In more recent years, there has been increasing research interest in exploiting the use of application specific hardware for solving optimisation problems. Examples of solvers that use specialised hardware are IBM's Quantum System One and D-wave's Quantum Annealer (QA) and Fujitsu's Digital Annealer (DA). These solvers have been developed to optimise problems faster than traditional meta-heuristics implemented on general purpose machines. Previous research has shown that these solvers (can optimise many problems much quicker than exact solvers such as GUROBI and CPLEX. Such conclusions have not been made when comparing hardware solvers with classical evolutionary algorithms. Making a fair comparison between traditional evolutionary algorithms, such as Genetic Algorithm (GA), and the DA (or other similar solvers) is challenging because the later benefits from the use of application specific hardware while evolutionary algorithms are often implemented on general-purpose machines. Moreover, quantum or quantum-inspired solvers are limited to solving problems in a specific format. A common formulation used is Quadratic Unconstrained Binary Optimisation (QUBO). Many optimisation problems are however constrained and have natural representations that are non-binary. Converting such problems to QUBO can lead to more problem difficulty and/or larger search space. The question addressed in this paper is whether quantum or quantum-inspired solvers can optimise QUBO transformations of combinatorial optimisation problems faster than classical evolutionary algorithms applied to the same problems in their natural representations. We show that the DA often present better average objective function values than GA on Travelling Salesman, Quadratic Assignment and Multi-dimensional Knapsack Problem instances.
In more recent years, there has been increasing research interest in exploiting the use of application specific hardware for solving optimisation problems. Examples of solvers that use specialised hardware are IBM's Quantum System One and D-wave's Quantum Annealer (QA) and Fujitsu's Digital Annealer (DA). These solvers have been developed to optimise problems faster than traditional meta-heuristics implemented on general purpose machines. Previous research has shown that these solvers (can optimise many problems much quicker than exact solvers such as GUROBI and CPLEX. Such conclusions have not been made when comparing hardware solvers with classical evolutionary algorithms. Making a fair comparison between traditional evolutionary algorithms, such as Genetic Algorithm (GA), and the DA (or other similar solvers) is challenging because the later benefits from the use of application specific hardware while evolutionary algorithms are often implemented on general-purpose machines. Moreover, quantum or quantum-inspired solvers are limited to solving problems in a specific format. A common formulation used is Quadratic Unconstrained Binary Optimisation (QUBO). Many optimisation problems are however constrained and have natural representations that are non-binary. Converting such problems to QUBO can lead to more problem difficulty and/or larger search space. The question addressed in this paper is whether quantum or quantum-inspired solvers can optimise QUBO transformations of combinatorial optimisation problems faster than classical evolutionary algorithms applied to the same problems in their natural representations. We show that the DA often present better average objective function values than GA on Travelling Salesman, Quadratic Assignment and Multi-dimensional Knapsack Problem instances.
Quantum and quantum-inspired optimisation algorithms are designed to solve problems represented in binary, quadratic and unconstrained form. Combinatorial optimisation problems are therefore often formulated as Quadratic Unconstrained Binary Optimisation Problems (QUBO) to solve them with these algorithms. Moreover, these QUBO solvers are often implemented using specialised hardware to achieve enormous speedups, e.g. Fujitsu's Digital Annealer (DA) and D-Wave's Quantum Annealer. However, these are single-objective solvers, while many real-world problems feature multiple conflicting objectives. Thus, a common practice when using these QUBO solvers is to scalarise such multi-objective problems into a sequence of single-objective problems. Due to design trade-offs of these solvers, formulating each scalarisation may require more time than finding a local optimum. We present the first attempt to extend the algorithm supporting a commercial QUBO solver as a multi-objective solver that is not based on scalarisation. The proposed multi-objective DA algorithm is validated on the bi-objective Quadratic Assignment Problem. We observe that algorithm performance significantly depends on the archiving strategy adopted, and that combining DA with non-scalarisation methods to optimise multiple objectives outperforms the current scalarised version of the DA in terms of final solution quality.
