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Ising on the Donut: Regimes of Topological Quantum Error Correction from Statistical Mechanics

Lucas H. English1,*, Sam Roberts2, Stephen D. Bartlett1, Andrew C. Doherty1,2, and Dominic J. Williamson1,2

  • 1Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia
  • 2PsiQuantum, Brisbane, Australia

  • *Contact author: lucas.english@sydney.edu.au
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PRX Quantum 7, 043004 – Published 6 October, 2026

DOI: https://doi.org/10.1103/4wvz-1p8m

Abstract

Utility-scale quantum computers require quantum error correcting codes with large numbers of physical qubits to achieve sufficiently low logical error rates. The performance of quantum error correction (QEC) is generally predicted through large-scale numerical simulations, used to estimate thresholds, finite-size scaling, and exponential suppression of logical errors below threshold. The connection of QEC to models from statistical mechanics provides an alternative tool for analyzing QEC performance. However, predicting the behavior of these models also requires large-scale numerical simulations, as analytic solutions are not generally known. Here we exploit an exact mapping, from a toric code under bit-flip noise that is post-selected on being syndrome free to the exactly solvable two-dimensional Ising model on a torus, to derive an analytic solution for the logical failure rate across its full domain of physical error rates. In particular, this mapping provides closed-form expressions for the logical failure rate in four distinct regimes: the path-counting, below-threshold (ordered), near-threshold (critical), and above-threshold (disordered) regimes. Our framework places a number of familiar and long-standing numerical observations on firm theoretical ground. It also motivates explicit ansätze for the conventional QEC setting of non-post-selected codes whose statistical mechanics mappings involve random-bond disorder. Specifically, we introduce an effective tension model for the below-threshold regime, and a new scaling ansatz for the near-threshold regime, derived from an analysis of the domain wall energy cost distributions. By bridging statistical mechanics theory and quantum error correction practice, our results offer a new toolkit for designing, benchmarking, and understanding topological codes beyond current computational limits.

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Popular Summary

Quantum computers rely on quantum error correcting codes to protect fragile quantum information from noise, but predicting how well these codes perform is usually possible only through large numerical simulations. In this work, we show that the performance of a widely used topological quantum error correcting code can instead be calculated exactly, without simulation, by mapping the problem to a simple and well-understood model from statistical physics. This approach allows us to compute logical error rates in closed form and directly connect the performance of the quantum code to the surface tension of a two-dimensional magnetic system. Our results provide new intuition for why topological codes work, and when they fail, while offering practical tools for benchmarking quantum hardware. More broadly, this work strengthens the bridge between quantum error correction and statistical mechanics, opening new routes to analytic performance predictions for future quantum technologies.

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References (125)

