⚛️ quantum physics

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

This paper introduces a partition function framework to estimate logical error curves in stabilizer codes by defining a ratio of partition functions that measures the success probability of maximum partition function decoding, demonstrating that this approach offers greater sample efficiency than traditional failure counting, particularly in low-noise regimes and for codes like the toric and color codes.

Original authors: Leon Wichette, Hans Hohenfeld, Elie Mounzer, Linnea Grans-Samuelsson

Published 2026-07-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Leon Wichette, Hans Hohenfeld, Elie Mounzer, Linnea Grans-Samuelsson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret message across a stormy ocean using a fleet of tiny, fragile boats. In the world of quantum computing, these boats are "qubits," and the storm is "noise"—random glitches that flip bits of information, turning a "yes" into a "no" before the message arrives. To survive the storm, scientists use "quantum error correction," which is like tying many small boats together into a giant, sturdy raft. If one boat gets swamped, the others hold the raft steady. But here's the tricky part: the storm isn't always the same. Sometimes it's a gentle drizzle; other times, it's a hurricane. And sometimes, the boats themselves are a bit wobbly, with some leaking faster than others.

To figure out how well these rafts will hold up, scientists use a clever trick borrowed from a completely different field: the physics of magnets and heat. They imagine the quantum error problem as a game of arranging magnets on a grid. In this game, "disorder" (the noise) tries to scramble the magnets, while "order" (the error correction) tries to keep them aligned. By studying how these magnets behave at different "temperatures," researchers can predict how likely the quantum raft is to sink. This paper dives deep into that game, not just to see if the raft sinks, but to find the most efficient way to calculate exactly how close it is to sinking, especially when the storm is very quiet and the boats are very far apart.

The Paper's Story: A New Way to Count the Storm

This paper introduces a new, super-efficient toolkit for predicting how well quantum error-correcting codes perform. The authors, a team of physicists and computer scientists, propose a method that treats the problem of decoding quantum errors like a statistical mechanics puzzle involving "partition functions." Think of a partition function as a giant, magical scorecard that counts every possible way a storm could hit your raft and how likely each scenario is.

The paper focuses on two main ways to read this scorecard, which correspond to two different decoding strategies:

The "Maximum Likelihood" Decoder (The Optimist): This strategy looks at the scorecard at a specific "Nishimori temperature" (a special setting where the math works out perfectly) and picks the single most likely path to save the message. It's like asking, "What is the one best way to fix this?"
The "Maximum Probability" Decoder (The Pragmatist): This strategy looks at the scorecard at "zero temperature" (the coldest, most rigid setting) and picks the single most probable error to fix, ignoring how many other equally probable errors might exist. It's like asking, "What is the most common mistake I see?"

The authors discovered that these two strategies are actually measuring different things. They defined a new metric called "Decoding Probability" to measure the success of the first strategy (the Optimist) and an existing metric called "Order Probability" to measure the second (the Pragmatist).

The Big Surprise: Counting is Harder Than You Think

The most exciting finding in the paper is about efficiency. Usually, to know how often a decoder fails, you have to simulate the storm thousands of times, watch the raft sink, and count the failures. This is like trying to measure the speed of a car by driving it across the country and counting every pothole you hit. It takes a long time and a lot of fuel (computing power).

The authors show that using their new "ratio" method (looking at the partition function scorecards directly) is like having a GPS that tells you the speed instantly. In their simulations of the toric code (a popular type of quantum raft) under bitflip noise, they found that the ratio method needed less than 3% of the samples to get the same level of accuracy as the traditional counting method. In the low-noise regime (where the storm is calm and the raft is very stable), this advantage is huge. It means scientists can predict how well a quantum computer will work in the future without running millions of expensive simulations.

What They Found About "Degeneracy" and "Ensembling"

The paper also explores a concept called degeneracy. Imagine you have a broken boat, and there are five different ways to patch it up that are all equally good. A "Maximum Probability" decoder might just pick one of those five at random. A "Degeneracy Enhanced" decoder (dMP) would realize there are five options and pick the patch that belongs to the group with the most options, giving it a better chance of being right.

The authors found that:

In uniform noise (where every boat is equally wobbly), this "degeneracy enhancement" helps a little bit, but mostly for small rafts with even numbers of boats.
In non-uniform noise (where some boats are wobblier than others), the degeneracy disappears because the "best" patch is now unique. However, they found that even here, a technique called ensembling (running the decoder multiple times with slight random tweaks) still helps. It's like asking five different mechanics to fix the boat; even if they all find the same best patch, asking them multiple times ensures you don't miss a subtle detail.

What They Ruled Out and What's Still Unknown

The paper explicitly argues against the idea that the "Order Probability" (the Pragmatist's score) is the same as the "Decoding Probability" (the Optimist's score). They show that these two numbers are different, and confusing them leads to the wrong estimate of how good a decoder is.

