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arXiv:quant-ph/0702169 · Journal of Statistical Mechanics Theory and Experiment

Quantum wavefunction annealing of spin glasses on ladders

Javier Rodríguez-Laguna

Published May 11, 2007• 4 citations
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Abstract

A technique inspired by quantum annealing is proposed in order to obtain the classical ground state of a spin glass by tracking the full wavefunction of a given system within the subspace of matrix product states, using the density matrix renormalization group. The technique is exemplified within the problem of obtaining the classical ground state of an Ising spin glass on ladder geometries. Its performance is evaluated and related to the entanglement entropy.

DOI
10.1088/1742-5468/2007/05/p05008

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A technique inspired by quantum annealing is proposed in order to obtain the classical ground state of a spin glass by tracking the full wavefunction of a given system within the subspace of matrix product states, using the density matrix renormalization group. The technique is exemplified within the problem of obtaining the classical ground state of an Ising spin glass on ladder geometries. Its performance is evaluated and related to the entanglement entropy.
We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Hamiltonian is best inferred at a finite temperature in the sense of smallest Hamming distance when the original ferromagnetic interactions are disturbed by noise. We use the numerical transfer-matrix method to establish the existence of an optimal finite temperature in one- and two-dimensional systems. Our numerical results are supported by mean-field calculations, which give an explicit expression of the optimal temperature to infer the spin-glass ground state as a function of variances of the distributions of the original interactions and the noise. The mean-field prediction is in qualitative agreement with numerical data. Implications on postprocessing of quantum annealing on a noisy device are discussed.
The density matrix renormalization group (DMRG) has been extended to study quantum phase transitions on random graphs of fixed connectivity. As a relevant example, we have analysed the random Ising model in a transverse field. If the couplings are random, the number of retained states remains reasonably low even for large sizes. The resulting quantum spin-glass transition has been traced down for a few disorder realizations, through the careful measurement of the entanglement entropy and some selected observables: spatial correlations, energy gap and spin-glass susceptibility, among others.
Quantum annealing is a quantum algorithm proposed recently for combinatorial optimization problems. It manipulates time evolution of a quantum mechanical state and obtain an approximate solution. In order to implement the algorithm in classical computers, we propose to apply the density matrix renormalization group method. Simulation of the time evolution of a quantum mechanical state becomes possible by the density matrix renormalization group method for problems of large size. We explain quantum annealing and the density matrix renormalization group method, and present results of numerical simulation using them.
We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HIH_IHI​ by implementing time evolution with a Hamiltonian H(t)=HI+g(t)HtH(t)=H_I+g(t) H_tH(t)=HI​+g(t)Ht​. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t)g(t)g(t) and a specific mixing Hamiltonian HtH_tHt​ that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HIH_IHI​, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time TTT and uncorrelated from the HIH_IHI​ annealing schedule. This proves the possibility for spin glasses of a faster than ∼1/log⁡βT\sim 1/\log^{\beta} T∼1/logβT energy relaxation.
Abstract It is rigorously shown that an appropriate quantum annealing for any finite-dimensional spin system has no quantum first-order transition in transverse magnetization. This result can be applied to finite-dimensional spin-glass systems, where the ground state search problem is known to be hard to solve. Consequently, it is strongly suggested that the quantum first-order transition in transverse magnetization is not fatal to the difficulty of combinatorial optimization problems in quantum annealing.