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

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

arXiv:2505.15758v3quant-ph

TL;DR

Optimal logical error curves are not measured by the order probability, despite its use in locating optimal-decoder thresholds. The paper introduces decoding probability alongside order probability to characterize different partition-function decoders and evaluates the framework across code and noise examples. These ratios can estimate logical performance with fewer samples than counting decoder failures, particularly at low noise.

  • Problem

    Order probability can locate an optimal-decoder threshold but does not measure the optimal decoder's success rate or full logical error curves.

    Source: §1 · p. 2

  • Method

    The paper defines decoding probability and order probability for maximum and probabilistic partition function decoders, respectively, and estimates them from partition functions.

    Source: §1 · p. 2

  • Results

    Partition-function ratios estimate decoder success with fewer samples than counting failures, especially at low noise, and the paper reports this framework across toric-code and color-code examples.

    Source: §1 · p. 3, §8 · p. 30

  • Takeaways & Limitations

    The ratios provide logical-performance estimates that can help filter codes and assess how far a fast decoder is from optimality.

    Source: §1 · p. 3

    The framework's extension depends on computing or approximating partition functions, and the paper notes that tensor-network methods are more limited in 3D than in 2D.

    Source: §8 · p. 30

Abstract

from arXiv · show

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.

1 Introduction

The paper develops partition-function ratios to estimate logical performance and distinguish optimal decoding success from the success measured by order probability. Across toric-code and color-code examples, these estimates can be more sample-efficient than counting decoder failures, especially at low noise.

1 Introduction

  • The order probability identifies the optimal-decoder threshold but does not measure optimal-decoder success; decoding probability measures optimal logical error curves on the Nishimori line.Order probability instead measures probabilistic partition function decoding, which corresponds to maximum probability decoding at zero temperature.Source: §1 · p. 2
  • Maximum partition function decoding and probabilistic partition function decoding have success rates measured by decoding probability and order probability, respectively.At zero temperature, they correspond to degeneracy-enhanced maximum probability decoding and maximum probability decoding, respectively.Source: §1 · p. 2
  • The framework distinguishes code performance from decoder implementation and can inform code selection and assessment of how far a decoder is from optimal.The paper also studies zero-temperature matching-based decoders, ensembling, boundary conditions, code-distance parity, bias, and additional noise settings.Source: §1 · p. 3
  • The distinction between the ratios leaves open whether a maximum partition function decodability boundary can differ from the statistical-mechanical phase boundary.The paper gives conditions for a vertical decodability boundary despite a reentrant phase boundary.Source: §1 · p. 3
  • For non-uniform toric-code bitflip noise, maximum probability and degeneracy-enhanced maximum probability decoding coincide, while the improvement of maximum likelihood over maximum probability decreases with greater noise-rate variation.Ensembling can still help if perturbations sample less probable errors as well as the most probable ones.Source: §1 · p. 3
  • Partition-function ratios estimate logical performance with fewer samples than counting decoder failures, with the advantage increasing in the low-noise regime.The paper reports this pattern for toric-code bitflip noise and color-code bitflip and depolarizing noise, though the latter depolarizing results are limited by accessible noise rates and distances.Source: §1 · pp. 3–4

2 Background

The background sets up stabilizer-code decoding in terms of error classes and maps their probabilities to partition functions in disordered statistical mechanics. For toric-code bitflip noise, the resulting model is the Random Bond Ising Model, whose partition functions can be computed efficiently by Pfaffian methods.

