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arXiv:2604.06324

Higher Nishimori Criticality and Exact Results at the Learning Transition of Deformed Toric Codes

Rushikesh A. Patil, Malte Pütz, Simon Trebst, Guo-Yi Zhu, Andreas W. W. Ludwig

cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elquant-phSubmitted Apr 7, 2026• Announced Apr 9, 2026• 0 citations
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Abstract

We revisit a learning-induced tricritical point, at which three phases with strong, weak, and broken $Z_2$ 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 $\textit{higher Nishimori line}$, 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 $R\rightarrow2$ limit, where disorder is averaged according to the Born rule. As such, the learning tricritical point is in fact a $\textit{higher Nishimori critical point}$. Using this identification, we obtain a number of $\textit{exact results}$ at this $\textit{higher}$ 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 $c_{\text{eff}}$ $\textit{decreases}$ under renormalization group (RG) flow from the $\textit{higher}$ 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 $c_{\text{eff}} = 0.522(1)$. 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 $\textit{higher}$ Nishimori criticality in general dimensions $D>1$.

Comments
35 pages, 14 figures, 1 table

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Conditional probability distributions describe the effect of learning an initially unknown classical state through Bayesian inference. Here we demonstrate the existence of a learning transition, having signatures in the long distance behavior of conditional correlation functions, in the two-dimensional classical Ising model. This transition, which arises when learning local energy densities, extends all the way from the infinite-temperature paramagnetic state down to the thermal critical state. The intersection of the line of learning transitions and the thermal Ising transition is a new tricritical point. Our model for learning also exactly describes the effects of weak measurements on ground states of frustration-free quantum Hamiltonians, which interpolate between the toric code and a paramagnet. Notably, the location of the above tricritical point implies that the quantum memory defined by the degenerate ground states in the topological phase is robust to weak measurement, even when the initial state is arbitrarily close to the quantum phase transition separating topological and trivial phases. Our analysis uses a replica field theory combined with the renormalization group, and we chart out the phase diagram using a combination of tensor network and Monte Carlo techniques. Our methods can be extended to study the more general effects of learning on both classical and quantum states. The learning induced critical states can be realized in classical or quantum devices.
The quantum error correction threshold is closely related to the Nishimori physics of random statistical models. We extend quantum information measures such as coherent information beyond the Nishimori line and establish them as sharp indicators of phase transitions over the full ppp-TTT plane. These generalized measures admit a natural operational interpretation as diagnostics of inference mismatch for decoders operating at an effective temperature. We derive exact inequalities for several generalized measures, demonstrating that each attains its extremum along the Nishimori line. As a direct application, we study these measures in the 2d ±J\pm J±J random-bond Ising model-corresponding to a surface code under bit-flip noise-and revisit the Nishimori multicritical point. Among all indicators, coherent information exhibits the weakest finite-size effects, enabling a high-precision estimate pc=0.1092212(4)p_c=0.1092212(4)pc​=0.1092212(4) and the associated critical exponents.
Abstract The Nishimori point of the random bond Ising model is a prototype of renormalization group fixed points with strong disorder. We show that the exact correlation length and crossover critical exponents at this point can be identified in two and three spatial dimensions starting from properties of the Nishimori line. These are the first exact exponents for frustrated random magnets, a circumstance to be also contrasted with the fact that the exact exponents of the Ising model without disorder are not known in three dimensions. Our considerations extend to higher dimensions and models other than Ising.