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IEEE International Conference on Quantum Software (QSW)

Quantum and Digital Annealing for the Quadratic Assignment Problem

Philippe Codognet, Daniel Díaz, Salvador Pinto Abreu

Published Jul 1, 2022• 12 citations
Publisher

Abstract

The Quadratic Assignment Problem is a a classical constrained optimization problem used to model many real-life applications. We present experiments in solving the Quadratic Assignment Problem by means of Quantum Annealing and Quantum-inspired Annealing. We describe how to model this classical combinatorial problem in terms of QUBO (Quadratic Unconstrained Binary optimization) for implementing it on hardware solvers based on quantum or quantum-inspired annealing (D-Wave, Fujitsu Digital Annealing Unit and Fixstars Amplify Annealing Engine). We present performance result for these implementations and compare them with well established metaheuristic solvers on classical hardware, such as Robust Tabu Search and External Optimization.

DOI
10.1109/qsw55613.2022.00016

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The Quadratic Assignment Problem is a a classical constrained optimization problem used to model many real-life applications. We present experiments in solving the Quadratic Assignment Problem by means of Quantum Annealing and Quantum-inspired Annealing. We describe how to model this classical combinatorial problem in terms of QUBO (Quadratic Unconstrained Binary optimization) for implementing it on hardware solvers based on quantum or quantum-inspired annealing (D-Wave, Fujitsu Digital Annealing Unit and Fixstars Amplify Annealing Engine). We present performance result for these implementations and compare them with well established metaheuristic solvers on classical hardware, such as Robust Tabu Search and External Optimization.
We present experiments in solving combinatorial optimization and constraint satisfaction problems by means of Quantum Annealing. We describe how to model classic constraint problems such as N-queens and Magic Square as well as hard combinatorial problems such as the Costas Array Problem or the Quadratic Assignment Problem in terms of QUBO (Quadratic Unconstrained Binary Optimization). QUBO is the input language of quantum computers based on quantum annealing such as the D-Wave systems and of the ”quantum-inspired” but classical devices such as Fujitsu’s Digital Annealing Unit or Hitachi’s CMOS Annealing Machine. We present preliminary results for solving these combinatorial optimization and constraint satisfaction problems by implementation on the D-Wave quantum computer.
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.
Abstract QUBO (quadratic unconstrained binary optimization) has become the modeling language for quantum annealing and quantum‐inspired annealing solvers. We present different approaches in QUBO for the magic square problem and the quadratic assignment problem (QAP), which can be modeled by linear equations and a permutation constraint over integer variables. Different ways of encoding integers by Booleans in QUBO amount to models, the implementation of which could have very different performance. Experiments performed on the Fixstars Amplify Annealer Engine, a quantum‐inspired annealing solver, show that, compared to the classical one‐hot encoding, using unary encoding for integers performs slightly better for the QAP and much better for magic square.
Quantum Annealing is an alternative type of computation in which problems are encoded in quantum Hamiltonians (energy functions) and quantum dynamics is used to find solutions (ground states of minimal energy). Quantum computers such as the D-Wave systems are indeed implementing those ideas in hardware, as well as "quantum-inspired" devices based on classical electronics such as Fujitsu’s Digital Annealing Unit. All those systems use the same modeling language: Quadratic Unconstrained Binary Optimization (QUBO). However, QUBO is a low-level language and for modeling combinatorial problems such as constraint satisfaction and constrained optimization problems, we need to introduce higher-level abstractions in order to define complex constraints.
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.
We present experiments in solving constrained optimization and constraint satisfaction problems by means of Quantum Annealing in terms of QUBO (Quadratic Unconstrained Binary Optimization). We investigate different QUBO formulations for the encoding of integers into Booleans and for the encoding of the At-Most-One constraint, which is used in combinatorial problems requiring capacity constraints. We compare four different QUBO models and report experiments done on the D-Wave annealing computer and the "quantum-inspired" Fixstars Amplify Annealing Engine. For this, we consider a simple example taken from the domain of Constraint Satisfaction Problems (CSP), the N-queens problem, as its formulation involves multiple instances of the At-Most-One constraints.