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Axioms

A Quantum Adiabatic Algorithm for Multiobjective Combinatorial Optimization

Benjamı́n Barán, Marcos Villagra

Published Mar 9, 2019• 1 citation
Publisher

Abstract

In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.

DOI
10.3390/axioms8010032

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In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.
In this work we propose what we consider the first quantum algorithm for multiobjective combinatorial optimization, at least to the best of our knowledge. The proposed algorithm is based on the adiabatic algorithm of Farhi et al. and it is constructed by mapping a multiobjective combinatorial optimization problem into a Hamiltonian using a convex combination among objectives. We present mathematical properties of the eigenspectrum of the associated Hamiltonian and prove that the quantum adiabatic algorithm can find Pareto-optimal solutions provided certain convex combinations of objectives are used and the underlying multiobjective problem meets certain restrictions.
This thesis studies multiobjective optimization problems in the context of quantum computing. Quantum computing is a computational paradigm based on the laws of quantum physics as superposition, interference and entanglement. New quantum algorithms have emerged that proved to be more efficient than classical algorithms. Particularly, Grover’s search algorithm can find a specific element out of a set of N elements with complexity O(√N). Applications of Grover’s algorithm to optimization problems are currently being studied by other researchers, and in this thesis, a new adaptive search method based on Grover’s algorithm applied to several biobjective optimization problems is introduced. This new algorithm is compared against one of the most cited multiobjective optimization algorithms known as NSGA-II. Experimental evidence suggests that the quantum optimization method proposed in this work is at least as effective as NSGA-II in average, considering an equal number of executions. The proposed quantum algorithm, however, only requires approximately the square root of the number of evaluations executed by NSGA-II. Also, two different types of oracles with regard to the proposed algorithm were considered and the experimental results have shown that one of this oracles has requiered less iterations for similar performance.
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.
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 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.
Abstract We consider a classical and superadiabatic version of an iterative quantum adiabatic algorithm to solve combinatorial optimization problems. This algorithm is deterministic because it is based on purely classical dynamics, that is, it does not rely on any stochastic approach to mimic quantum dynamics. Moreover, we use the exact shortcut to adiabaticity for stationary states of classical spin systems, and thus the final state of an annealing process does not depend on the annealing time. We apply this algorithm to a certain class of hard instances of the 3-SAT problem, which is specially hard for purely adiabatic algorithms. We find that more than 90% of such 64-bits hard instances, which we try to solve, can be resolved by a few iteration. Our approach can also be used to analyze properties of instances themselves apart from stochastic uncertainty and shortage of adiabaticity.