2.1 Preliminaries

  • A decoder uses the measured syndrome to choose a correction, succeeding when correction and error belong to the same logical equivalence class.Errors with a given syndrome are partitioned into classes under stabilizer multiplication.Source: §2.1 · pp. 4–5
  • Maximum likelihood decoding selects the most probable logical error class, whereas maximum probability decoding selects an individual most probable error.Degeneracy-enhanced maximum probability decoding selects the class containing the largest number of maximum-probability errors; it reduces to maximum probability decoding when those classes have equal counts.Source: §2.1 · pp. 5–6

2.2 Error Class Probabilities in the Statistical Mechanical Mapping

  • The statistical-mechanics mapping assigns spins to stabilizer generators so that an error's partition function gives its error-class probability.The mapping connects optimal decoding thresholds to order-disorder transitions and extends to independent Pauli noise in stabilizer codes.Source: §2.2 · p. 6
  • For toric-code bitflip noise, the mapped Hamiltonian is that of the Random Bond Ising Model, and its quenched-disorder partition function can be computed efficiently by Pfaffian methods.The spins correspond to Z-type stabilizers and qubits occupy lattice edges.Source: §2.2 · p. 7
  • At the Nishimori temperature, choosing the correction with the largest partition function is maximum likelihood decoding for uniform bitflip noise.The background fixes the couplings and Nishimori temperature according to the uniform error rate.Source: §2.2 · p. 8

3 Success rates for partition function decoders

The section relates partition-function ratios to the success rates of two decoding strategies, distinguishing maximum partition function decoding from probabilistic partition function decoding. Their behavior connects ML, MP, and degeneracy-aware MP decoding to different temperatures, while their performance can differ at finite distance and away from the Nishimori temperature.

3.1 The maximum partition function decoder and the probabilistic partition function decoder

  • At the Nishimori temperature, maximum partition function decoding is ML; at zero temperature, it is dMP, while probabilistic partition function decoding is MP.The zero-temperature partition function retains degeneracy information that an energy-only analysis loses.Source: §3.1 · p. 8

3.2 The decoding probability and the order probability

  • At finite temperatures the two probabilities generally differ, and at zero temperature they agree only when the maximum-probability classes have equal numbers of maximum-probability errors.Both approach 1/N at infinite temperature, where N is the number of equivalence classes per syndrome.Source: §3.2 · p. 9
  • The decoding probability measures the success rate of a maximum partition function decoder, while the order probability measures that of a probabilistic partition function decoder.The ratios are expected to give narrower confidence intervals than counting decoder successes.Source: §3.2 · pp. 9–11

3.3 Performance comparisons

  • A maximum partition function decoder can remain optimal away from the Nishimori temperature if a most-likely class has greater partition function than every less-likely class.Under this condition the decodability boundary is vertical; it is distinct from the order-disorder phase boundary.Source: §3.3 · pp. 11–12
  • At finite distance, maximum and probabilistic partition function decoders can have different success rates even in the ordered phase, although they increasingly agree with distance there.For reentrant phase boundaries, MP and dMP thresholds can also differ.Source: §3.3 · p. 12

4 Partition function based estimates of decoder performance in the toric code under bitflip noise

For the toric code under bitflip noise, the authors compare partition-function-ratio estimates with direct decoder-failure counts and examine decoder performance at the Nishimori temperature and near zero temperature. Ratio estimates can require far fewer samples, while comparisons reveal reentrant boundaries and differences among ML, MP, and dMP performance.

4 Partition function based estimates of decoder performance in the toric code under bitflip noise

  • Counting and ratio estimates agree within error bars at both the Nishimori temperature and 0.1TNish.The authors use 0.1TNish as a stand-in for zero temperature in subsequent partition-function computations.Source: §4 · p. 13
  • The toric code under bitflip noise has a reentrant decodability boundary as well as a reentrant phase boundary.The criterion for optimality of maximum partition function decoding away from the Nishimori temperature is not fulfilled in this setting.Source: §4 · p. 13
  • At 0.1TNish, dMP and MP failure rates are nearly identical at odd distances, while dMP performs better at small even distances.At the Nishimori temperature, maximum partition function decoding generally outperforms probabilistic partition function decoding without a visibly changed threshold.Source: §4 · p. 13
  • dMP closely matches ML at low distances or low error rates, but ML increasingly outperforms dMP as distance and error rate increase.The study computes RBIM partition functions mainly with the FKT algorithm and also adapts Wang-Landau simulation for zero-temperature estimates.Source: §4 · pp. 13–14
  • Less than 3% of the nearly two-million-sample count was needed for ratio estimates to match its precision at p=0.01, d=5.The sample-size reduction depends on noise rate and distance, and the ratio method is increasingly favorable at low noise.Source: §4 · p. 14

