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This research introduces a partition function framework that distinguishes between maximum likelihood decoding at the Nishimori temperature and probabilistic strategies for handling error degeneracy in stabilizer codes. By mapping quantum error correction to disordered statistical mechanics models, the study demonstrates more sample-efficient methods for estimating logical performance boundaries without relying solely on failure counting. These findings offer a rigorous theoretical approach to optimizing decoder accuracy under various noise regimes, directly advancing our understanding of fault-tolerant quantum computation limits.
A partition function framework for estimating logical error curves in stabilizer codes
arXiv:2505.15758v3 Announce Type: replace Abstract: Based on the mapping between stabilizer quantum error correcting codes and disordered statistical mechanics models, we define a ratio of partition functions that measures the success probability for maximum partition function decoding, which at the Nishimori temperature corresponds to maximum likelihood (ML) decoding. We show that this ratio differs from the similarly defined order probability and describe the decoding strategy whose success rate is described by the order probability. We refer to the latter as a probabilistic partition function decoding and show that it is the strategy that at zero temperature corresponds to maximum probability (MP) decoding. Based on the difference between the two decoders, we discuss the possibility of a maximum partition function decodability boundary outside the order-disorder phase boundary. At zero temperature, the difference between the two ratios measures to what degree MP decoding can be improved by accounting for degeneracy among maximum probability errors, through methods such as ensembling. We consider in detail the example of the toric code under bitflip noise, which maps to the Random Bond Ising Model. We demonstrate that estimation of logical performance through decoding probability and order probability is more sample efficient than estimation by counting failures of the corresponding decoders, especially in the regime of low noise. We consider both uniform noise and noise where qubits are given individual error rates. The latter noise model lifts the degeneracy among maximum probability errors, but we show that ensembling remains useful as long as it also samples less probable errors. We also consider, in less detail, the color code under bitflip and depolarizing noise.
Abstract:Based on the mapping between stabilizer quantum error correcting codes and disordered statistical mechanics models, we define a ratio of partition functions that measures the success probability for maximum partition function decoding, which at the Nishimori temperature corresponds to maximum likelihood (ML) decoding. We show that this ratio differs from the similarly defined order probability and describe the decoding strategy whose success rate is described by the order probability. We refer to the latter as a probabilistic partition function decoding and show that it is the strategy that at zero temperature corresponds to maximum probability (MP) decoding. Based on the difference between the two decoders, we discuss the possibility of a maximum partition function decodability boundary outside the order-disorder phase boundary. At zero temperature, the difference between the two ratios measures to what degree MP decoding can be improved by accounting for degeneracy among maximum probability errors, through methods such as ensembling. We consider in detail the example of the toric code under bitflip noise, which maps to the Random Bond Ising Model. We demonstrate that estimation of logical performance through decoding probability and order probability is more sample efficient than estimation by counting failures of the corresponding decoders, especially in the regime of low noise. We consider both uniform noise and noise where qubits are given individual error rates. The latter noise model lifts the degeneracy among maximum probability errors, but we show that ensembling remains useful as long as it also samples less probable errors. We also consider, in less detail, the color code under bitflip and depolarizing noise.
37 pages, 21 figures. v2: 38 pages, 21 figures, made minor edits and added links to github for source code. v3: 45 pages, 30 figures. Added material: color code under bitflip and depolarizing, surface code under hook errors and readout, further analysis of sample efficiency
arXiv:2505.15758 [quant-ph]
(or arXiv:2505.15758v3 [quant-ph] for this version)
Full text under the publisher's open licence · 314 words
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