  1. J. Preskill, Beyond NISQ: The megaquop machine, ACM Trans. Quantum Comput. 6, 1 (2025).
  2. D. Poulin, Optimal and efficient decoding of concatenated quantum block codes, Phys. Rev. A 74, 052333 (2006).
  3. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. 43, 4452 (2002).
  4. S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014).
  5. H. G. Katzgraber, H. Bombín, and M. A. Martin-Delgado, Error threshold for color codes and random three-body Ising models, Phys. Rev. Lett. 103, 090501 (2009).
  6. C. T. Chubb, General tensor network decoding of 2D Pauli codes, arXiv:2101.04125.
  7. C. Wang, J. Harrington, and J. Preskill, Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory, Ann. Phys. 303, 31 (2003).
  8. F. H. E. Watson and S. D. Barrett, Logical error rate scaling of the toric code, New J. Phys. 16, 093045 (2014).
  9. C. T. Chubb and S. T. Flammia, Statistical mechanical models for quantum codes with correlated noise, Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8, 269 (2021).
  10. D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, in Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC ’97 (ACM Press, New York, NY, United States, 1997), pp. 176–188.
  11. J. Preskill, Reliable quantum computers, Proc. R. Soc. Lond., Ser. A 454, 385 (1998).
  12. E. Knill, R. Laflamme, and W. H. Zurek, Resilient quantum computation, Science 279, 342 (1998).
  13. P. Aliferis, D. Gottesman, and J. Preskill, Quantum accuracy threshold for concatenated distance-3 code, Quantum Inf. Comput. 6, 97 (2006).
  14. H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, Strong resilience of topological codes to depolarization, Phys. Rev. X 2, 021004 (2012).
  15. Y. Xiao, B. Srivastava, and M. Granath, Exact results on finite size corrections for surface codes tailored to biased noise, Quantum 8, 1468 (2024).
  16. Y. Li and M. P. A. Fisher, Statistical mechanics of quantum error correcting codes, Phys. Rev. B 103, 104306 (2021).
  17. D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, On the stability of topological order in tensor network states, Phys. Rev. B 104, 235151 (2021).
  18. H. Song, J. Schönmeier-Kromer, K. Liu, O. Viyuela, L. Pollet, and M. A. Martin-Delgado, Optimal thresholds for fracton codes and random spin models with subsystem symmetry, Phys. Rev. Lett. 129, 230502 (2022).
  19. G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s cat: Stable long-range entanglement from finite-depth unitaries and weak measurements, Phys. Rev. Lett. 131, 200201 (2023).
  20. Y. Li, N. O’Dea, and V. Khemani, Perturbative stability and error-correction thresholds of quantum codes, PRX Quantum 6, 010327 (2025).
  21. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  22. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  23. L. H. English, D. J. Williamson, and S. D. Bartlett, Thresholds for postselected quantum error correction from statistical mechanics, Phys. Rev. Lett. 135, 120603 (2025).
  24. H. Chen, D. Xu, G. M. Sommers, D. A. Huse, J. D. Thompson, and S. Gopalakrishnan, Scalable accuracy gains from postselection in quantum error correcting codes, arXiv:2510.05222.
  25. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  26. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2012).
  27. D. Gottesman, Stabilizer codes and quantum error correction, arXiv:quant-ph/9705052.
  28. W. L. McMillan, Domain-wall renormalization-group study of the two-dimensional random Ising model, Phys. Rev. B 29, 4026 (1984).
  29. D. S. Fisher and D. A. Huse, Equilibrium behavior of the spin-glass ordered phase, Phys. Rev. B 38, 386 (1988).

  30. The constant of proportionality reflects that codes which admit materialized symmetries, that is, non-empty subsets of stabilizer generators whose product is the identity induces gauge degrees of freedom in the spins which overcounts physical errors with a multiplicity equivalent in each logical coset [122]. That is, by flipping all spins of a given symmetry, we map back to the same physical error. These multiplicities cancel when taking fractions of partition functions.

