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arXiv:2609.21886

Ensemble Dependence of the Critical Exponent at a Quantum Error Correction Threshold

Idan Dror, Moshe Goldstein

quant-phSubmitted Sep 18, 2026• Announced Sep 21, 2026• 0 citations
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Abstract

In thermodynamics it is common to assume that the choice of ensemble (e.g., micro-canonical, canonical, or grand-canonical) should not affect the underlying physics in the thermodynamics limit. We show that this does not necessarily hold for critical exponents. Examining a simplified model of quantum error correction (single step encoding and decoding by a random unitary) and the behavior of both the fidelity and magic at the corresponding threshold, we find different exponents when using generic channels or supposedly equivalent quantum trajectories. Interestingly, the obtained exponents saturate a recently derived information theoretic bound by Feldman et al. (2026), which we extend from the grand-canonical to the canonical case, including intermediate ensembles which we define. Moreover, even the existence of the transition is shown to be ensemble-dependent.

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In thermodynamics it is common to assume that the choice of ensemble (e.g., micro-canonical, canonical, or grand-canonical) should not affect the underlying physics in the thermodynamics limit. We show that this does not necessarily hold for critical exponents. Examining a simplified model of quantum error correction (single step encoding and decoding by a random unitary) and the behavior of both the fidelity and magic at the corresponding threshold, we find different exponents when using generic channels or supposedly equivalent quantum trajectories. Interestingly, the obtained exponents saturate a recently derived information theoretic bound by Feldman et al. (2026), which we extend from the grand-canonical to the canonical case, including intermediate ensembles which we define. Moreover, even the existence of the transition is shown to be ensemble-dependent.
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. 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. We show that the discrepancy survives the vanishing of relative fluctuations (σ/μ → 0) because fidelity is a large-deviation tail quantity, and we compute the associated entropy and capacity budgets. We further derive the statistical cost of resolving ensemble-dependent exponents: distinguishing two candidate exponents at distance Δp from threshold requires a sample count scaling as Δp⁻⁴ and system sizes scaling as Δp⁻², a compound burden that we quantify under stated assumptions. 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. The results clarify when ensemble choice is physically consequential rather than a matter of convenience in QEC threshold physics. ## 1. Introduction Quantum error correction protects quantum information from decoherence by en Full text and updates: papers.qnfo.org/papers/ensemble-dependence-of-critical-exponents-at-quantum-error-correction-thresholds/
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
Universal scaling of entanglement estimators of critical quantum systems has drawn a lot of attention in the past. Recent studies indicate that similar universal properties can be found for bipartite information estimators of classical systems near phase transition, opening a new direction in the study of critical phenomena. We explore this subject by studying the information estimators of classical spin chains with general mean-field interactions. In our explicit analysis of two different bipartite information estimators in the canonical ensemble we find that, away from criticality, both the estimators remain finite in the thermodynamic limit. On the other hand, along the critical line there is a logarithmic divergence with increasing system size. The coefficient of the logarithm is fully determined by the mean-field interaction and it is the same for the class of models we consider. The scaling function, however, depends on the details of each model. In addition, we study the information estimators in the micro-canonical ensemble, where they are shown to exhibit a different universal behavior. We verify our results using numerical calculation for two specific cases of the general Hamiltonian.
It is common knowledge that the microcanonical, canonical, and grand-canonical ensembles are equivalent in thermodynamically large systems. Here, we study finite-size effects in the latter two ensembles. We show that contrary to naive expectations, finite-size errors are exponentially small in grand canonical ensemble calculations of translationally invariant systems in unordered phases at finite temperature. Open boundary conditions and canonical ensemble calculations suffer from finite-size errors that are only polynomially small in the system size. We further show that finite-size effects are generally smallest in numerical linked cluster expansions. Our conclusions are supported by analytical and numerical analyses of classical and quantum systems.
We show that the probability distribution of the error exponent in i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the work by Dalai [1] and the recent ones by Renes [2] and Li and Yang [3], implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate regime the threshold we derive coincides with the ensemble-average of the exponent, that is, the typical random coding (TRC) exponent [4].
The properties of an infinite system at a continuous phase transition are characterised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems are deduced starting from the well-established scaling relations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling are suppressed in the microcanonical ensemble compared to the corresponding corrections in the canonical ensemble.