5 Bias, degeneracy and ensembling in matching based decoding

Matching-based decoders can be biased by which equivalent matchings they return, and weak ensembling only approximates degeneracy-aware decoding. The performance effects vary with code distance parity, boundaries, and noise model.

5.1 Matching based decoding and ensembling

  • Ensembling perturbs matching edge weights and selects the most frequently returned equivalence class; weak ensembling samples minimum-weight matchings, while strong ensembling also samples higher-weight matchings.Weak ensembling need not sample minimum-weight matchings fairly: the illustrated blue path is returned 50% of the time, despite a fair rate of 20%.Source: §5 · p. 14, §5.1 · p. 18, Figure 8 · p. 19

5.2 Comparison of MP and dMP to matching based decoding in the toric code

  • At even toric-code distances beyond d = 4, MWPM has noticeably higher logical failure rates than MP, while weakly ensembled MWPM closely tracks dMP at both parities.For odd distances, MWPM and MP differ only slightly; ensembling may be robust because it need only identify the most-degenerate error classes, not their exact degeneracies.Source: §5.2 · p. 19

5.3 Boundary and parity dependence of performance effects from bias and weak ensembling

  • For uniform bitflip noise, toric-code degeneracy can affect the leading logical-error term at even distances, but only the first subleading term at odd distances.At even distance, the lowest-weight even logical representative has weight d; at odd distance, its weight is 2d, suppressing degeneracy effects as d grows.Source: §5.3 · p. 21
  • For the toric code, degeneracy effects are expected to be suppressed at odd distances; ensembling shows no improvement for odd rotated surface codes, but modest gains for other boundary-parity combinations.The surface-code gains may also reflect ensembling's robustness to decoder bias, rather than degeneracy enhancement alone.Source: §5.3 · p. 21

5.4 Degeneracy and bias in other noise models

  • Under split bitflip hook errors, ensembling significantly improves performance at even distances, but d = 4 reverses the trend; with bitflip and readout noise, its difference from MWPM is modest.At d = 4, the matching decoder's bias apparently aligns with correlated-error patterns; the split decoding graph does not model those correlations.Source: §5.4 · p. 22

6 Strong ensembling, and performance estimates in the toric code under non-uniform bitflip noise

Under truncated-Gaussian bitflip noise, non-uniformity generally improves logical performance but lifts degeneracies, limiting weak ensembling; strong ensembling helps mainly at even distances and near threshold.

6 Strong ensembling, and performance estimates in the toric code under non-uniform bitflip noise

  • The truncated-Gaussian model independently draws each qubit's bitflip rate from a normal distribution with mean p and standard deviation σp, truncated to [10^-4, 1/2].This non-uniformity lifts ground-state degeneracies, which is expected to reduce dMP to MP for small nonzero σp.Source: §6 · p. 25
  • Simulations compare MP, dMP, and ML over σp ∈ {0.005, 0.06, 0.12}, using 10^4 error samples and a low-temperature approximation for MP/dMP.The approximation uses T = 0.01 T_Nish and 9999-bit precision to suppress non-ground-state contributions.Source: §6 · p. 25

6.1 The effect of non-uniformity on the difference between MP and ML performance

  • Across tested standard deviations, ML outperforms MP and dMP at higher physical error rates, leaving a performance gap that weak ensembling does not close.For odd distances, dMP performs only as well as MP; for even distances, this holds already with slight non-uniformity.Source: §6.1 · p. 25, Figure 14 · p. 26
  • Increasing noise standard deviation generally lowers logical failure rates and raises thresholds for each decoding strategy, while shrinking the ML–dMP/MP gap slightly.At sufficiently low physical error rates and high standard deviation, the strategies become indistinguishable.Source: §6.1 · p. 25, Figure 14 · p. 26