They also investigated whether the "decodability boundary" (the point where the decoder stops working) is different from the "phase boundary" (the point where the magnets in the statistical model lose their order). In their simulations of the toric code, these boundaries appeared to be the same, suggesting that for this specific code, the phase transition perfectly predicts the decoder's failure. However, they leave it as an open question whether this is true for all quantum codes or if there are exotic cases where the decoder fails outside the phase boundary.

The Bottom Line

This paper doesn't claim to have built a perfect quantum computer. Instead, it provides a much sharper magnifying glass for looking at how quantum error correction works. By using these "partition function ratios," researchers can estimate the performance of quantum codes with far fewer samples than before. This is a crucial step for the future, because as we build larger and more complex quantum computers, we need to know exactly how well they will handle the noise without spending an eternity running simulations. The authors suggest that this method is particularly useful when the noise is low and the codes are large—the exact regime we need to reach for large-scale quantum computing to become a reality.

Technical Summary: A Partition Function Framework for Estimating Logical Error Curves in Stabilizer Codes

Problem Statement

Estimating the logical performance of quantum error correcting codes (QECCs) under realistic noise models is critical for scaling quantum computers. While the optimal threshold of stabilizer codes is known to correspond to order-disorder phase transitions in disordered statistical mechanics models (specifically the Random Bond Ising Model, RBIM), determining the full logical error curves remains computationally challenging.

Traditional methods for estimating logical error rates rely on counting the number of decoding failures in Monte Carlo simulations. This approach suffers from poor sample efficiency, particularly in the low-noise regime where logical failure rates are exponentially small, requiring prohibitively large numbers of samples to achieve narrow confidence intervals. Furthermore, existing statistical mechanics mappings have primarily focused on determining thresholds (phase boundaries) rather than the full success rates of specific decoding strategies, such as Maximum Likelihood (ML), Maximum Probability (MP), and degeneracy-enhanced MP (dMP) decoding. A distinction has also been lacking between the success rates of "optimal" decoders and those of decoders that sample corrections based on partition function probabilities.

Methodology

The authors propose a framework that maps stabilizer codes to disordered statistical mechanics models to define two key metrics: decoding probability and order probability. These metrics are defined as ratios of partition functions, allowing for the estimation of decoder success rates without directly simulating decoder failures.

Theoretical Framework

Statistical Mechanics Mapping: The paper utilizes the established mapping where error class probabilities correspond to partition functions ( $Z$ $Z$ ) of a statistical mechanics model at specific temperatures.
- Nishimori Temperature ( $T_{Nish}$ ): Corresponds to Maximum Likelihood (ML) decoding.
- Zero Temperature ( $T=0$ ): Corresponds to Maximum Probability (MP) decoding.
Decoder Definitions:
- Maximum Partition Function Decoder: Returns a correction from the class with the largest partition function at a given temperature. At $T_{Nish}$ , this is the ML decoder; at $T=0$ , it is the dMP decoder (which accounts for degeneracy among maximum probability errors).
- Probabilistic Partition Function Decoder: Returns a correction from a class with probability proportional to its partition function. At $T=0$ , this corresponds to the MP decoder.
Key Metrics:
- Decoding Probability ( $P_d$ ): Defined as the average ratio of the partition function of the chosen maximum class to the sum of all partition functions. The paper proves that $P_d$ measures the success rate of the maximum partition function decoder.
- Order Probability ( $P_o$ ): Defined as the average ratio of the partition function of the actual error class to the sum of all partition functions. The paper proves that $P_o$ measures the success rate of the probabilistic partition function decoder.

Numerical Implementation

The authors implement this framework using two primary methods for computing partition functions:

Fisher-Kasteleyn-Temperley (FKT) Algorithm: Used for the toric code under bitflip noise (mapping to the RBIM). This allows efficient computation of partition functions via Pfaffians. The authors approximate the zero-temperature limit using $T = 0.1 T_{Nish}$ with high-precision arithmetic (4096 bits).
Tensor Network Methods: Used for the color code under bitflip and depolarizing noise, where FKT is not applicable. The authors employ the SweepContractor.jl package.
Wang-Landau (WL) Simulation: Used as a complementary method to estimate density of states and partition functions at zero temperature, validating the FKT approximations.

The framework is applied to:

Toric Code: Under uniform and non-uniform (truncated Gaussian) bitflip noise.
Surface Codes: Unrotated and rotated planar surface codes.
Color Code: Under bitflip and depolarizing noise.

