arXiv subjects

Malte Pütz

Publications and source records attributed to Malte Pütz.

4 recordsLinked to original sources

arXiv2026

Learning Potts Models and $Z_3$ Toric Codes: Higher and Ordinary Nishimori Criticality

Motivated by a previous Ising study, we identify a ${\it higher}$ Nishimori line in the learning phase diagram of the $2D$ $q$-state Potts model $(2 < q\leq 4)$ under bond-energy measurements. This ${\it higher}$ Nishimori line meets the critical temperature line of the Potts model, in a ${\it higher}$ 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 ${\it decrease}$ 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 ${\it Elitzur's \; theorem}$ that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed $\mathbb{Z}_q$ toric code where the tricritical ${\it higher}$ Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.

cond-mat.stat-mech↗

arXiv2026

Learning transitions in classical Ising models and deformed toric codes

Conditional probability distributions describe the effect of learning an initially unknown classical state through Bayesian inference. Here we demonstrate the existence of a \textit{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.

cond-mat.stat-mech↗

arXiv2026

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

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 $D>1$.

cond-mat.stat-mech↗

arXiv2025

Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss

In circuit-based quantum state preparation, qubit loss and coherent errors are circuit imperfections that imperil the formation of long-range entanglement beyond a certain threshold. The critical theory at the threshold is a continuous entanglement transition known to be described by a (2+0)-dimensional non-unitary conformal field theory which, for the two types of imperfections of certain circuits, is described by either percolation or Nishimori criticality, respectively. Here we study the threshold behavior when the two types of errors simultaneously occur and show that, when moving away from the Clifford-regime of projective stabilizer measurements, the percolation critical point becomes unstable and the critical theory flows to Nishimori universality. We track this critical renormalization group (RG) crossover flow by mapping out the entanglement phase diagrams, parametrized by the probability and strength of random weak measurements, of two dual protocols preparing surface code or GHZ-class cat states from a parent cluster state via constant-depth circuits. Extensive numerical simulations, using hybrid Gaussian fermion and tensor network / Monte Carlo sampling techniques on systems with more than a million qubits, demonstrate that an infinitesimal deviation from the Clifford regime leads to a sudden, strongly non-monotonic entanglement growth at the incipient non-unitary RG flow. We argue that spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit multifractality. For percolation, we provide the exact (non-quadratic) multifractal spectrum of exponents, while for the Nishimori fixed point we show high-precision numerical results for five leading exponents characterizing multifractality.

quant-ph↗