Interface Between Quantum Information and Statistical Physics

The following sections are included:

• PREFACE

• SYMPOSIUM ON INTERFACE BETWEEN QUANTUM INFORMATION AND STATISTICAL PHYSICS

• LIST OF PARTICIPANTS

• ORGANIZING COMMITTEE

• CONTENTS

We briefly review the theoretical formulation of bosons in an optical lattice subject to a effective magnetic field. Starting from the Bose-Hubbard Hamiltonian, we can reduce the problem into the classical frustrated system in the two limiting cases, i.e., the Josephson junction regime and the hard-core limit at finite temperatures. The former can be described by the uniformly frustrated XY model, which is accessible in usual BEC experiments in a deep optical lattice. The latter introduces the extended XY model with the fluctuation of the amplitude of pseudospins, referred to as the gauged CP¹ model. The common features of the two systems are that the thermodynamic properties are sensitive to the magnetic flux piecing the plaquette. We show the results of the Monte Carlo simulations for the nontrivial latter case. Despite the presence of the particle number fluctuation, the thermodynamic properties are qualitatively similar to those of the frustrated XY model, where only the phase is a dynamical variable.

Exciton-polaritons in semiconductor microcavities have been recently observed to undergo dynamical Bose-Einstein condensation. This opens the door for many of the analogous experiments to be performed on a semiconductor chip that have been performed in atomic systems. One of the future applications of quantum technology is quantum simulation, where quantum many body problems are directly simulated in the laboratory. Recent advances towards the realization of quantum simulators using exciton-polaritons will be reviewed. In particular, we discuss in detail one application of the ideas of quantum simulation, where Bose-Einstein condensates are used to accelerate the speed of finding the ground state of a given optimization problem. The method uses the principle of indistinguishability, in contrast to the principle of superposition which is traditionally used to obtain a quantum speedup. By taking advantage of bosonic final state stimulation, the convergence to the ground state can be accelerated by a factor of N, where N is the number of bosons in the BEC.

The following sections are included:

• Introduction

• An approach using ytterbum

• Production of quantum degenerate Yb atoms

• Superfluid-Mott insulator transition

• Strongly-correlated phases in Bose-Fermi mixtures

• Prospect

• Acknowledgement

• References

Integrable models in two dimensions are well-studied. Their appearance proved to be so universal in various kinds of topics including 2D conformal field theory, 3D Chern-Simons theory, to name a few. We present how 4D supersymmetric gauge theory also gets related to spin-chain models.

More precisely, we study Baxter's T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU(N)/SU(2)^N-3 correspondence emerges. That is, two kinds of 4D superconformal field theories having the above different gauge groups are encoded simultaneously in one Baxter's T-Q equation which captures their spectral curves. For example, while one is SU(N_c) with N_f = 2N_c flavors the other turns out to be SU(2)^N_c-3 with N_c hyper-multiplets (N_c > 3). It is seen that the corresponding Seiberg-Witten differential supports our proposal.

The correlation functions of quantum integrable models have been investigated extensively for the past twenty years. We recently studied correlation functions of the integrable higher spin XXZ chain in the massive regime. The analysis is based on the algebraic Bethe ansatz method. This article gives a simple introduction to the algebraic Bethe ansatz method, taking the spin-1/2 XXZ chain as an example. We also present the results of one point functions of integrable spin-1 XXZ chain.

One of the remarkable notions in the recent development of quantum physics is the weak value related to weak measurements. We emulate it as a two-time conditional expectation in a classical stochastic model. We use the well known symmetrized form of the master equation, which is formally equivalent to the wave equation in quantum mechanics apart from the fact that wave functions are always real. The origin of the unusual behaviors of the weak value such as the negative probability and the abnormal enhancement of some expectations becomes clearer in the present case, where the two-time conditional probability has no ambiguity of imaginary/complex values.

Various scaling relations of the entanglement entropy are reviewed. Based on the scaling, I would like to point out similarity of mathematical formulation among recent topics in wide research area. In particular, the scaling plays crucial roles in identifying a quantum system with a physically different classical system. Close connection between the scaling and hyperbolic geometry and contrast between bulk/edge correspondence and compactification for the identification are also addressed.

First I give a brief description of the classical Hopfield model introducing the fundamental concepts of patterns, retrieval, pattern recognition, neural dynamics, capacity and describe the fundamental results obtained in this field by Amit, Gutfreund and Sompolinsky,¹ using the non rigorous method of replica and the rigorous version given by Pastur, Shcherbina, Tirozzi² using the cavity method. Then I give a formulation of the theory of Quantum Neural Networks (QNN) in terms of the XY model with Hebbian interaction. The problem of retrieval and storage is discussed. The retrieval states are the states of the minimum energy. I apply the estimates found by Lieb³ which give lower and upper bound of the free-energy and expectation of the observables of the quantum model. I discuss also some experiment and the search of ground state using Monte Carlo Dynamics applied to the equivalent classical two dimensional Ising model constructed by Suzuki et al.⁶ At the end there is a list of open problems.

In terms of the stochastic process of a quantum-mechanical variant of Markov chain Monte Carlo method based on the Suzuki-Trotter decomposition, we analytically derive deterministic flows of order parameters such as magnetization in infinite-range (a mean-field like) quantum spin systems. Under the static approximation, differential equations with respect to order parameters are explicitly obtained from the Master equation that describes the microscopic-law in the corresponding classical system. We discuss several possible applications of our approach to several research topics, say, image processing and neural networks. This paper is written as a self-review of two papers^1,2 for Symposium on Interface between Quantum Information and Statistical Physics at Kinki University in Osaka, Japan.

It is an important topic to investigate nature of the phase transition in wide area of science such as statistical physics, materials science, and computational science. Recently it has been reported the efficiency of quantum adiabatic evolution/ quantum annealing for systems which exhibit a phase transition, and we cannot obtain a good solution in such systems. Thus, to control the nature of the phase transition has been also attracted attention in quantum information science. In this paper we review nature of the phase transition and how to control the order of the phase transition. We take the Ising model, the standard Potts model, the Blume-Capel model, the Wajnflasz-Pick model, and the Potts model with invisible states for instance. Until now there is no general method to avoid the difficulty of annealing method in systems which exhibit a phase transition. It is a challenging problem to propose a method how to erase or how to control the nature of the phase transition in the target system.

Quantum annealing method has been widely attracted attention in statistical physics and information science since it is expected to be a powerful method to obtain the best solution of optimization problem as well as simulated annealing. The quantum annealing method was incubated in quantum statistical physics. This is an alternative method of the simulated annealing which is well-adopted for many optimization problems. In the simulated annealing, we obtain a solution of optimization problem by decreasing temperature (thermal fluctuation) gradually. In the quantum annealing, in contrast, we decrease quantum field (quantum fluctuation) gradually and obtain a solution. In this paper we review how to implement quantum annealing and show some quantum fluctuation effects in frustrated Ising spin systems.