6.2 Strong ensembling for non-uniform noise

  • For even distances, strong ensembling improves logical error rates by up to about 0.03, most around threshold and at small σp; the gain vanishes at σp = 0.15.For odd distances, gains are absent or very small.Source: Figure 15 · p. 27, §6.2 · p. 28

7 The color code under bitflip noise and depolarizing noise

The color-code extension uses tensor-network contractions on the Nishimori line to estimate logical performance under bitflip and depolarizing noise. Ratio estimates agree with decoder-failure counting and can require substantially fewer samples, especially at low noise.

7 The color code under bitflip noise and depolarizing noise

  • The color-code study applies the framework to bitflip and depolarizing noise using tensor networks because FKT is unavailable for these settings.The evaluated distances are limited by computational requirements.Source: §7 · p. 28
  • Tensor-network contraction sums error configurations within each logical class, but its code-structure-based construction restricts this study to the Nishimori temperature.Consequently, it cannot investigate MP or dMP decoding at other temperatures.Source: §7 · p. 28
  • Truncation can produce unphysical negative logical-class probabilities unless the maximum bond dimension is adapted to p and d.The required χmax values ranged from 32 to 128, with larger values needed for depolarizing noise at low p and large d.Source: §7 · p. 28
  • Under both noise models, the ratio method's efficiency advantage increases in the low-p regime, though depolarizing-noise gains are less dramatic.Higher required bond dimensions under depolarizing noise limit the accessible p range.Source: §7 · p. 28, Figure 18 · p. 30
  • Ratio-method and maximum-partition-function counting estimates give compatible color-code logical performance and threshold estimates.The thresholds are consistent with about 10.9% for bitflip noise [49] and 18.9% for depolarizing noise [12].Source: Figure 16 · p. 29
  • For d = 7, the ratio method needs 6.3% of the sample set under bitflip noise and 26% under depolarizing noise to match counting-method precision.The examples use p = 0.025 and p = 0.02, respectively.Source: Figure 17 · p. 29

8 Discussion

The discussion presents a partition-function framework for comparing logical performance across codes and decoding approaches, while emphasizing that its reach depends on computing or approximating partition functions. The numerical scope and noise models leave important extensions and generality questions open.

8 Discussion

  • The framework extends beyond toric-code bitflip noise when partition functions can be computed or approximated, as illustrated by color-code bitflip and depolarizing noise.The color-code distances considered are more limited; qLDPC and circuit-level settings are proposed as broader applications using other computational methods.Source: §8 · p. 30
  • Generalization depends on feasible partition-function or logical-class-probability computation; tensor networks are promising for 2D but so far more limited in 3D.The discussion leaves open whether Wang–Landau sampling could be preferable for qLDPC codes lacking tensor-network-friendly structure.Source: §8 · p. 30
  • FKT is less likely than tensor networks or Wang–Landau to apply to realistic noise, while the worm algorithm covers matchable structures including surface-code bitflip and readout noise.The cited work combines it with correlated decoding to improve the surface-code depolarizing threshold over correlated matching [50].Source: §8 · p. 30
  • Ratio-based estimation is especially useful at low p and large distance, where conventional failure sampling struggles and low logical error rates matter for large-scale computing.The framework also enables apples-to-apples code comparisons without confounding from code-specific decoder implementations.Source: §8 · p. 31
  • The bitflip model is far from realistic, and normally distributed non-uniform noise is unlikely, so qualitative findings need testing under more realistic noise.Correlations could change both performance and the observed ensembling gains.Source: §8 · p. 31
  • Degeneracy effects may vary with boundaries and distance parity, and their broader impact remains to be quantified.Ensembling could yield larger gains in codes or noise models with stronger degeneracy effects.Source: §8 · p. 31
  • For the studied temperatures, maximum-partition-function and probabilistic-partition-function threshold estimates agree, but whether their decodability and phase boundaries differ elsewhere remains open.The simulations focus on the Nishimori temperature and the zero-temperature limit.Source: §8 · p. 31

9 Author contributions

The author-contribution statement assigns simulation, analysis, theory, implementation, and supervision roles across the team.

