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A partition function framework for estimating logical error curves in stabilizer codes

Leon Wichette, Hans Hohenfeld, Elie Mounzer, and Linnea Grans-Samuelsson

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AbstractBased 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.► BibTeX data@article{Wichette2026partitionfunction, doi = {10.22331/q-2026-07-28-2175}, url = {https://doi.org/10.22331/q-2026-07-28-2175}, title = {A partition function framework for estimating logical error curves in stabilizer codes}, author = {Wichette, Leon and Hohenfeld, Hans and Mounzer, Elie and Grans-Samuelsson, Linnea}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2175}, month = jul, year = {2026} }► References [1] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002). https://doi.org/10.1063/1.1499754 [2] Christopher T. Chubb and Steven T. Flammia. ``Statistical mechanical models for quantum codes with correlated noise''. Annales de l'Institut Henri Poincaré D 8, 269–321 (2021). https://doi.org/10.4171/aihpd/105 [3] Manuel Rispler, Davide Vodola, Markus Müller, and Seyong Kim. ``The random coupled-plaquette gauge model and the surface code under circuit-level noise'' (2024). arXiv:2412.14004. arXiv:2412.14004 [4] Christophe Piveteau, Christopher T. Chubb, and Joseph M. Renes. ``Tensor Network Decoding Beyond 2D''. PRX Quantum 5, 040303 (2024). https://doi.org/10.1103/PRXQuantum.5.040303 [5] Jan Behrends and Benjamin Béri. ``Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors''. PRX Quantum 6, 040305 (2025). https://doi.org/10.1103/gskb-t5ql [6] Jan Behrends and Benjamin Béri. ``The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology''. PRX Quantum 6, 040350 (2025). https://doi.org/10.1103/psf5-b6j2 [7] Davide Vodola, Manuel Rispler, Seyong Kim, and Markus Müller. ``Fundamental thresholds of realistic quantum error correction circuits from classical spin models''. Quantum 6, 618 (2022). https://doi.org/10.22331/q-2022-01-05-618 [8] Florian Venn, Jan Behrends, and Benjamin Béri. ``Coherent-Error Threshold for Surface Codes from Majorana Delocalization''.

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Mathematical Programming Computation 1, 43–67 (2009). https://doi.org/10.1007/s12532-009-0002-8 [58] Balázs Dezső, Alpár Jüttner, and Péter Kovács. ``LEMON–an open source C++ graph template library''. Electronic notes in theoretical computer science 264, 23–45 (2011). https://doi.org/10.1016/j.entcs.2011.06.003 [59] Donald R Jones, Cary D Perttunen, and Bruce E Stuckman. ``Lipschitzian optimization without the Lipschitz constant''. Journal of optimization Theory and Applications 79, 157–181 (1993). https://doi.org/10.1007/BF00941892Cited by[1] Cory T. Aitchison and Benjamin Béri, "Spacetime Spins: Statistical mechanics for error correction with stabilizer circuits", arXiv:2512.21991, (2025). [2] Zac Tobias, Nikolas P. Breuckmann, and Benedikt Placke, "Optimal Decoding with the Worm", arXiv:2603.05428, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-07-28 14:00:31). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-07-28 14:00:30: Could not fetch cited-by data for 10.22331/q-2026-07-28-2175 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractBased 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.► BibTeX data@article{Wichette2026partitionfunction, doi = {10.22331/q-2026-07-28-2175}, url = {https://doi.org/10.22331/q-2026-07-28-2175}, title = {A partition function framework for estimating logical error curves in stabilizer codes}, author = {Wichette, Leon and Hohenfeld, Hans and Mounzer, Elie and Grans-Samuelsson, Linnea}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2175}, month = jul, year = {2026} }► References [1] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. Journal of Mathematical Physics 43, 4452–4505 (2002). https://doi.org/10.1063/1.1499754 [2] Christopher T. Chubb and Steven T. Flammia. ``Statistical mechanical models for quantum codes with correlated noise''. Annales de l'Institut Henri Poincaré D 8, 269–321 (2021). https://doi.org/10.4171/aihpd/105 [3] Manuel Rispler, Davide Vodola, Markus Müller, and Seyong Kim. ``The random coupled-plaquette gauge model and the surface code under circuit-level noise'' (2024). arXiv:2412.14004. arXiv:2412.14004 [4] Christophe Piveteau, Christopher T. Chubb, and Joseph M. Renes. ``Tensor Network Decoding Beyond 2D''. PRX Quantum 5, 040303 (2024). https://doi.org/10.1103/PRXQuantum.5.040303 [5] Jan Behrends and Benjamin Béri. ``Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors''. PRX Quantum 6, 040305 (2025). https://doi.org/10.1103/gskb-t5ql [6] Jan Behrends and Benjamin Béri. ``The surface code beyond Pauli channels: Logical noise coherence, information-theoretic measures, and errorfield-double phenomenology''. PRX Quantum 6, 040350 (2025). https://doi.org/10.1103/psf5-b6j2 [7] Davide Vodola, Manuel Rispler, Seyong Kim, and Markus Müller. ``Fundamental thresholds of realistic quantum error correction circuits from classical spin models''. Quantum 6, 618 (2022). https://doi.org/10.22331/q-2022-01-05-618 [8] Florian Venn, Jan Behrends, and Benjamin Béri. ``Coherent-Error Threshold for Surface Codes from Majorana Delocalization''.

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A partition function framework for estimating logical error curves in stabilizer codes

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