the-art real-time decoders either use lookup tables, which are not scalable, or use approximate decoding, which significantly increases logical error rates.
Fault-tolerant quantum computing relies on Quantum Error Correction (QEC), which encodes logical qubits into data and parity qubits. Error decoding is the process of translating the measured parity bits into types and locations of errors. To prevent a backlog of errors, error decoding must be performed in real-time (i.e., within 1μs on superconducting machines). Minimum Weight Perfect Matching (MWPM) is an accurate decoding algorithm for surface code, and recent research has demonstrated real-time implementations of MWPM (RT-MWPM) for a distance of up to 9. Unfortunately, beyond d=9, the number of flipped parity bits in the syndrome, referred to as the Hamming weight of the syndrome, exceeds the capabilities of existing RT-MWPM decoders. In this work, our goal is to enable larger distance RT-MWPM decoders by using adaptive predecoding that converts high Hamming weight syndromes into low Hamming weight syndromes, which are accurately decoded by the RT-MWPM decoder.
The error rates of quantum devices are orders of magnitude higher than what is needed to run most quantum applications. To close this gap, Quantum Error Correction (QEC) encodes logical qubits and distributes information using several physical qubits. By periodically executing a syndrome extraction circuit on the logical qubits, information about errors (called syndrome) is extracted while running programs. A decoder uses these syndromes to identify and correct errors in real time, which is necessary to prevent accumulation of errors. Unfortunately, software decoders are slow and hardware decoders are fast but less accurate. Thus, almost all QEC studies so far have relied on offline decoding.
Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.
The Minimum-Weight Perfect Matching (MWPM) decoder is widely used in Quantum Error Correction (QEC) decoding. Despite its high accuracy, existing implementations of the MWPM decoder cannot catch up with quantum hardware, e.g., 1 million measurements per second for superconducting qubits. They suffer from a backlog of measurements that grows exponentially and as a result, cannot realize the power of quantum computation. We design and implement a fast MWPM decoder, called Parity Blossom, which reaches a time complexity almost proportional to the number of defect measurements. We further design and implement a parallel version of Parity Blossom called Fusion Blossom. Given a practical circuit-level noise of 0.1%, Fusion Blossom can decode a million measurement rounds per second up to a code distance of 33. Fusion Blossom also supports stream decoding mode that reaches a 0.7 ms decoding latency at code distance 21 regardless of the measurement rounds.
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.
Quantum Error Correction (QEC) decoding faces a fundamental accuracy-efficiency tradeoff. Classical methods like Minimum Weight Perfect Matching (MWPM) exhibit variable performance across noise models and suffer from polynomial complexity, while tensor network decoders achieve high accuracy but at prohibitively high computational cost. Recent neural decoders reduce complexity but lack the accuracy needed to compete with computationally expensive classical methods. We introduce SAQ-Decoder, a unified framework combining transformer-based learning with constraint aware post-processing that achieves both near Maximum Likelihood (ML) accuracy and linear computational scalability with respect to the syndrome size. Our approach combines a dual-stream transformer architecture that processes syndromes and logical information with asymmetric attention patterns, and a novel differentiable logical loss that directly optimizes Logical Error Rates (LER) through smooth approximations over finite fields. SAQ-Decoder achieves near-optimal performance, with error thresholds of 10.99% (independent noise) and 18.6% (depolarizing noise) on toric codes that approach the ML bounds of 11.0% and 18.9% while outperforming existing neural and classical baselines in accuracy, complexity, and parameter efficiency. Our findings establish that learned decoders can simultaneously achieve competitive decoding accuracy and computational efficiency, addressing key requirements for practical fault-tolerant quantum computing systems.