This is a short review on an interdisciplinary field of quantum information science and statistical mechanics. We first give a pedagogical introduction to the stabilizer formalism, which is an efficient way to describe an important class of quantum states, the so-called stabilizer states, and quantum operations on them. Furthermore, graph states, which are a class of stabilizer states associated with graphs, and their applications for measurement-based quantum computation are also mentioned. Based on the stabilizer formalism, we review two interdisciplinary topics. One is the relation between quantum error correction codes and spin glass models, which allows us to analyze the performances of quantum error correction codes by using the knowledge about phases in statistical models. The other is the relation between the stabilizer formalism and partition functions of classical spin models, which provides new quantum and classical algorithms to evaluate partition functions of classical spin models.

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We survey some recent work where, motivated by traffic inference, we design in parallel two concurrent models, an Ising and a Gaussian ones, with the constraint that they are suitable for ``belief-propagation'' (BP) based inference. In order to build these model, we study how a Bethe mean-field solution to inverse problems obtained with a maximum spanning tree (MST) of pairwise mutual information, can serve as a reference point for further perturbation procedures. We consider three different ways along this idea: the first one is based on an explicit natural gradient formula; the second one is a link by link construction based on iterative proportional scaling (IPS); the last one relies on a duality transformation leading to a loop correction propagation algorithm on a dual factor graph.

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The optimal nonlinearity for readout a weak signal embedded in strong noise is investigated. In practical devices, a high signal-to-noise ratio (SNR) is preferred more than the Bayesian estimation, which generally requires complicated calculations. The optimal nonlinearity is determined by the noise statistics including its temporal correlations, and provides the highest SNR. According to the result, the existence of nonlinear devices that exhibit SNR higher than linear devices is shown under non-Gaussian noise. In contrast, linear devices exhibit the highest SNR in the presence of Gaussian noise. Using the resultant nonlinearity, the Cramér–Rao lower bound is realized by an easy linear estimator. Our derivation of the optimal characteristics of the filter gives the criterion to designed practical devices in the presence of noise.

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Proteins spontaneously fold into a specific native structure, and thereby function in living cell. If a large protein has metastable structures, it is expected that misfolding occurs. In contrast, for a short part cut out from a large protein, the misfolding is often presumed to be absent. For example, the so-called intrinsically disordered (ID) region is experimentally known to take a fluctuating state. On the other hand, many low energy structures of a short ID region is obtained by the molecular dynamics simulation and therefore the short ID region is expected to exhibit misfolding. The shortness of the ID region may resolve this inconsistency. To shed light on the effect of the shortness, we investigate a short polypeptide into which the simulated structures are embedded as rigid structures by employing the associative memory model. We show that the misfolding occurs even in the short polypeptide because the free energy barriers between the embedded metastable states are sufficiently high. This result indicates that the shortness of the polypeptide is not sufficient to explain the fluctuation observed in the experiment. Therefore we expect that all the simulated structures are not so rigid and the short ID region actually fluctuates largely from these structures.

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Time series in physical and information sciences often show nonstationary trends and are beyond the scope of the conventional methods under assumption of stationarity. Especially, if infinitely many possible trends can occur unpredictably, it is difficult to tackle them with a single algorithm without previous knowledge. However, it is possible to estimate interesting statistical parameters from the data with unpredictable drifts for some specific semiparametric statistical models. In this paper, with brain signals in mind we consider a semiparametric, mixture of Gaussian models where the trend distribution is not restricted at all. We derive an estimator of the covariance matrix for multivariate time series and demonstrate that it works robustly against any unpredictable temporal drift in signals (means) while the conventional cross-correlogram leads to spurious correlations contaminated by the drift.

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A relationship between frustration and the transition point at zero temperature of the Ising spin glasses is reported. We find that the concentration of antiferromagnetic bonds in the system is in good agreement with the critical concentration at zero temperature when the derivative of the average number of frustrated plaquettes with respect to the average number of antiferromagnetic bonds is equal to unity. This relation is confirmed in the Ising spin glasses with binary couplings on the two-dimensional lattices, the hierarchical lattices, and the three-body Ising spin glasses with binary couplings on the two-dimensional lattices. The same argument in the Sherrington–Kirkpatrick model yields a point that is identical to the replica-symmetric solution of the transition point at zero temperature.

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We propose a new approach to solve the problem of the prime factorization, formulating the problem as a ground state searching problem of statistical mechanics Hamiltonian. This formulation is expected to give a new insight to this problem. Especially in the context of computational complexity, one would expect to yield the information which leads to determination of the typical case computational complexity of the factorizing process. With this perspective, first we perform simulation with replica exchange Monte Carlo method. We investigated the first passage time that the correct form of prime factorization is found and observed the behavior which seems to indicate exponential computational hardness. As a secondary purpose, we also expected that this method may become a new efficient algorithm to solve the factorization problem. But for now, our method seems to be not efficient comparing to the existing method; number field sieve.