Quantum and quantum-inspired optimisation algorithms are designed to solve problems represented in binary, quadratic and unconstrained form. Combinatorial optimisation problems are therefore often formulated as Quadratic Unconstrained Binary Optimisation Problems (QUBO) to solve them with these algorithms. Moreover, these QUBO solvers are often implemented using specialised hardware to achieve enormous speedups, e.g. Fujitsu’s Digital Annealer (DA) and D-Wave’s Quantum Annealer. However, these are single-objective solvers, while many real-world problems feature multiple conflicting objectives. Thus, a common practice when using these QUBO solvers is to scalarise such multi-objective problems into a sequence of single-objective problems. Due to design trade-offs of these solvers, formulating each scalarisation may require more time than finding a local optimum.We present the first attempt to extend the algorithm supporting a commercial QUBO solver as a multi-objective solver that is not based on scalarisation. The proposed multi-objective DA algorithm is validated on the bi-objective Quadratic Assignment Problem. We observe that algorithm performance significantly depends on the archiving strategy adopted, and that combining DA with non-scalarisation methods to optimise multiple objectives outperforms the current scalarised version of the DA in terms of final solution quality.
Permutation problems, such as the Travelling Salesman Problem (TSP) and the Quadratic Assignment Problem (QAP), form a well-established class of combinatorial optimisation problems with broad applications in routing, assignment, and scheduling. Metaheuristics, particularly Evolutionary Algorithms (EAs), have been widely demonstrated to yield high-quality solutions for these problems. In parallel, classical commercial solvers based on Mixed-Integer Programming (MIP), including CPLEX and Gurobi, have also been extensively applied. In recent years, there has been growing interest in quantum, quantum-inspired, and quantum-hybrid approaches for solving combinatorial optimisation problems. This paper presents a comparative analysis of six optimisation solvers: Genetic Algorithm (GA), Estimation of Distribution Algorithm (EDA), CPLEX, Gurobi, Digital Annealer (DA), and a quantum-hybrid Nonlinear solver (NL), applied to permutation problems. We evaluate solver performance in terms of runtime, solution quality relative to optimality, and scalability across instances of TSP and QAP. Our results highlight the promising performance of the quantum-hybrid Nonlinear solver across all evaluation metrics.
Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.
Quantum computers are early-stage technology, but are progressing rapidly. Here, we compare metaheuristics with a quantum annealer (QA), a quantum computer tailored to solve combinatorial optimisation problems formulated as a Quadratic Unconstrained Binary Optimisation (QUBO) problem. Formulating non-trivial optimisation problems as QUBOs is a challenge, so it is to ensure that the direct comparison of classical and quantum search algorithms is conducted fairly. This paper illustrates the use of a streamlined methodology to derive a QUBO for a classical optimisation problem – the Steiner tree problem – and provides comprehensive experiments comparing D-WAVE pure QA and hybrid QA with classical methods including Simulated Annealing and Genetic Algorithms. We find that the pure QA suffers from considerable qubit noise but the hybrid QA is on a par with traditional methods.
For various optimization problems, the classical time to solution is super-polynomial and intractable to solve with classical bit-based computing hardware to date. Digital and quantum annealers have the potential to identify near-optimal solutions for such optimization problems using a quadratic unconstrained binary optimization (QUBO) problem formulation. This work benchmarks two use cases to evaluate the utility of QUBO solvers for combinatorial optimization problems, in order to determine if a QUBO formulation and annealing-based algorithms have an advantage over classical mixed-integer programming (MIP) and constraint programming (CP) solvers. Various QUBO and solver metrics such as problem mapping, quantitative interconnectivity, penalty structure, solver minimum cost (obtained optimal value) and solver time to solution have been applied to evaluate different QUBO problems. Constrained and unconstrained QUBO solvers are compared including the Fujitsu digital annealer (DA), various D-Wave hybrid quantum annealing solvers (QA, HQA), and the classical MIP/CP solvers HiGHS, Gurobi, SCIP, and CP-SAT. The two industrially relevant use cases are reaction network pathway analysis and mRNA codon selection. For reaction pathway analysis, classical MIP/CP solvers (especially Gurobi and CP-SAT) are observed to solve the problem to optimality in reasonable time frames. For mRNA codon selection, Gurobi outperformed all other solvers in time to solution for all problem sizes, followed by CP-SAT and the D-Wave Nonlinear (NL) HQA solver.