  31. H. Nishimori, Internal energy, specific heat and correlation function of the bond-random Ising model, Prog. Theor. Phys. 66, 1169 (1981).
  32. P. Iyer and D. Poulin, Hardness of decoding quantum stabilizer codes, IEEE Trans. Inf. Theory 61, 5209 (2015).
  33. R. Fisch and A. K. Hartmann, Ground-state and domain-wall energies in the spin-glass region of the two-dimensional ±J random-bond Ising model, Phys. Rev. B 75, 174415 (2007).
  34. C. Gidney, M. Newman, P. Brooks, and C. Jones, Yoked surface codes, Nat. Commun. 16, 4498 (2025).
  35. H. Bombín, M. Pant, S. Roberts, and K. I. Seetharam, Fault-tolerant postselection for low-overhead magic state preparation, PRX Quantum 5, 010302 (2024).
  36. S. C. Smith, B. J. Brown, and S. D. Bartlett, Mitigating errors in logical qubits, Commun. Phys. 7, 386 (2024).
  37. O. Melchert and A. K. Hartmann, Scaling behavior of domain walls at the T=0 ferromagnet to spin-glass transition, Phys. Rev. B 79, 184402 (2009).
  38. R. Fisch, Finite-size scaling of the domain wall entropy distributions for the 2D ±J Ising spin glass, J. Stat. Phys. 125, 793 (2006).
  39. H. Khoshbakht and M. Weigel, Domain-wall excitations in the two-dimensional Ising spin glass, Phys. Rev. B 97, 064410 (2018).
  40. C. Amoruso and A. K. Hartmann, Domain-wall energies and magnetization of the two-dimensional random-bond Ising model, Phys. Rev. B 70, 134425 (2004).
  41. C. Amoruso, E. Marinari, O. C. Martin, and A. Pagnani, Scalings of domain wall energies in two dimensional Ising spin glasses, Phys. Rev. Lett. 91, 087201 (2003).
  42. N. Kawashima and H. Rieger, Finite-size scaling analysis of exact ground states for ±J spin glass models in two dimensions, Europhys. Lett. 39, 85 (1997).
  43. K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  44. R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, Tricolored lattice gauge theory with randomness: Fault tolerance in topological color codes, New J. Phys. 13, 083006 (2011).
  45. M. Rispler, D. Vodola, M. Müller, and S. Kim, The random coupled-plaquette gauge model and the surface code under circuit-level noise, npj Quantum Inf. 12, 127 (2026).
  46. A. Pandey, A. Mahadevan, A. A. Middleton, and D. S. Fisher, Low-temperature transition of 2D random-bond Ising model and quantum infinite randomness, arXiv:2603.02308.
  47. Z.-Q. Wan, X.-D. Dai, and G.-Y. Zhu, Revisiting Nishimori multicriticality through the lens of information measures, Phys. Rev. Res. 8, 023059 (2026).
  48. B. Kaufman, Crystal statistics. II. Partition function evaluated by spinor analysis, Phys. Rev. 76, 1232 (1949).
  49. M.-C. Wu and C.-K. Hu, Exact partition functions of the Ising model on M×N planar lattices with periodic–aperiodic boundary conditions, J. Phys. A 35, 5189 (2002).
  50. B. Kastening, Simplified transfer matrix approach in the two-dimensional Ising model with various boundary conditions, Phys. Rev. E 66, 057103 (2002).
  51. N. S. Izmailian, Finite-size effects for anisotropic 2D Ising model with various boundary conditions, J. Phys. A 45, 494009 (2012).
  52. P. Di Francesco, P. Mathieu, and D. Sénéchal, The two-dimensional Ising model, in Conformal Field Theory (Springer New York, New York, NY, 1997), pp. 439–486.
  53. S. Bravyi and A. Vargo, Simulation of rare events in quantum error correction, Phys. Rev. A 88, 062308 (2013).
  54. V. Privman, Fluctuating interfaces, surface tension, and capillary waves: An introduction, Int. J. Mod. Phys. C 03, 857 (1992).
  55. F. Schmitz, P. Virnau, and K. Binder, Logarithmic finite-size effects on interfacial free energies: Phenomenological theory and Monte Carlo studies, Phys. Rev. E 90, 012128 (2014).
  56. M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, New York, 1999).
  57. B. M. McCoy and T. T. Wu, Asymptotic expansion of ⟨σ0,0σM,N⟩, in The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA, 1973), pp. 284–312.
  58. E. V. Ivashkevich, N. S. Izmailian, and C.-K. Hu, Kronecker’s double series and exact asymptotic expansions for free models of statistical mechanics on torus, J. Phys. A 35, 5543 (2002).
  59. H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60, 252 (1941).
  60. R. Baxter, Exactly solved models in statistical mechanics, in Dover Books on Physics (Dover Publications, New York, 2007).
  61. A. I. Bugrij and V. N. Shadura, Duality of a two-dimensional nonhomogeneous Ising model on a finite lattice, Phys. Rev. B 55, 11045 (1997).
  62. C. Gidney, N. Shutty, and C. Jones, Magic state cultivation: Growing T states as cheap as CNOT gates, arXiv:2409.17595.
  63. S.-H. Lee, F. Thomsen, N. Fazio, B. J. Brown, and S. D. Bartlett, Low-overhead magic state distillation with color codes, PRX Quantum 6, 030317 (2025).
  64. O. Higgott and C. Gidney, PyMatching v2, https://github.com/oscarhiggott/PyMatching (2022).
  65. T. M. Stace and S. D. Barrett, Error correction and degeneracy in surface codes suffering loss, Phys. Rev. A 81, 022317 (2010).
  66. H. Cao, S. Zhao, D. Feng, Z. Shen, H. Yan, T. Su, W. Sun, H. Xu, F. Pan, H. Yu, and P. Zhang, Exact decoding of quantum error-correcting codes, Phys. Rev. Lett. 134, 190603 (2025).
  67. C. K. Thomas and A. A. Middleton, Numerically exact correlations and sampling in the two-dimensional Ising spin glass, Phys. Rev. E 87, 043303 (2013).
  68. D. Cimasoni, A generalized Kac–Ward formula, J. Stat. Mech. Theory Exp. 2010, P07023 (2010).
  69. A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C 7, 1671 (1974).
  70. F. Parisen Toldin, A. Pelissetto, and E. Vicari, Strong-disorder paramagnetic-ferromagnetic fixed point in the square-lattice ±J Ising model, J. Stat. Phys. 135, 1039 (2009).
  71. M. Hasenbusch, F. P. Toldin, A. Pelissetto, and E. Vicari, Universal dependence on disorder of two-dimensional randomly diluted and random-bond ±J Ising models, Phys. Rev. E 78, 011110 (2008).
  72. P. Le Doussal and A. B. Harris, ϵ expansion for the Nishimori multicritical point of spin glasses, Phys. Rev. B 40, 9249 (1989).
  73. P. Le Doussal and A. B. Harris, Location of the Ising spin-glass multicritical point on Nishimori’s line, Phys. Rev. Lett. 61, 625 (1988).
  74. I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Random-bond Ising model in two dimensions: The Nishimori line and supersymmetry, Phys. Rev. B 63, 104422 (2001).
  75. S. Cho and M. P. A. Fisher, Criticality in the two-dimensional random-bond Ising model, Phys. Rev. B 55, 1025 (1997).
  76. F. Merz and J. T. Chalker, Two-dimensional random-bond Ising model, free fermions, and the network model, Phys. Rev. B 65, 054425 (2002).
  77. G. Delfino, Critical exponents at the Nishimori point, J. Stat. Mech. Theory Exp. 2025, 043203 (2025).
  78. B. Placke and N. P. Breuckmann, Random-bond Ising model and its dual in hyperbolic spaces, Phys. Rev. E 107, 024125 (2023).
  79. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  80. M. E. Beverland, B. J. Brown, M. J. Kastoryano, and Q. Marolleau, The role of entropy in topological quantum error correction, J. Stat. Mech. Theory Exp. 2019, 073404 (2019).
  81. D. A. Huse and C. L. Henley, Pinning and roughening of domain walls in Ising systems due to random impurities, Phys. Rev. Lett. 54, 2708 (1985).