9 Author contributions

  • LW and HH performed and analyzed simulations; LGS developed the theory, while EM and LGS guided and supervised numerical work.EM developed the initial CUDA code, which LW later took over; all authors contributed to discussion and writing.Source: §9 · p. 32

A Zero Temperature limit

The authors find that FKT estimates at T = 0.1TNish with 4096-bit precision suffice to estimate low-temperature performance.

A Zero Temperature limit

  • Matching performance estimates within error bars support T = 0.1TNish and 4096-bit precision as sufficient FKT parameters for low-temperature performance.The comparison also matched FKT estimates at T = 0.01TNish with 9192-bit precision and Wang-Landau estimates at T = 0.1TNish with zero-temperature estimates.Source: A Zero Temperature limit · p. 32

B FKT algorithm

The FKT method maps the toric-code bitflip-noise model to Pfaffians of a decorated dual-lattice graph. Replica-exchange Wang-Landau estimates the density of states, allowing partition functions to be calculated at arbitrary temperatures, including zero.

B FKT algorithm

  • The toric-code bitflip model maps to an Ising model whose partition function is computed from Pfaffians of Kasteleyn matrices on a decorated dual lattice.The matrix encodes graph connectivity, edge weights, and orientation.Source: B FKT algorithm · pp. 32, 34
  • Spin configurations correspond to relative domain walls plus one fixed face spin, while the decorated-graph dimer mapping is not one-to-one.Dimer coverings can encode multiple configurations, and the map is invariant under global spin flip.Source: B FKT algorithm · p. 32, Figure 21 · p. 34, Figure 22 · p. 35
  • On the torus, four boundary orientations cancel contributions from nonphysical domain-wall loops, giving 2Z as the sum of four Pfaffians.Only the even-even wrapping component corresponds to physical spin configurations.Source: B FKT algorithm · p. 34

C Replica-exchange Wang-Landau algorithm

  • Replica-exchange Wang-Landau estimates the density of states using inverse-degeneracy random walks that produce flat energy histograms.The density of states enables partition-function calculations at arbitrary temperatures, including zero, unlike the finite-temperature FKT method.Source: C Replica-exchange Wang-Landau algorithm · p. 34
  • Wang-Landau updates density estimates and histograms after spin flips, reducing its modification factor after each flat-histogram stage until a precision threshold is reached.The run parameter controls precision and the total number of spin flips.Source: C Replica-exchange Wang-Landau algorithm · p. 35

D Examples of confidence interval ratios

The ratio and counting methods’ confidence-interval widths differ by a sample-size-independent scale factor, with fluctuations that vanish as sample size increases.

D Examples of confidence interval ratios

  • The width ratio r(p, d) is independent of total sample size, while sample-dependent fluctuations vanish with increased sample size.The comparison covers toric-code bitflip noise and color-code bitflip and depolarizing noise.Source: D Examples of confidence interval ratios · p. 36, Figure 23 · p. 37, Figure 24 · p. 37
  • The sample-size invariance of r(p, d) underlies interpolation of ratio-method confidence intervals and estimates of required sample sizes for a target interval width.The plotted estimates average subset results using more than 0.1 of the full sample; error bars are the corresponding standard deviations.Source: D Examples of confidence interval ratios · p. 36

E Minimum-Weight Perfect Matching Decoder

The section describes minimum-weight perfect matching as a maximum-probability decoder that selects a correction consistent with the observed syndrome. It constructs a weighted matching graph from the error model and identifies success by the correction’s effect on logical observables.

