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.
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.
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 ceffdecreases 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.
Learning Potts Models and Z3 Toric Codes: Higher and Ordinary Nishimori Criticality
Motivated by a previous Ising study, we identify a higher Nishimori line in the learning phase diagram of the 2Dq-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.
Quantum data learning (QDL) provides a framework for extracting physical insights directly from quantum states, bypassing the need for any identification of the classical observable of the theory. A central challenge in many-body physics is that the identity of quantum phases, especially those with topological order, are often inaccessible through local observables or simple symmetry-breaking diagnostics. Here, we apply QDL techniques to the 2+1-dimensional toric-code loop-gas model in a magnetic field. Ground states are generated across multiple lattice sizes using a parametrized loop-gas circuit (PLGC) with a variational quantum-eigensolver (VQE) approach. We then train a quantum convolutional neural network (QCNN) across the full field-parameter range to perform phase classification and capture the overall phase structure. We also employ a physics-aware training protocol that excludes the near-critical region (0.2 <= x <= 0.4)) around (x_c = 0.25), the phase-transition point estimated by quantum Monte Carlo, reserving this window for testing to evaluate the ability of the model to learn the phase transition. In parallel, we implement an unsupervised quantum k-means method based on state overlaps, which partitions the dataset into two phases without prior labeling. Our supervised QDL approach recovers the phase structure and accurately locates the phase transition, in close agreement with previously reported values; the unsupervised QDL approach recovers the phase structure and locates the phase transition with a small offset as expected in finite volumes; both QDL methods outperform classical alternatives. These findings establish QDL as an effective framework for characterizing topological quantum matter, studying finite volume effects, and probing phase diagrams of higher-dimensional systems.
The Boltzmann distribution encodes our subjective knowledge of the configuration in a classical lattice model, given only its Hamiltonian. If we acquire further information about the configuration from measurement, our knowledge is updated according to Bayes' theorem. We examine the resulting “conditioned ensembles,” finding that they show many new phase transitions and new renormalization-group fixed points. (Similar conditioned ensembles also describe “partial quenches” in which some of the system's degrees of freedom are instantaneously frozen, while the others continue to evolve.) After describing general features of the replica field theories for these problems, we analyze the effect of measurement on illustrative critical systems, including: critical Ising and Potts models, which show surprisingly rich phase diagrams, with RG fixed points at weak, intermediate, and infinite measurement strength; various models involving free fields, XY spins, or flux lines in 2D or 3D; and geometrical models such as polymers or clusters. We also give a formalism for measurement of classical stochastic processes. We use this to make connections with quantum dynamics, in particular with “charge sharpening” in 1D, for which we give a purely hydrodynamic derivation of the known effective field theory. We discuss qualitative differences between RG flows for the above measured systems, described by N → 1 replica limits, and those for disordered systems, described by N → 0 limits. In addition to discussing measurement of critical states, we give a unifying treatment of a family of inference problems for noncritical states. These are related to the Nishimori line in the phase diagram of the random-bond Ising model, and are relevant to various quantum error correction problems. We describe distinct physical interpretations of conditioned ensembles and note interesting open questions.
Many aspects of the well-known mapping between the partition function of a classical spin model and the quantum entangled state have been studied in recent years. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state have been mostly ignored. In this paper, we consider this problem for an important example of the Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. We show that the temperature on the classical side is mapped to bit-flip noise on the quantum side. It is then shown that a transition from a coherent superposition of a given quantum state to a noncoherent mixture corresponds exactly to paramagnetic-ferromagnetic phase transition in the Ising model. To identify such a transition further, we define an order parameter to characterize the decoherence of such a mixture and show that it behaves similar to the order parameter (magnetization) of the 2D Ising model, a behavior that is interpreted as a robust coherence in the toric code model. Furthermore, we consider other properties of the noisy toric code model exactly at the critical point. We show that there is a relative stability to noise for the toric code state at the critical noise which is revealed by a relative reduction in susceptibility to noise.
We investigate the quantum phase diagram of the K-layer Ising toric code corresponding to K layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays Z2K topological order originating from the toric codes in each layer, the system shows Z2 topological order in the high-Ising limit. The latter is demonstrated for general K by deriving an effective low-energy model in Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases K=3 and K=4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both K separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.