  82. More generally, in the generalized Ising model [95] of decoding, duality and GKS inequalities [123, 124] imply that the conditional MLD success probability is maximized in the clean (fully postselected) case. Equivalently, in the convention used here, the corresponding domain wall free energy cost is at most as large as its clean model value. See Eq. (18) and Appendix A of Ref. [125].

  83. M. Kardar and Y.-C. Zhang, Scaling of directed polymers in random media, Phys. Rev. Lett. 58, 2087 (1987).
  84. Y. Li, S. Vijay, and M. P. A. Fisher, Entanglement domain walls in monitored quantum circuits and the directed polymer in a random environment, PRX Quantum 4, 010331 (2023).
  85. C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159, 151 (1994).
  86. C. A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177, 727 (1996).
  87. P. L. Ferrari and R. Frings, Finite time corrections in KPZ growth models, J. Stat. Phys. 144, 1123 (2011).
  88. L. Dumaz and B. Virág, The right tail exponent of the Tracy–Widom β distribution, Ann. Inst. H. Poincaré Probab. Statist. 49, 915 (2013).
  89. M. Kardar, Roughening by impurities at finite temperatures, Phys. Rev. Lett. 55, 2923 (1985).

  90. Under the statistical mechanical mapping, the domain wall ground state energy cost is directly proportional to the logical gap. Concretely, at zero temperature each violated bond contributes an energy cost of 2J, so the ground state energy cost of inserting a nontrivial domain wall is 2×(logical gap). In this sense the logical gap is the combinatorial counterpart of the domain wall energy.

