Mathematical foundation of quantum annealing

Eliahu Cohen, Boaz Tamir

Shu Tanaka

Satoshi Morita, Hidetoshi Nishimori

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.

Hadi Salloum, Mohammad Reza Bahrami, Artem Matevosian +3 more

This paper explores the integration of quantum computing, specifically quantum annealing, into robotics for inspecting electrical transmission lines. By using quantum annealing's computational power, we address the dynamics of robot inspectors on transmission line conductors, enhancing the computational efficiency of managing the movements of robot inspectors on transmission line conductors. The use of these robots is crucial for performing efficient and safe inspections. We discuss the mathematical foundations of quantum annealing, transforming differential equations governing robot dynamics into Ising model Hamiltonians suitable for quantum annealing devices. Through experiments using D-Wave's ‘Advantage’ quantum annealer, we demonstrate the viability of quantum annealing in solving complex robotics problems, paving the way for advanced applications in the field.

Akseli Partanen

Jincheng Zhang

Quantum-Enhanced Iterative Complexity (QICE) presents a novel approach to optimization, employing quantum annealing to accelerate the discovery and improvement of complex mathematical models. This research focuses on utilizing quantum annealing to overcome the inherent limitations of classical computational methods in solving intricate problems. The core mechanism centers around a combination of quantum annealing and iterative refinement algorithms, enabling a significant expansion of the solution space and ultimately enhancing solution accuracy. QICE offers a fundamentally new paradigm in optimization, capitalizing on quantum mechanics to surpass classical computational capabilities. This paper details the theoretical foundations, experimental implementation, and preliminary results demonstrating the effectiveness of QICE in optimizing challenging mathematical models.