E Minimum-Weight Perfect Matching Decoder

  • Minimum-weight perfect matching is a maximum-probability decoder that finds a syndrome-consistent correction operator.The discussion is limited to toric and unrotated and rotated planar surface codes, with Pauli-X and Pauli-Z decoding performed independently.Source: E Minimum-Weight Perfect Matching… · p. 38
  • The error model is specified by detector matrix H, logical-observable matrix O, and individual error probabilities p; more probable errors receive lower graph-edge weights.The matching graph uses stabilizer measurements as nodes and error mechanisms as edges.Source: E Minimum-Weight Perfect Matching… · p. 38
  • The syndrome graph connects defect nodes by shortest paths in the matching graph, and a minimum-weight perfect matching yields a consistent correction.A perfect matching covers every syndrome-graph node with no shared endpoints and minimizes the total edge weight.Source: E Minimum-Weight Perfect Matching… · p. 38
  • Decoding succeeds when the correction and the unknown error have the same effect on logical observables; otherwise a logical error occurs.The simulations use PyMatching 2, which implements sparse blossom.Source: E Minimum-Weight Perfect Matching… · p. 38

F Further results for MWPM Decoding and Ensembling under other noise models

The appendix gives further surface-code results for ensembling under split-hook and combined bitflip/readout noise. It also illustrates how syndrome defects and shortest paths are used to construct a minimum-weight matching correction.

F Further results for MWPM Decoding and Ensembling under other noise models

  • For the rotated surface code under split-hook error bitflip noise, Fig. 26 shows ensembling at d = 4; d = 3 is omitted because no combination of its two error locations leads to a logical error.Source: F Further results for MWPM Decoding… · p. 38
  • For combined bitflip and readout noise, Figure 27 uses d rounds of stabilizer measurements, with readout noise applied only to Z-type stabilizers.The figure compares even and odd code distances.Source: F Further results for MWPM Decoding… · p. 38
  • The syndrome graph connects defect nodes using shortest paths, and a minimum-weight perfect matching corresponds to a most probable correction consistent with the syndrome.The matching graph is extended by boundary nodes before constructing the syndrome graph.Source: Figure 25 · p. 39

G Optimization of Ensembling Parameters

The section optimizes edge-weight perturbations for ensembling and reports approximate settings for uniform and non-uniform noise. The reported ensembling effects vary across the tested noise models and decoding setups.

G Optimization of Ensembling Parameters

  • For uniform noise, the best results consistently occur at σξ ∈ [10^-6, 10^-2] across four tested error rates and distances d = 8 and d = 16.Larger perturbations monotonically increase the observed logical error rate, while considerably smaller values cause slight increases.Source: G Optimization of Ensembling Parameters · p. 39, Figure 28 · p. 41
  • The parameter search covers toric-code distances d = 8 and d = 16, four error rates, and 10,000 sampled configurations per optimization step.The coarse-grained global search tests perturbation standard deviations within [0, 0.5]; computational cost limits the search scope.Source: G Optimization of Ensembling Parameters · p. 39
  • At d = 4 under split bitflip hook errors, MWPM significantly outperforms ensembled MWPM and perturbed MPWM.The reported results use 50 random perturbations for ensembling and one perturbation for the “Perturbation” results.Source: Figure 26 · p. 40
  • Under combined bitflip and readout noise, ensembling modestly improves performance at both even and odd code distances.The observed threshold is compatible with the known value pth ∼ 2.9% [10].Source: Figure 27 · p. 40
  • Across tested error-rate standard deviations, the approximate optimum for non-uniform noise is σΞ = 0.025 at both d = 8 and d = 16.This value is used for the non-uniform-noise MWPM ensembling simulations.Source: G Optimization of Ensembling Parameters · p. 39, Figure 29 · p. 41, Figure 30 · p. 42
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