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We propose a framework to describe nonequilibrium dynamics with respect to a set of macroscopic variables by assuming that microscopic states that have the same corresponding macroscopic variables appear with equal-probability, i.e., the nonequilibrium state is treated as an extended microcanonical ensemble. The method is numerically examined by the relaxation dynamics of the two dimensional Potts model with temperature fixed, and it is found that the two-variable (energy and magnetization) description gives much more quantitatively accurate prediction than the one-variable (energy) description, which is sufficient for equilibrium states. This means that the transient states are difficult to be approximated by the equilibrium state fixing temperature, but possible by the one fixing temperature and magnetic field.

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We present here a simple equation explicitly incorporating non-locality, which reproduces quantized energy levels of the bound states for the square well potentials. Introduction of this equation is motivated by studies of differential equation with time delayed feedback, which can be viewed as describing temporal non-locality.

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We show several calculations to identify the critical point in the ground state in random spin systems including spin glasses on the basis of the duality analysis. The duality analysis is a profound method to obtain the precise location of the critical point in finite temperature even for spin glasses. We propose a single equality for identifying the critical point in the ground state from several speculations. The equality can indeed give the exact location of the critical points for the bond-dilution Ising model on several lattices and provides insight on further analysis on the ground state in spin glasses.

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On the basis of analogy between Bayesian inference and statistical mechanics, we construct a method of wave-front reconstruction in remote sensing using the synthetic aperture radar (SAR) interferometry. Here, we use the maximizer of the posterior marginal (MPM) estimate for phase unwrapping and maximum entropy for noise reduction from unwrapped wave-fronts. Next, we investigate static property of the MPM estimate from a phase diagram described by using Monte Carlo simulation for a wave-front which is typical in remote sensing using the SAR interferometry. The phase diagram clarifies that phase unwrapping is accurately realized by the MPM estimate using an appropriate model prior under the constraint of surface-consistency condition, and that the MPM estimate smoothly carries out phase unwrapping utilizing fluctuations around the MAP solution. Also, using the Monte Carlo simulations, we clarify that the method of maximum entropy using an appropriate model prior succeeds in reducing noises from the unwrapped wave-front obtained by the MPM estimate.

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Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one of the most difficult problems of enumerative combinatorics. Once we give up calculating the exact number of them, however, we have a chance to apply powerful computational methods of statistical mechanics to this problem. In this paper, we develop a statistical enumeration method for SAWs using the multicanonical Monte Carlo method. A key part of this method is to expand the configuration space of SAWs to random walks, the exact number of which is known. Using this method, we estimate a number of N-step SAWs on a square lattice, c_N, up to N=256. The value of c₂₅₆ is 5.6(1)× 10¹⁰⁸ (the number in the parentheses is the statistical error of the last digit) and this is larger than one googol (10¹⁰⁰).

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We study the large deviation function for the empirical measure of diffusing particles at one fixed position. We find that the large deviation function exhibits anomalous system size dependence in systems with translational symmetry if and only if they satisfy the following conditions: (i) there exists no macroscopic flow, and (ii) their space dimension is one or two. We analyze the relation between this anomaly and the so-called long-time tail behavior on the basis of phenomenological arguments. We also investigate this anomaly by using a contraction principle.

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The recent work, Nemoto and Sasa [Phys. Rev. E, 83: 030105(R) (2011)], has shown that large deviations of the current characterizing a nonequilibrium system are obtained by observing the typical current for a modified system specified by a variational principle. In the present study, we will give a generalized version of the Nemoto–Sasa study by extracting a hidden mathematical structure from the fluctuation-response relation which is well-known in statistical mechanics. Here, the minimization of the Kullback–Leibler divergence plays an essential role.

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In this paper, we review how to obtain the central charge of a critical entanglement Hamiltonian through the nested entanglement entropy which was first introduced in Ref. [1]. The critical phenomena of the entanglement Hamiltonian can be identified by the central charge obtained by the nested entanglement entropy. We review our previous studies [1, 2] in which we investigated certain entanglement nature of two-dimensional valence-bond-solid (VBS) state and quantum hard-square models on square and triangle ladders using the nested entanglement entropy.

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The relative Fisher information is less acknowledged by the general physics compared with the relative entropy (Kullback–Leibler divergence). We recapitulate the author's recent work on its use in the consideration of an evaluation of the gradient of the dissipated work in phase space during nonequilibrium operations from the viewpoint of information theory. The probability distributions in thermal equilibrium corresponding to the forward and backward processes are assumed. A profound constraint is found to be obtained thanks to the logarithmic Sobolev inequality. We see that both the relative entropy and the Fisher information of the canonical distributions provide a possible lower bound for the phase space structure of the associated process.

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Fields of experts (FoE) model, which is regarded as a higher-order Markov random field whose clique potentials are modeled by the products of experts, matches spatial structures of natural images well, and therefore, it is an efficient prior of natural images. However, the FoE model does not readily admit efficient inferences because of the complexity of landscape of its energy function. In this paper, we propose an efficient mean field approximation for the FoE model by using a perturbative expansion in statistical mechanics. Our proposed mean field approximation can be applied to the FoE under general settings and can be solved in linear time with respect to the number of pixels. In the latter part of this paper, we apply our method to the image inpainting problem, and we show it gives results being the same or better than ones given by a simple gradient method proposed in the original work.

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