In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier--Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models /
Z2 stabilizer codes in spatial dimensions
d=2−5, and extends to Potts variables with
q≤4. The main application of our projection scheme is to
Zq surface codes, whose decoding problem maps to the disordered
q-state clock model. For
q≥5 the clean clock model has \textit{two} Berezinskii--Kosterlitz--Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-
Zq-toric-code numerics, but are found to satisfy the Gilbert--Varshamov self-dual entropy relation
lnq≃Hq(T1∗)+Hq(T2∗), although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.
Motivated by a previous Ising study, we identify a
higher Nishimori line in the learning phase diagram of the
2D q-state Potts model
(2<q≤4) under bond-energy measurements. This
higher Nishimori line meets the critical temperature line of the Potts model, in a
higher Nishimori critical point -- a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete
q-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic
decrease along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on
Elitzur′stheorem that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed
Zq toric code where the tricritical
higher Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.
We investigate the critical behavior of the random-bond $\ifmmode\pm\else\textpm\fi{}J$ Ising model on a square lattice at the multicritical Nishimori point in the $T\text{\ensuremath{-}}p$ phase diagram, where
T is the temperature and
p is the disorder parameter (
p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by $2p\ensuremath{-}1=\text{tanh}(1/T)$, along which the multicritical point lies. The multicritical Nishimori point is located at ${p}^{\ensuremath{\ast}}=0.890\text{ }81(7)$, ${T}^{\ensuremath{\ast}}=0.9528(4)$, and the renormalization-group dimensions of the operators that control the multicritical behavior are
y1=0.655(15) and
y2=0.250(2); they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.
We revisit a learning-induced tricritical point, at which three phases with strong, weak, and broken
Z2 symmetry meet, in the phase diagram of a deformed toric code wavefunction subjected to weak measurements. This setting is exactly dual to a classical Bayesian inference phase diagram of the
2D classical Ising model. Here we demonstrate that this tricritical point lies on a distinct
higher Nishimori line, which has an emergent gauge-invariant formulation, just like the ordinary Nishimori line but with a higher replica symmetry as a replica stat-mech model in the replica number
R→2 limit, where disorder is averaged according to the Born rule. As such, the learning tricritical point is in fact a
higher Nishimori critical point. Using this identification, we obtain a number of
exact results at this
higher Nishimori critical point; e.g., we show that the power-law exponent of the Edwards-Anderson correlation function is exactly equal to that of the spin correlation function at the unmeasured Ising critical point and verify this in numerical simulations. Using the tools of the proof of a
c-effective theorem [arXiv:2507.07959], we show that the Casimir effective central charge
ceff decreases under renormalization group (RG) flow from the
higher Nishimori critical point to the unmeasured
2D Ising critical point, and is thus greater than
1/2. This is corroborated by extensive numerical simulations finding
ceff=0.522(1). The analytical result also explains, with a physically motivated assumption, the numerically observed increase of the Casimir effective central charge under the RG flow from the ordinary Nishimori critical point to the clean Ising critical point in the random-bond Ising model. We also discuss
higher Nishimori criticality in general dimensions
D>1.
We investigate the statistical physics of quantum error correction in
U(1) symmetry-enriched topological quantum memories. Starting from a phenomenological error model of charge-conserving noise, we study the optimal decoder assuming the local charges of each anyon can be measured. The error threshold of the optimal decoder corresponds to a continuous phase transition in a disordered two-dimensional integer loop model on the Nishimori line. Using an effective replica field theory analysis and Monte Carlo numerics, we show that the optimal decoding transition exhibits Berezinskii-Kosterlitz-Thouless universality with a modified universal jump in winding number variance. We further generalize the model beyond the Nishimori line, which defines a large class of suboptimal decoders. At low nonzero temperatures and strong disorder, we find numerical evidence of a disorder-dominated loop-glass phase which corresponds to a "confidently incorrect" decoder. The zero-temperature limit defines the minimum-cost flow decoder, which serves as the
U(1) analog of minimum-weight perfect matching in
Z2 topological codes. Both the optimal and minimum-cost flow decoders are shown to dramatically outperform the charge-agnostic optimal decoder in symmetry-enriched topological codes.
Random projective measurements are an effective tool to model the interlinked phenomenology of monitoring, inferring, and decoding highly entangled states such as quantum error-correcting codes. At the same time, the study of such monitored dynamics has opened a new arena for critical phenomena, renewing longstanding questions in disordered statistical mechanics, including percolation, random-bond models, and Nishimori physics. Here we investigate the robustness of stabilizer quantum error-correcting codes under a single round of independently sampled single-qubit Pauli measurements and demonstrate that their threshold behavior falls into two broad categories - geometric and homological percolation transitions, with the toric code and color code being the principal examples for each class. We establish the distinct character of these two threshold theories by analyzing the logical-support criteria of these codes and by providing numerical estimates of their respective critical exponents obtained from optimized stabilizer simulations of large-scale systems with over one million qubits. Our framework extends to non-CSS, subsystem, higher-dimensional, and Floquet stabilizer codes and allows us to determine their respective threshold phase diagrams. We exemplify how the latter provide quantitative and qualitative guidance for the computationally more demanding problems of tunable learning and maximum-likelihood decoding. As a practical application of our framework, we discuss the robustness of the gross code.