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Interdisciplinary Information Sciences

Study on Quantum Annealing Using the Density Matrix Renormalization Group

Sei Suzuki, Masato Okada

Published Jan 1, 2007• 1 citation
Publisher

Abstract

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.

DOI
10.4036/iis.2007.49

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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.
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This paper investigates the application of quantum annealing to solve combinatorial optimization problems. The core claim is that quantum annealing algorithms can effectively enhance the efficiency and accuracy of solving such problems. The underlying mechanism involves simulating the quantum annealing process, leveraging the quantum tunneling effect to explore the solution space and identify optimal solutions. This work presents a novel approach by applying quantum annealing to combinatorial optimization, offering a fresh perspective on problem-solving. We explore the theoretical foundations, model the annealing process, and discuss potential challenges and future research directions. The focus is on demonstrating the feasibility and potential benefits of this approach rather than presenting results from experimental implementations.
Simulating large quantum circuits with a classical computer requires enormous computational power, because the number of quantum states increases exponentially as a function of number of qubits. The density matrix renormalization group (DMRG) was introduced by White to study the properties of relatively large‐scale one‐dimensional quantum systems. This method corresponds to an efficient data compression for one‐dimensional quantum systems. In this study, we applied the DMRG method to quantum circuit simulation, and found that a simple quantum circuit based on Grover’s algorithm can be simulated with a classical computer.
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Quantum annealing is a generic algorithm using quantum-mechanical fluctuations to search for the solution of an optimization problem. The present paper first reviews the fundamentals of quantum annealing and then reports on preliminary results for an alternative method. The review part includes the relationship of quantum annealing with classical simulated annealing. We next propose a novel quantum algorithm which might be available for hard optimization problems by using a classical-quantum mapping as well as the Jarzynski equality introduced in nonequilibrium statistical physics.