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arXiv:2202.08939 · Mathematics

Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems

Evangelos Stogiannos, Christos Papalitsas, Theodore Andronikos

Published Apr 13, 2022• 17 citations
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Abstract

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

DOI
10.3390/math10081294

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This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.
The traveling salesman problem is a well-known NP-hard problem in combinatorial optimization. This paper shows how to solve it on an Ising Hamiltonian based quantum annealer by casting it as a quadratic unconstrained binary optimization (QUBO) problem. Results of practical experiments are also presented using D-Wave’s 5,000 qubit Advantage 1.1 quantum annealer and the performance is compared to a classical solver. It is found the quantum annealer can only handle a problem size of 8 or less nodes and its performance is subpar compared to the classical solver both in terms of time and accuracy.
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Combinatorial problems pose significant challenges, especially for large-scale instances. A promising approach to addressing these challenges involves quadratic unconstrained binary optimization (QUBO) models, a key framework in quantum computing. QUBO formulations require tailored approaches owing to their unique characteristics and the hardware constraints of quantum systems. This paper focuses on the QUBO formulations of the Hamiltonian cycle problem (HCP). We analyse three QUBO formulations for the HCP, including two inspired by Lagrangian relaxation techniques, where penalties may result in negative costs for certain solutions but are balanced within the model. A preprocessing methodology is also introduced. To evaluate these models, we employ D-Wave quantum annealers, which feature a fixed qubit topology and impose strict resource constraints. The mapping of QUBOs on to this architecture highlights the importance of efficient model design. Through experimental studies, we analyse the performance of the proposed QUBOs with and without preprocessing. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.
This study compares the performance of D-Wave’s two quantum-classical hybrid optimization solvers, which integrate quantum computing principles with classical optimization techniques to solve complex optimization problems. It focuses on the Binary Quadratic Model (BQM) solver, which encodes optimization problems using a binary quadratic formulation, and the Constrained Quadratic Model (CQM) solver, which can incorporate constraints into the model. The solvers were tested on five combinatorial optimization problems: Traveling Salesman Problem (TSP), Quadratic Knapsack Problem (QKP), Quadratic Assignment Problem (QAP), Weighted Max-Cut Problem (WMP), and Maximum Clique Problem (MCP). The goal is to quantify the accuracy of solutions compared to the optimal ones. Experiments on multiple instances with known optimal solutions examined solver behavior and solution quality. At default execution durations, the CQM solver generally outperforms BQM with lower error and higher consistency. Across all test problem instances in TSP, QKP, QAP, and WMP, the CQM solver achieved lower average absolute errors by 48.38, 27.83, 21.29, and 0.05 percentage points, respectively, compared to the BQM solver. An exception was found in MCP, where the BQM solver outperformed the CQM solver by 0.75 percentage points. These findings provide a guide for end users to understand the characteristics of the solvers and to select the appropriate solver for their specific optimization problems.
A classical algorithm is tested against D-Wave hybrid quantum annealers for special cases of the Traveling Salesman Problem where the weights of the edges in a graph form an arithmetic progression. The classical algorithm, Christofides, Nearest Neighbor, Simulated Annealing, Ant Colony Optimization, Genetic Algorithm, and D-Wave hybrid quantum annealers are also tested on the Euclidean Traveling Salesman Problem. D-Wave never outperformed the classical algorithm within its specified scope but outperformed all classical algorithms and heuristics for the Euclidean Traveling Salesman Problem before being overtaken at larger problem sizes.
Hybrid quantum-classical algorithms can help mitigating the physical limitations of current quantum devices, particularly the low qubit count and the reduced topological connectivity. In this paper, we propose a hybrid technique to solve a well-known NP-hard optimization problem: the Traveling Salesperson Problem (TSP). Our approach is based on a graph contraction technique that removes most of the dimensionality of the original problem instance, producing a sub-TSP of a size suitable to be efficiently solved by a quantum device. The performance of our approach is first demonstrated on classical quantum simulation using Path Integral Monte Carlo, and then run on a D-Wave quantum annealer.