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arXiv:2210.11193 · Journal of Statistical Mechanics Theory and Experiment

Efficient inference in the transverse field Ising model

Eduardo Domínguez, Hilbert J. Kappen

Published Mar 1, 2023• 0 citations
arXivHTMLPDF

Abstract

Abstract In this paper we introduce an approximate method to solve the quantum cavity equations for transverse field Ising models. The method relies on a projective approximation of the exact cavity distributions of imaginary time trajectories (paths). A key feature, novel in the context of similar algorithms, is the explicit separation of the classical and quantum parts of the distributions. Numerical simulations show accurate results in comparison with the sampled solution of the cavity equations, the exact diagonalization of the Hamiltonian (when possible) and other approximate inference methods in the literature. The computational complexity of this new algorithm scales linearly with the connectivity of the underlying lattice, enabling the study of highly connected networks, as the ones often encountered in quantum machine learning problems.

DOI
10.1088/1742-5468/acba02

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Abstract In this paper we introduce an approximate method to solve the quantum cavity equations for transverse field Ising models. The method relies on a projective approximation of the exact cavity distributions of imaginary time trajectories (paths). A key feature, novel in the context of similar algorithms, is the explicit separation of the classical and quantum parts of the distributions. Numerical simulations show accurate results in comparison with the sampled solution of the cavity equations, the exact diagonalization of the Hamiltonian (when possible) and other approximate inference methods in the literature. The computational complexity of this new algorithm scales linearly with the connectivity of the underlying lattice, enabling the study of highly connected networks, as the ones often encountered in quantum machine learning problems.
The cavity method is a well-established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann et al. [Phys. Rev. B 78, 134424 (2008)] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary-time formalism. Here we show how to take analytically the continuous imaginary-time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path-integral representation of the imaginary-time dynamics. As a side result we also show how this procedure can be used in simple heat bath Monte Carlo simulations of generic quantum spin models. The replica symmetric continuous-time quantum cavity method is formulated for a wide class of models and applied as a simple example on the Bethe lattice ferromagnet in a transverse field. The results of the methods are confronted with various approximation schemes in this particular case. On this system we performed quantum Monte Carlo simulations that confirm the exactness of the cavity method in the thermodynamic limit.
We propose a generalization of the cavity method to quantum spin glasses on fixed connectivity lattices. Our work is motivated by the recent refinements of the classical technique and its potential application to quantum computational problems. We numerically solve for the phase structure of a connectivity q=3q=3q=3 transverse field Ising model on a Bethe lattice with $\ifmmode\pm\else\textpm\fi{}J$ couplings and investigate the distribution of various classical and quantum observables.
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schr\"odinger equation numerically for small size systems with various exchange interactions. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum annealing leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule.
We study the problem of approximating the partition function of the transverse-field Ising model (TFIM), a widely studied quantum many-body model with important applications in quantum simulation and quantum annealing. Despite its fundamental importance, the algorithmic landscape for computing the TFIM partition function has remained poorly understood beyond restricted parameter regimes. We provide a precise characterization of the temperature regimes in which efficient approximation is possible, establishing a sharp computational phase transition. Let J denote the symmetric interaction matrix and Δ(J) = λ_{max}(J)-λ_{min}(J) denote its spectral width. We show that, for all inverse temperatures β ∈ [0,1/Δ(J)], there exists an efficient classical randomized algorithm that approximates the partition function tr(e^{-β H}) to within an arbitrarily small multiplicative factor. To obtain this result, we apply the standard Trotter decomposition to map the quantum model to a classical spin system, and then leverage new techniques in Markov chain analysis to derive an efficient algorithm that samples from and computes the partition function of the resulting distribution. This temperature threshold is tight: for β > 1/Δ(J), we show that there exists interaction matrix J for which approximating the partition function, even within an exponential factor, is NP-hard and thus is unlikely to admit an efficient classical or quantum algorithm.
In this study, we present an alternative analytical approach to solving large-scale Ising problems by reformulating the discrete Ising Hamiltonian into a continuous framework. This transformation enables us to derive exact solutions for a nontrivial class of fully connected Ising models. To validate our method, we conducted numerical experiments comparing our analytical solutions with those obtained from a quantum-inspired Ising algorithm and a quantum Ising machine. The results demonstrate that the quantum-inspired algorithm and brute-force method successfully align with our solutions, while the quantum Ising machine exhibits notable deviations. Our method offers promising avenues for analytically solving diverse Ising problem instances, while the class of Ising problems addressed here provides a robust framework for assessing the fidelity of Ising machines.
We present a self-contained numerical study of the one-dimensional transverse-field Ising model (TFIM), tracing its ground-state entanglement structure from exact diagonalization at small system size (L=8,14,20) through density-matrix renormalization group (DMRG) calculations up to L=100. A single, consistent methodology - identical field grid, identical convergence diagnostic (independent runs at bond dimensions chi=100 and chi=200 at every point) - is used across all seven system sizes, and we show explicitly where exact diagonalization and DMRG must agree exactly rather than merely approximately. Along the way we document and resolve a bond-dimension convergence artifact that produced a spurious discontinuity in an earlier, less systematic dataset, as a worked cautionary example for practitioners. Using the resolved dataset we extract the central charge of the transition via two complementary routes - the leading logarithmic finite-size scaling of the mid-chain entropy, and the full Calabrese-Cardy formula applied to every bond of every system size simultaneously - obtaining c_eff to 0.51-0.52 as short-distance lattice corrections are systematically excluded, consistent with the exact two-dimensional Ising value c=1/2. We review the quantum-classical (Suzuki-Trotter) correspondence that underlies this agreement and discuss how the Widom-Kadanoff scaling hypothesis transplants from the classical to the quantum problem. The manuscript is intended as both a physics result and a worked methodological example of careful finite-size numerics.