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arXiv:1001.0836

Quantum Annealing with Jarzynski Equality (CCP2009)

Masayuki Ohzeki, Hidetoshi Nishimori

Published Jan 6, 2010• 0 citations
arXivHTMLPDF

Abstract

We show a practical application of an well-known nonequilibrium relation, the Jarzynski equality, in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. It has been disclosed that the ordinary quantum computational algorithm to solve a kind of hard optimization problems, has a bottleneck that its computational time is restricted to be extremely slow without relevant errors. However, by our novel strategy shown in the present study, we might overcome such a difficulty.

DOI
10.48550/arxiv.1001.0836

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We show a practical application of an well-known nonequilibrium relation, the Jarzynski equality, in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. It has been disclosed that the ordinary quantum computational algorithm to solve a kind of hard optimization problems, has a bottleneck that its computational time is restricted to be extremely slow without relevant errors. However, by our novel strategy shown in the present study, we might overcome such a difficulty.
We show a practical application of the Jarzynski equality in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. We consider to incorporate the Jarzynski equality into quantum annealing, which is one of the generic algorithms to solve the combinatorial optimization problem. The ordinary quantum annealing suffers from nonadiabatic transitions whose rate is characterized by the minimum energy gap Δmin of the quantum system under consideration. The quantum sweep speed is therefore restricted to be extremely slow for the achievement to obtain a solution without relevant errors. However, in our strategy shown in the present study, we find that such a difficulty would not matter.
Optimization deals with problem of finding the minimum of a given cost function (the relationship between the total cost of production and the quantity of a product produced) and in the combinatorial optimisation problems the cost function depends on a large number of variables and hard problems are those for which the computational time is not bound by any polynomial in the problem size [1]. To solve hard optimization problems is a challenging task. Several techniques have been developed to get to the solution(s) of such problems. Here we will discuss about some of the problems and also the techniques that have already been implemented.
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.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
This paper investigates the application of quantum annealing for solving complex combinatorial optimization problems. The core claim is that quantum annealing leverages its global search capabilities to efficiently address such problems. The proposed mechanism involves formulating the combinatorial optimization problem as a quantum annealing problem and utilizing the simulated annealing process within quantum annealing to conduct a search for the optimal solution. This research explores the potential of quantum annealing in practical applications, acknowledging that its application in combinatorial optimization is still in its exploratory phase. The algorithm is designed to minimize the energy of a system described by a Hamiltonian, which evolves over time according to the time-dependent Schrödinger equation. The core of the algorithm lies in mapping the problem's constraints and objective function into this Hamiltonian. The search process is governed by quantum fluctuations, represented by the operator H^\hat{H}H^, which encourages transitions to lower energy states, effectively navigating the solution space. The effectiveness of the algorithm is evaluated based on the quality of the solution found and the computational resources required. This work provides a foundational exploration of quantum annealing's potential within the field of combinatorial optimization.
This paper investigates the application of scalable quantum annealing to solve computationally challenging combinatorial optimization problems. The core idea is to map these problems into a quantum annealing framework, leveraging quantum tunneling to efficiently explore the solution space and ultimately converge to the global optimum. The research addresses the existing limitations in the scalability of quantum annealing, proposing a novel approach to tackle NP-hard problems effectively. We will demonstrate the potential of this methodology through a theoretical analysis and formulation of the quantum annealing algorithm, focusing on key parameters and their impact on solution quality and runtime. The ultimate goal is to provide a foundational understanding of applying quantum annealing to combinatorial optimization, paving the way for future developments and practical implementations. This work introduces a new framework for approaching complex optimization problems by harnessing the unique properties of quantum mechanics.