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arXiv:0812.0694 · Physical Review A

Quantum annealing and the Schrödinger-Langevin-Kostin equation

Diego de Falco, Dario Tamascelli

Published Jan 15, 2009• 12 citations
arXivHTMLPDF

Abstract

We show, in the context of quantum combinatorial optimization, or quantum annealing, how the nonlinear Schr\"odinger-Langevin-Kostin equation can dynamically drive the system toward its ground state. We illustrate, moreover, how a frictional force of Kostin type can prevent the appearance of genuinely quantum problems such as Bloch oscillations and Anderson localization which would hinder an exhaustive search.

DOI
10.1103/physreva.79.012315

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We show, in the context of quantum combinatorial optimization, or quantum annealing, how the nonlinear Schr"odinger-Langevin-Kostin equation can dynamically drive the system toward its ground state. We illustrate, moreover, how a frictional force of Kostin type can prevent the appearance of genuinely quantum problems such as Bloch oscillations and Anderson localization which would hinder an exhaustive search.
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Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.
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Quantum annealing (QA) is a promising calculation method for obtaining a better solution to combinatorial optimization problems. QA is a natural computation technique proposed in the context of statistical physics in 1998. In 2011, the world's first commercial machine realizing QA experimentally was released. After that, the development of hardware and software has been exhaustive, and the studies on application search have been done using commercial QA machines. In this paper, we first explain the basics of QA without assuming the knowledge of statistical physics and quantum physics. We also show a couple of examples of application studies on QA. In addition, existing problems and overcoming other methods are explained.
The dynamics of nonlinear systems qualitatively change depending on their parameters, which is called bifurcation. A quantum-mechanical nonlinear oscillator can yield a quantum superposition of two oscillation states, known as a Schrödinger cat state, via quantum adiabatic evolution through its bifurcation point. Here we propose a quantum computer comprising such quantum nonlinear oscillators, instead of quantum bits, to solve hard combinatorial optimization problems. The nonlinear oscillator network finds optimal solutions via quantum adiabatic evolution, where nonlinear terms are increased slowly, in contrast to conventional adiabatic quantum computation or quantum annealing, where quantum fluctuation terms are decreased slowly. As a result of numerical simulations, it is concluded that quantum superposition and quantum fluctuation work effectively to find optimal solutions. It is also notable that the present computer is analogous to neural computers, which are also networks of nonlinear components. Thus, the present scheme will open new possibilities for quantum computation, nonlinear science, and artificial intelligence.