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International Conference on Information and Communication Technology Convergence (ICTC)

PyMatching-based Decoding Simulation of Rotated Surface Codes in Classical Quantum Simulator

Sangmin Lee, Ki‐Sung Jin, Gyu‐Il Cha

Published Oct 19, 2022• 0 citations
Publisher

Abstract

PyMatching provides a method for decoding quantum errors using the minimum-weight perfect matching algorithm. However, although the PyMatching supports various quantum error correction codes, no deep study has been conducted on the rotated planar surface codes that are receiving the latest attention. In this study, we define a procedure for detecting errors using PyMatching for rotated planar surface codes and analyze the correctness of error correction results. To compare error correction behavior by logical qubit size, we used logical qubit with distances of 3, 5, and 7, respectively. Furthermore, we have conducted comparative empirical studies by comparing the theoretical predictions of PyMatching with the actual execution results of classical state vector simulators.

DOI
10.1109/ictc55196.2022.9952682

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PyMatching provides a method for decoding quantum errors using the minimum-weight perfect matching algorithm. However, although the PyMatching supports various quantum error correction codes, no deep study has been conducted on the rotated planar surface codes that are receiving the latest attention. In this study, we define a procedure for detecting errors using PyMatching for rotated planar surface codes and analyze the correctness of error correction results. To compare error correction behavior by logical qubit size, we used logical qubit with distances of 3, 5, and 7, respectively. Furthermore, we have conducted comparative empirical studies by comparing the theoretical predictions of PyMatching with the actual execution results of classical state vector simulators.
Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about 74 % the number of qubits used by the unrotated code to achieve a logical error rate of p L = 10 − 12 at the operational physical error rate of p = 10 − 3 . The ratio remains ≈ 75 % for p values within a factor of two of p = 10 − 3 for all useful p L . Our work finds the low- p L scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.
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.
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.
Abstract Quantum error correction is essential for realizing fault-tolerant quantum computing, where both the efficiency and accuracy of the decoding algorithms play critical roles. In this work, we introduce the implementation of the Planar algorithm, a software framework designed for fast and exact decoding of quantum codes with a planar structure. The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model. Then it utilizes the exact Kac–Ward formula to solve it. In this way, Planar offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure, including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise. Unlike traditional minimum-weight decoders such as minimum-weight perfect matching (MWPM), Planar achieves theoretically optimal performance while maintaining polynomial-time efficiency. In addition, to demonstrate its capabilities, we exemplify the implementation using the rotated surface code, a commonly used quantum error correction code with a planar structure, and show that Planar achieves a threshold of approximately p uc ≈ 0.109 under the depolarizing error model, with a time complexity scaling of O(N 0.69), where N is the number of spins in the Ising model.
Quantum error correction (QEC) is crucial for achieving fault-tolerant quantum computing by addressing errors in noisy qubits. The surface code, a leading QEC framework, traditionally employs the Minimum Weight Perfect Matching (MWPM) decoder. We present a step-by-step methodology that formulates surface code decoding as a supervised machine learning classification problem within a two-level framework, where low-level modules address physical qubit errors and highlevel modules correct logical errors. We implement Feedforward Neural Networks (FFNN) and Graph Neural Networks (GNN), which are individually employed to perform low-level physical qubit error correction and high-level logical error correction for rotated surface codes of distancesd=3,d=5d=3, d=5d=3,d=5, andd=7d=7d=7. Under depolarizing noise, both FFNN and GNN achieve lower logical error rates than MWPM, with GNN yielding up to 28% improvement atd=7d=7d=7. GNN achieves pseudothresholds of 0.1084, 0.1311, and 0.1441 ford=3,d=5d=3, d=5d=3,d=5, andd=7d=7d=7, surpassing MWPM's 0.0914, 0.1024, and 0.1126. Crossover points further indicate GNN's robustness at higher noise levels (e.g., 0.1632 vs. 0.1291 for MWPM). By learning error distributions in a two-level setting, both FFNN and GNN outperform MWPM, demonstrating their potential as scalable, high-performance decoders for near-term quantum devices.
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 % .