While we are still far from Fault Tolerant Quantum Computers (FTQCs) which are required to obtain a significant advantage in some algorithms, for benchmarking purposes, it can be interesting to try and solve problems with Noisy Intermediate Scale Quantum (NISQ) era Quantum Computers (QCs) on some of the future applications of FTQC. In this paper, our aim is to assess the performance of D-Wave Quantum Processing Units (QPUs) in solving systems of linear equations. This can also be used as a first reference point to compare NISQ era QPUs between them. To that end, we illustrate the transformation of these systems into Quadratic Unconstrained Binary Optimization (QUBO) problems, which can be tackled by these QPUs. We evaluate their performance in their default configurations and demonstrate some usual way to enhance their performance. Finally, we introduce a novel approach to solving these problems that does not result in an exponential growth of their coefficients. Data-points from these experiments are meant to be integrated in the BACQ benchmark [2].
While we are still far from Fault Tolerant Quantum Computers (FTQCs) which are required to obtain a significant advantage in some algorithms, for benchmarking purposes, it can be interesting to try and solve problems with Noisy Intermediate Scale Quantum (NISQ) era Quantum Computers (QCs) on some of the future applications of FTQC. In this paper, our aim is to assess the performance of D-Wave Quantum Processing Units (QPUs) in solving systems of linear equations. This can also be used as a first reference point to compare NISQ era QPUs between them. To that end, we illustrate the transformation of these systems into Quadratic Unconstrained Binary Optimization (QUBO) problems, which can be tackled by these QPUs. We evaluate their performance in their default configurations and demonstrate some usual way to enhance their performance. Finally, we introduce a novel approach to solving these problems that does not result in an exponential growth of their coefficients. Data-points from these experiments are meant to be integrated in the BACQ benchmark [2].
Quantum annealing has the potential to outperform classical transistor-based computer technologies in tackling intricate combinatorial optimization problems. However, ongoing scientific debates cast doubts on whether quantum annealing devices (or quantum annealers) can genuinely provide better problem-solving capabilities than classical computers. The question of whether quantum annealing algorithms (QAAs) running on quantum annealers have computational advantages over classical algorithms (CAs) running on classical computers still remains unclear. This paper aims to clarify the question by classifying and benchmarking QAAs that utilize quadratic unconstrained binary optimization (QUBO) formulas to solve NP-hard problems. It proposes a four-class classification of QUBO formulas and exemplifies each class by QUBO formulas used by QAAs for solving specific NP-hard problems, such as the subset sum, maximum cut, vertex cover, 0/1 knapsack, graph coloring, Hamiltonian cycle, traveling salesperson, and job shop scheduling problems. The classification is based on the following two criteria: (i) Does the number of QUBO variables scale linearly with the problem input size? (ii) Does the QUBO formula have both the constraint term and the optimization term? QAAs are implemented and run on a D-Wave quantum annealer for benchmarking. They are benchmarked against related classical algorithms (CAs) running on classical computers in terms of the quality of the solution and the time to the solution. The benchmarking results reveal which classes of QUBO formulas are likely to provide advantages to QAAs over CAs. Furthermore, based on the benchmarking results, observations and suggestions are given for each class of QUBO formulas, facilitating the adoption of appropriate actions to improve the performance of QAAs.
This study systematically benchmarks several non-fault-tolerant quantum computing algorithms across four distinct optimization problems: max-cut, number partitioning, knapsack, and quantum spin glass. Our benchmark includes noisy intermediate-scale quantum (NISQ) algorithms, such as the variational quantum eigensolver, quantum approximate optimization algorithm, quantum imaginary time evolution, and imaginary time quantum annealing, with both ansatz-based and ansatz-free implementations, alongside tensor network methods and direct simulations of the imaginary-time Schrödinger equation. For comparative analysis, we also utilize classical simulated annealing and quantum annealing on D-Wave devices. Employing default configurations, our findings reveal that no single non-FTQC algorithm performs optimally across all problem types, underscoring the need for tailored algorithmic strategies. This work provides an objective performance baseline and serves as a critical reference point for advancing NISQ algorithms and quantum annealing platforms.
Abstract Quantum computers have strict requirements for the problems that they can efficiently solve. One of the principal limiting factor for the performances of noisy intermediate-scale quantum (NISQ) devices is the number of qubits required by the running algorithm. Several combinatorial optimization problems can be solved with NISQ devices once that a corresponding quadratic unconstrained binary optimization (QUBO) form is derived. Numerous techniques have been proposed to achieve such reformulations and, depending on the method chosen, the number of binary variables required, and therefore of qubits, can vary considerably. The aim of this work is to drastically reduce the variables needed for these QUBO reformulations in order to unlock the possibility to efficiently obtain optimal solutions for a class of optimization problems with NISQ devices. This goal is achieved by introducing novel tools that allow an efficient use of slack variables, even for problems with non-linear constraints, without the need to approximate the starting problem. We divide our new techniques in two independent parts, called the iterative quadratic polynomial and the master-satellite methods. Hence, we show how to apply our techniques in case of an NP-hard optimization problem inspired by a real-world financial scenario called Max-Profit Balance Settlement . We follow by submitting several instances of this problem to two D-wave quantum annealers, comparing the performances of our novel approach with the standard methods used in these scenarios. Moreover, this study allows to appreciate several performance differences between the D-wave Advantage and next-generation Advantage2 quantum annealers. We show that the adoption of our techniques in this context allows to obtain QUBO formulations with significantly fewer slack variables, i.e. around 90% less, and D-wave annealers provide considerably higher correct solution rates, which moreover do not decrease with the input size as fast as when adopting standard techniques.
Quantum computers can be much faster than any classical computers in solving a certain class of problems. One of the interesting problems is the combinatorial optimization problem, which is a challenge for classical computer systems. We encode these problems into the Quadratic Unconstrained Binary Optimization (QUBO) format and then solve them with quantum solvers. In this study, we compare the performance of D-Wave’s quantum annealing system with classical solvers, namely, Gurobi and Fixstars. We demonstrate the current capabilities of the D-Wave system.
Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.
Many quantum computing algorithms are being developed with the advent of quantum computers. Solving linear systems is one of the most fundamental problems in almost all of science and engineering. HHL algorithm, a monumental quantum algorithm for solving linear systems on the gate model quantum computers, was invented and several advanced variations have been developed. However, HHL-based algorithms have a lot of limitations in spite of their importance. We address solving linear systems on a D-Wave quantum annealing device. To formulate a quadratic unconstrained binary optimization (QUBO) model for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate this QUBO model on the D-Wave system and discuss the results.