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Quantum Information and Computation

Bang-bang control as a design principle for classical and quantum optimization algorithms

BapatAniruddha, JordanStephen

Published May 1, 2019• 1 citation
Publisher

Abstract

Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It h...
DOI
10.5555/3370251.3370255

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Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It has been argued that quantum properties such as tunneling may give quantum algorithms advantage in finding ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm (QAOA) have shown promising results on various problem instances that are considered classically hard. Here, building on previous observations from \cite{mcclean2016, Yang2017}, we argue that the type of control strategy used by the optimization algorithm may be crucial to its success. Along with SA, QAO, and QAOA, we define a new, bang-bang version of simulated annealing, BBSA, and study the performance of these algorithms on two well-studied problem instances from the literature. Both classically and quantumly, the successful control strategy is found to be bang-bang, exponentially outperforming the quasistatic analogues on the same instances. Lastly, we construct O(1)-depth QAOA protocols for a class of symmetric cost functions, and provide an accompanying physical picture.
Quantum annealers aim at solving nonconvex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists of designing a classical energy function whose ground states are the sought optimal solutions of the original optimization problem and add a controllable quantum transverse field to generate tunneling processes. A key challenge is to identify classes of nonconvex optimization problems for which quantum annealing remains efficient while thermal annealing fails. We show that this happens for a wide class of problems which are central to machine learning. Their energy landscapes are dominated by local minima that cause exponential slowdown of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.
Simulated Annealing (SA) is often used to provide satisfactory solutions to solving combinatorial problems. However, the SA is limited by its tendency to get stuck in local minima, which often prevents the global optimal solution from being reached for large or very complex problems. To overcome this obstacle, we present a new method inspired by the properties of degenerate quantum systems, named Degenerate Quantum Simulated Annealing (SA-DQS). This innovative approach exploits the coexistence of several quantum states of the same energy; this multiplicity of states is advantageous because it allows to finding of several solutions with identical energies, thus increasing the chances of discovering and building a promising solution among them, and of getting out of the traps of local minima more efficiently. The traveling salesman problem (TSP) is used as an application benchmark to validate the effectiveness of SA-DQS. Compared to standard simulated annealing (SA-S), SA-DQS showed superior performance on multiple instances of TSP. On average, the solutions provided by SA-DQS are equal to or closer to the optimum known in TSPLIB, in addition, it also offers a shorter computation time compared to that of other heuristic methods.
Abstract Quantum Monte Carlo has found a new application far afield from the solution of the Schrodinger equation for many-body problems. In this paper the authors report a method for solving multidimensional optimization problems that is superior, in many cases, to other methods such as conjugate gradient methods, the simplex method, direction-set methods, genetic methods, and classical simulated annealing. The new approach, aptly titled “quantum annealing,” is closely related to classical annealing, in which a physical system such as a metal cluster is slowly cooled and finds its lowest-energy configuration as the temperature drops to zero. In classical annealing the system follows classical-mechanical dynamics as the temperature drops. In quantum annealing a collection of configurations evolves in the the same way as walkers undergoing diffusion and multiplication in diffusion QMC with the added feature of a slowly dropping value for Planck’s constant h. Classical annealing allows the escape from local minima with the aid of thermal fluctuations. Quantum annealing allows such escape by delocalization and tunneling. Since a QMC walker can multiply in number on entering a low-energy region, a single walker can populate an entire region.