  91. H. Bombín, C. Dawson, N. Nickerson, M. Pant, and J. Sullivan, Increasing error tolerance in quantum computers with dynamic bias arrangement, arXiv:2303.16122.
  92. T. Hillmann, G. Dauphinais, I. Tzitrin, and M. Vasmer, Single-shot and measurement-based quantum error correction via fault complexes, Phys. Rev. A 112, L040401 (2025).
  93. F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259 (1971).
  94. C. Song, Kramers–Wannier duality and random-bond Ising model, Entropy 26, 636 (2024).
  95. Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Phys. Rev. A 90, 062320 (2014).
  96. R. Acharya et al., Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  97. G. P. Gehér, C. McLauchlan, E. T. Campbell, A. E. Moylett, and O. Crawford, Error-corrected Hadamard gate simulated at the circuit level, Quantum 8, 1394 (2024).
  98. G. Kishony and A. Fowler, Surface code off-the-hook: Diagonal syndrome-extraction scheduling, arXiv:2602.09099.
  99. Y. Hirai, S. Ikari, Y. Ueno, and Y. Suzuki, No more hooks in the surface code: Distance-preserving syndrome extraction for arbitrary layouts at minimum depth, arXiv:2603.01628.
  100. C. Gidney, Stim: A fast stabilizer circuit simulator, Quantum 5, 497 (2021).
  101. L. Wichette, H. Hohenfeld, E. Mounzer, and L. Grans-Samuelsson, A partition function framework for estimating logical error curves in stabilizer codes, Quantum 10, 2175 (2026).
  102. D. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, Surface code quantum computing by lattice surgery, New J. Phys. 14, 123011 (2012).
  103. A. Márton, L. Colmenarez, L. Bödeker, and M. Müller, Lattice surgery-based logical state teleportation via noisy links, Phys. Rev. Res. 7, 033238 (2025).
  104. A. G. Fowler, Accurate simulations of planar topological codes cannot use cyclic boundaries, Phys. Rev. A 87, 062320 (2013).
  105. H. Bombín, C. Dawson, R. V. Mishmash, N. Nickerson, F. Pastawski, and S. Roberts, Logical blocks for fault-tolerant topological quantum computation, PRX Quantum 4, 020303 (2023).
  106. H. Bombín and M. A. Martin-Delgado, Optimal resources for topological two-dimensional stabilizer codes: Comparative study, Phys. Rev. A 76, 012305 (2007).
  107. M. Newman, L. A. de Castro, and K. R. Brown, Generating fault-tolerant cluster states from crystal structures, Quantum 4, 295 (2020).
  108. H. Bombín, C. Dawson, T. Farrelly, Y. Liu, N. Nickerson, M. Pant, F. Pastawski, and S. Roberts, Fault-tolerant complexes, arXiv:2308.07844.
  109. H. Bombín, Topological order with a twist: Ising anyons from an Abelian model, Phys. Rev. Lett. 105, 030403 (2010).
  110. S. Jin, A. Sen, and A. W. Sandvik, Ashkin-Teller criticality and pseudo-first-order behavior in a frustrated Ising model on the square lattice, Phys. Rev. Lett. 108, 045702 (2012).
  111. R. Agrawal, L. F. Cugliandolo, L. Faoro, L. B. Ioffe, and M. Picco, Geometric phase transition of the three-dimensional z2 lattice gauge model, Phys. Rev. Lett. 135, 120601 (2025).
  112. S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature (London) 627, 778 (2024).
  113. K. Chen, Y. Liu, Y. Zhang, Z. Liang, Y.-A. Chen, K. Liu, and H. Song, Anyon theory and topological frustration of high-efficiency quantum low-density parity-check codes, Phys. Rev. Lett. 135, 076603 (2025).
  114. Z. Liang, K. Liu, H. Song, and Y.-A. Chen, Generalized toric codes on twisted tori for quantum error correction, PRX Quantum 6, 020357 (2025).
  115. D. A. Huse and D. S. Fisher, Pure states in spin glasses, J. Phys. A 20, L997 (1987).
  116. D. S. Fisher and D. A. Huse, Absence of many states in realistic spin glasses, J. Phys. A 20, L1005 (1987).
  117. C. Villani, The Wasserstein distances, in Optimal Transport: Old and New (Springer Berlin Heidelberg, Berlin, Heidelberg, 2009), pp. 93–111.
  118. J. Haah, Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices, J. Math. Phys. 62, 012201 (2021).
  119. H. Bombín, G. Duclos-Cianci, and D. Poulin, Universal topological phase of two-dimensional stabilizer codes, New J. Phys. 14, 073048 (2012).
  120. H. Bombín and M. A. Martin-Delgado, Statistical mechanical models and topological color codes, Phys. Rev. A 77, 042322 (2008).
  121. A. J. Landahl, J. T. Anderson, and P. R. Rice, Fault-tolerant quantum computing with color codes, arXiv:1108.5738.
  122. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  123. R. B. Griffiths, Correlations in Ising ferromagnets. I, J. Math. Phys. 8, 478 (1967).
  124. D. G. Kelly and S. Sherman, General Griffiths’ inequalities on correlations in Ising ferromagnets, J. Math. Phys. 9, 466 (1968).
  125. A. A. Kovalev, S. Prabhakar, I. Dumer, and L. P. Pryadko, Numerical and analytical bounds on threshold error rates for hypergraph-product codes, Phys. Rev. A 97, 062320 (2018).

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PRX Quantum 7, 043004– Published 6 October, 2026
  • Received 13 January 2026
  • Revised 15 May 2026
  • Accepted 15 September 2026
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