Solving discrete NP-hard problems is an important part of scientific discoveries and operations research as well as many commercial applications. A commonly used metric to compare meta-heuristic solvers is the time required to obtain an optimal solution, known as time to solution. However, for some applications it is desirable to have a set of high-quality and diverse solutions, instead of a single optimal one. For these applications, time to solution may not be informative of the performance of a solver, and another metric would be necessary. In particular, it is not known how well quantum solvers sample the configuration space in comparison to their classical counterparts. Here, we apply a recently introduced collective distance measure in solution space to quantify diversity by Mohseni et. al. and, based on that, we employ time-to-diversity as a metric for evaluation of meta-heuristics solvers. We use this measure to compare the performance of the D-Wave quantum annealing processor with a few classical heuristic solvers on a set of synthetic problems and show that D-Wave quantum annealing processor is indeed a competitive heuristic, and on many instances outperforms state-of-the-art classical solvers, while it remains on par on other instances. This suggests that a portfolio solver that combines quantum and classical solutions may win over all solvers.
Solving discrete NP-hard problems is an important part of scientific discoveries and operations research as well as many commercial applications. A commonly used metric to compare meta-heuristic solvers is the time required to obtain an optimal solution, known as time to solution. However, for some applications it is desirable to have a set of high-quality and diverse solutions, instead of a single optimal one. For these applications, time to solution may not be informative of the performance of a solver, and another metric would be necessary. In particular, it is not known how well quantum solvers sample the configuration space in comparison to their classical counterparts. Here, we apply a recently introduced collective distance measure in solution space to quantify diversity by Mohseni et. al. and, based on that, we employ time-to-diversity as a metric for evaluation of meta-heuristics solvers. We use this measure to compare the performance of the D-Wave quantum annealing processor with a few classical heuristic solvers on a set of synthetic problems and show that D-Wave quantum annealing processor is indeed a competitive heuristic, and on many instances outperforms state-of-the-art classical solvers, while it remains on par on other instances. This suggests that a portfolio solver that combines quantum and classical solutions may win over all solvers.
Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. To test this metric, we compare the sampling power of various quantum annealing strategies. We show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both Time-To-Solution (TTS) and Time-To-Diversity (TTD) for finding rare solutions. Using Path-Integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that non-equilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin-glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.
Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.
The recent availability of the first commercial quantum computers has provided a promising tool to tackle NP hard problems which can only be solved heuristically with present techniques. However, it is unclear if the current state of quantum computing already provides a quantum advantage over the current state of the art in classical computing. This article assesses the performance of the D-Wave 2X quantum annealer on two NP hard graph problems, in particular clique finding and graph partitioning. For this, we provide formulations as Qubo and Ising Hamiltonians suitable for the quantum annealer and compare a variety of quantum solvers (Sapi, QBSolv, QSage provided by D-Wave Sys, Inc.) to current classical algorithms (METIS, Simulated Annealing, third-party clique finding and graph splitting heuristics) on certain test sets of graphs. We demonstrate that for small graph instances, classical methods still outperform the quantum annealer in terms of computing time, even though the quality of the best solutions obtained is comparable. Nevertheless, due to the limited problem size which can be embedded on the D-Wave 2X chip, the aforementioned finding applies to most of problems of general nature solvable on the quantum annealer. For instances specifically designed to fit the D-Wave 2X architecture, we observe substantial speed-ups in computing time over classical approaches.
Quantum annealing (QA) has the potential to significantly improve solution quality and reduce time complexity in solving combinatorial optimization problems compared to classical optimization methods. However, due to the limited number of qubits and their connectivity, the QA hardware did not show such an advantage over classical methods in past benchmarking studies. Recent advancements in QA with more than 5000 qubits, enhanced qubit connectivity, and the hybrid architecture promise to realize the quantum advantage. Here, we use a quantum annealer with state-of-the-art techniques and benchmark its performance against classical solvers. To compare their performance, we solve over 50 optimization problem instances represented by large and dense Hamiltonian matrices using quantum and classical solvers. The results demonstrate that a state-of-the-art quantum solver has higher accuracy (~0.013%) and a significantly faster problem-solving time (~6561×) than the best classical solver. Our results highlight the advantages of leveraging QA over classical counterparts, particularly in hybrid configurations, for achieving high accuracy and substantially reduced problem solving time in large-scale real-world optimization problems.
The Kemeny Rank Aggregation (KRA) problem is a well-studied problem in the field of Social Choice with a variety of applications in many different areas like databases and search engines. Intuitively, given a set of votes over a set of candidates, the problem asks to find an aggregated ranking of candidates that minimizes the overall dissatisfaction concerning the votes. Recently, a diverse version of KRA was considered which asks for a sufficiently diverse set of sufficiently good solutions. The framework of diversity of solutions is a young and thriving topic in the field of artificial intelligence. The main idea is to provide the user with not just one, but with a set of different solutions, enabling her to pick a sufficiently good solution that satisfies additional subjective criteria that are hard or impossible to model. In this work, we use a quantum annealer to solve the KRA problem and to compute a representative set of solutions. Quantum annealing is a meta search heuristic that does not only show promising runtime behavior on currently existing prototypes but also samples the solutions space in an inherently different way, making use of quantum effects. We describe how KRA instances can be solved by a quantum annealer and provide an implementation as well as experimental evaluations. As existing quantum annealers are still restricted in their number of qubits, we further implement two different data reduction rules that can split an instance into a set of smaller instances. In our evaluation, we compare classical heuristics that allow to sample multiple solutions such as simulated annealing and local search with quantum annealing performed on a physical quantum annealer. We compare runtime, quality of solution, and diversity of solutions, with and without applying preceding data reduction rules.
Combinatorial optimization is widely regarded as a primary application for near-term quantum processors, although a definitive demonstration of the practical quantum advantage remains elusive. Recent studies have reported that both gate-based quantum circuits and quantum annealers can outperform state-of-the-art classical heuristics on multi-objective optimization (MO-MaxCut) problems. However, these studies did not fully account for the substantial pre- and post-processing overheads intrinsic to quantum solvers, leading to incomplete comparisons between quantum and classical approaches. In this work, we re-examine the same benchmark suite using GPU-based quantum-annealing-inspired algorithms (QAIAs), which, analogously to quantum processors, generate probabilistic samples and thus serve as formidable classical contenders. Our results show that QAIAs can sample candidate solutions approximately two orders of magnitude faster than previously studied quantum processors. In terms of end-to-end runtime, QAIAs also surpass industry-leading classical solvers, thereby establishing themselves as the superior performers among the quantum and classical solvers evaluated thus far for the MO-MaxCut instances.