Key Contributions

1. Distinction Between Decoding Strategies and Probabilities

The paper rigorously distinguishes between decoding probability and order probability, showing they measure the success rates of two distinct families of decoders:

Decoding Probability quantifies the performance of the maximum partition function decoder (optimal at $T_{Nish}$ , dMP at $T=0$ ).
Order Probability quantifies the performance of the probabilistic partition function decoder (MP at $T=0$ ).
The authors demonstrate that while these probabilities converge in the infinite distance limit, they differ at finite distances, particularly in the zero-temperature limit where degeneracy plays a role.

2. Sample Efficiency via Ratio Estimation

A major contribution is the demonstration that estimating logical error curves via partition function ratios is significantly more sample-efficient than counting decoder failures.

Variance Reduction: Counting failures yields a Bernoulli variable (0 or 1), whereas partition function ratios yield values in the continuous interval $[0, 1]$ .
Quantitative Gain: In the low-noise regime (e.g., $p=0.01$ for the toric code), the ratio method requires less than 3% of the sample size needed by the counting method to achieve the same confidence interval width. This efficiency gain increases as the noise rate decreases.

3. Analysis of Degeneracy and Bias

The paper investigates the role of degeneracy in decoding performance:

Uniform Noise: In the toric code under uniform bitflip noise, dMP decoding (approximated by weak ensembling of Minimum Weight Perfect Matching, MWPM) outperforms MP decoding primarily at small, even code distances. This is linked to the existence of ground state degeneracies that depend on the parity of the code distance and boundary conditions.
Non-Uniform Noise: When qubit error rates are non-uniform (lifting ground state degeneracies), the performance difference between MP and dMP vanishes. However, ML decoding still outperforms both, suggesting room for improvement via methods that sample less probable errors (strong ensembling).
Bias in MWPM: The authors show that standard MWPM implementations (e.g., PyMatching) suffer from bias, particularly at even distances, leading to suboptimal performance compared to the unbiased MP decoder. Weak ensembling helps mitigate this bias, aligning MWPM performance closer to dMP.

4. Phase Boundaries vs. Decodability Boundaries

The paper explores the relationship between the phase boundary of the statistical mechanics model and the "decodability boundary" (the threshold of the maximum partition function decoder). While the phase boundary for the toric code under bitflip noise is known to be reentrant, the authors find that the decodability boundary is also reentrant in this specific case. They discuss conditions under which these boundaries might diverge, suggesting that maximum partition function decoders could theoretically succeed outside the standard phase boundary, though this was not observed in the toric code examples.

Results

Toric Code (Uniform Noise):
- The decoding probability (ML/dMP) and order probability (MP) curves were estimated and found to match within error bars when compared to direct counting methods.
- At $T=0$ , dMP decoding outperforms MP decoding for small even distances, while they are nearly identical for odd distances.
- The ratio method achieved high precision with significantly fewer samples than counting failures, especially at low error rates ( $p < 0.05$ ).
Toric Code (Non-Uniform Noise):
- Introducing non-uniformity (standard deviation $\sigma_p$ ) lifted ground state degeneracies, causing MP and dMP performance to converge.
- ML decoding continued to outperform MP/dMP, indicating that degeneracy enhancement alone is insufficient to reach optimality; sampling of less probable errors is required.
- Ensembling (strong perturbation) showed performance gains over regular MWPM, particularly around the threshold and for small standard deviations, but gains diminished as non-uniformity increased.
Surface and Color Codes:
- Similar trends were observed in unrotated and rotated surface codes regarding the impact of boundary conditions and distance parity on degeneracy effects.
- For the color code under depolarizing noise, tensor network methods successfully estimated logical error curves, confirming the efficiency gains of the ratio method, though the gains were less dramatic than in the bitflip case due to limitations in bond dimension.

Significance and Claims

The paper claims that the proposed partition function framework provides a flexible and sample-efficient tool for estimating logical performance across different decoding schemes and noise models.

Efficiency: The primary significance is the ability to estimate logical error curves in the low-noise regime (critical for large-scale quantum computing) with drastically reduced sampling overhead compared to traditional failure counting.
Decoder Characterization: The framework allows for the precise characterization of the performance gap between suboptimal decoders (like MP or biased MWPM) and optimal decoders (ML), as well as the specific contribution of degeneracy handling (dMP).
Theoretical Insight: By distinguishing between decoding probability and order probability, the authors provide a clearer understanding of how different decoding strategies relate to the underlying statistical mechanics phase transitions. They suggest that this distinction could be useful for characterizing phase diagrams of more complex statistical mechanics models associated with quantum codes.

The authors remain modest regarding the generalization of their findings, noting that the specific behaviors observed (e.g., the magnitude of degeneracy effects or the shape of decodability boundaries) may vary in more realistic noise models (e.g., circuit-level noise) or different code families. They position this work as a foundational step toward more efficient performance estimation and a deeper theoretical understanding of the interplay between decoding strategies and statistical mechanics.