Ensemble Dependence of Critical Exponents at Quantum Error Correction Thresholds: Analytic Mechanisms and Statistical Consequences

In the paper's words

Ensemble equivalence — the expectation that microcanonical, canonical, and grand-canonical descriptions of a system agree in the thermodynamic limit — is a cornerstone of statistical mechanics, yet it can fail for observables that probe exponentially rare events. Recent work on a simplified quantum error correction (QEC) model reports that critical exponents governing fidelity and magic at the error-correction threshold depend on the noise ensemble chosen, and that even the existence of the transition can be ensemble-dependent [1,2]. Here we develop a minimal, fully analytic treatment of this phenomenon using a random-unitary encoding model subject to depolarizing noise.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Noise entropy at $n = 100$: $H = 100 \times 0.0807931$ = 8.07931 bits .

   from 4. Analysis

2. Holevo-type capacity bound per qubit: $1 - 0.0807931 = 0.9192069 \,\mathrm{bits}$; over 100 qubits: 91.92069 bits .

   from 4. Analysis

3. Entropy budgets: $h_{2}(0.01) = 0.0807931 \,\mathrm{bits}/\mathrm{qubit}$; GC noise entropy $H = 8.07931 \,\mathrm{bits}$; canonical location entropy 6.643856 bits; difference 1.435454 bits attributable to number fluctuations.

   from 5. Results

4. Capacity bound: 91.92069 bits over 100 qubits at $p = 0.01$ (Holevo-type upper bound 0.9192069 bits/qubit).

   from 5. Results

5. We quantified the information-theoretic asymmetry between ensembles (8.07931 versus 6.643856 bits of noise entropy) and showed that the canonical ensemble in this model exhibits a jump rather than a critical transition, reproducing in closed form the ensemble dependence of the transition's existence reported in [1,2].

   from 7. Conclusion

6. We further derived the statistical cost of confirming ensemble-dependent exponents: a sample count scaling as $\Delta p^{-4}$ and system sizes scaling as $\Delta p^{-2}$, with a compound burden of ~4 × $10^{16}$ qubit-samples at $\Delta p = 10^{-3}$ under stated assumptions.

   from 7. Conclusion

How it is done

1. We derive explicit expressions for the decoding fidelity in grand-canonical (independent per-qubit errors) and canonical (fixed error number) ensembles, and exhibit concrete numerical instances of the discrepancy: for a distance-3 code with 100 physical qubits and error rate $p = 0.01$, the grand-canonical fidelity is 0.735762 while the canonical fidelity is exactly 1.

   from Abstract

2. Using a simplified model — single-step encoding and decoding by a random unitary — it reports (i) different exponents for generic channels versus quantum trajectories (unravelings of the same channel into stochastic pure-state evolutions conditioned on measurement records), (ii) saturation of a recently derived information-theoretic bound by Feldman et al.

   from 1. Introduction

3. The arXiv query record and abstract for 2609.21886 [1] report that in a simplified QEC model — a single encoding/decoding step by a random unitary — the critical exponents of fidelity and magic at threshold differ between generic channels and supposedly equivalent quantum trajectories, that the exponents saturate an information-theoretic bound due to Feldman et al.

   from 2. Background and Related Work

4. One logical qubit is encoded into n physical qubits by a Haar-random unitary U (the "random encoding" of [1,2]).

   from 3. Methods

What would falsify it

1. We relate the mechanism to the known ensemble dependence of the random transverse-field Ising chain [3], discuss saturation of the information-theoretic bound of Feldman et al. as reported in [1,2], and outline falsifiable predictions.

   from Abstract

2. Third, to derive the statistical consequences of the claim: what sample sizes and system sizes are required to confirm or falsify ensemble-dependent exponents, and what failure modes arise at finite size.

   from 1. Introduction

3. Standard self-averaging arguments would therefore predict ensemble equivalence.

   from 4. Analysis

Where the claim stops

1. Our analysis uses a distance-3 code at a single parameter point for Part I, and assumed scaling parameters for Part II; the full exponent extraction of [1,2] requires finite-size scaling across n, which we have not performed.

   from 6. Discussion

2. (2026) bound is cited in [1,2] but is not in our bibliography; we cannot verify its value, form, or existence beyond the source abstract.

   from 6. Discussion

3. Our Step 9 result sharpens (c): at $\Delta p = 10^{-2}$ with $n = 10^{3}$, the quadratic term is $10^{-4} \times A$, below the finite-size rounding $A/n = 10^{-3}$, so a simulation at that scale cannot see $\beta _{2} = 2$ and might report spurious effective exponents.

   from 6. Discussion

Where to read

1. Abstract278 words · 1 min
2. 1. Introduction377 words · 2 min
3. 2. Background and Related Work689 words · 3 min
4. 3. Methods342 words · 1 min
5. 4. Analysis1,213 words · 5 min
6. 5. Results311 words · 1 min
7. 6. Discussion554 words · 2 min
8. 7. Conclusion250 words · 1 min
9. Appendix A. Divergence report582 words · 3 min
10. Appendix B. Single-source claim registry146 words · 1 min