The application of quantum annealing to the optimization of continuous-variable functions is a relatively unexplored area of research. We test the performance of quantum annealing applied to a one-dimensional continuous-variable function with a rugged energy landscape. After domain-wall encoding to map a continuous variable to discrete Ising variables, we first benchmark the performance of the real hardware, the D-Wave 2000Q, against several state-of-the-art classical optimization algorithms designed for continuous-variable problems to find that the D-Wave 2000Q matches the classical algorithms in a limited domain of computation time. Beyond this domain, classical global optimization algorithms outperform the quantum device. Next we examine several optimization algorithms that are applicable to the Ising formulation of the problem, such as the time-evolving block decimation (TEBD) to simulate ideal coherent quantum annealing, simulated annealing, simulated quantum annealing, and spin-vector Monte Carlo. The data show that TEBD's coherent quantum annealing achieves far better results than the other approaches, demonstrating the effectiveness of coherent tunneling. From these two types of benchmarks, we conclude that the hardware realization of quantum annealing has the potential to significantly outperform the best classical algorithms if thermal noise and other imperfections are sufficiently suppressed and the device operates coherently, as demonstrated in recent short-time quantum simulations.
The application of quantum annealing to the optimization of continuous-variable functions is a relatively unexplored area of research. We test the performance of quantum annealing applied to a one-dimensional continuous-variable function with a rugged energy landscape. After domain-wall encoding to map a continuous variable to discrete Ising variables, we first benchmark the performance of the real hardware, the D-Wave 2000Q, against several state-of-the-art classical optimization algorithms designed for continuous-variable problems to find that the D-Wave 2000Q matches the classical algorithms in a limited domain of computation time. Beyond this domain, classical global optimization algorithms outperform the quantum device. Next we examine several optimization algorithms that are applicable to the Ising formulation of the problem, such as the time-evolving block decimation (TEBD) to simulate ideal coherent quantum annealing, simulated annealing, simulated quantum annealing, and spin-vector Monte Carlo. The data show that TEBD's coherent quantum annealing achieves far better results than the other approaches, demonstrating the effectiveness of coherent tunneling. From these two types of benchmarks, we conclude that the hardware realization of quantum annealing has the potential to significantly outperform the best classical algorithms if thermal noise and other imperfections are sufficiently suppressed and the device operates coherently, as demonstrated in recent short-time quantum simulations.
In this article, we experimentally test the performance of the recently proposed domain-wall encoding of discrete variables Chancellor, 2019, on Ising model flux qubit quantum annealers. We compare this encoding with the traditional one-hot methods and find that they outperform the one-hot encoding for three different problems at different sizes of both the problem and the variables. From these results, we conclude that the domain-wall encoding yields superior performance against a variety of metrics furthermore; we do not find a single metric by which one hot performs better. We even find that a 2000Q quantum annealer with a drastically less connected hardware graph but using the domain-wall encoding can outperform the next-generation Advantage processor if that processor uses one-hot encoding.
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Quantum annealing is a computational paradigm in which optimisation problems are mapped onto the energy landscape of an interacting quantum system and explored through its dynamical evolution. By continuously transforming a simple initial Hamiltonian into one whose ground state encodes the solution, the system traverses a complex landscape via a combination of quantum fluctuations, tunnelling processes, and dissipative dynamics. Unlike gate-based quantum computing, quantum annealing is a specialised and near-term approach aimed primarily at discrete optimisation and sampling tasks. While it is not expected to provide polynomial-time solutions to NP-hard problems in the worst case, it offers a physically motivated heuristic for navigating rugged energy landscapes that arise across science and engineering. Modern quantum annealers realise programmable spin systems with thousands of qubits, placing them among the largest controllable quantum devices currently available. As a result, their significance extends beyond optimisation: they also function as experimental platforms for studying non-equilibrium many-body quantum dynamics in regimes that are difficult to access using classical simulation. In this review we present an accessible introduction to the principles of quantum annealing, describe the main hardware platforms and algorithmic techniques, and analyse how tunnelling, spectral gaps, and open-system effects shape computational performance. We survey applications ranging from optimisation and machine learning to quantum simulation and many-body physics, and discuss the central challenges in benchmarking, scaling, and control. These perspectives position quantum annealing as a distinctive framework at the interface of optimisation, stochastic sampling, and programmable quantum dynamics, with a role that is complementary to both classical algorithms and gate-based quantum computing.
We propose Quantum Enhanced Simulated Annealing (QESA), a novel hybrid optimization framework that integrates quantum annealing (QA) into simulated annealing (SA) to tackle continuous optimization problems. While QA has shown promise in solving binary problems such as those expressed in Ising or QUBO form, its direct applicability to real-valued domains remains limited. QESA bridges this gap by using QA to select discrete search directions that guide SA through the continuous solution space, enabling the use of quantum resources without requiring full problem discretization. We demonstrate QESA’s effectiveness on box-constrained quadratic programming (QP) problems, a class of non-convex optimization tasks that frequently arise in practice. Experimental results show that QESA consistently outperforms classical baselines in solution quality, particularly on larger and more ill-conditioned problems, while maintaining competitive runtime. As quantum annealing hardware matures, QESA offers a scalable and flexible strategy for leveraging quantum capabilities in continuous optimization.
Over the past decade, the usefulness of quantum annealing hardware for combinatorial optimization has been the subject of much debate. Thus far, experimental benchmarking studies have indicated that quantum annealing hardware does not provide an irrefutable performance gain over state-of-the-art optimization methods. However, as this hardware continues to evolve, each new iteration brings improved performance and warrants further benchmarking. To that end, this work conducts an optimization performance assessment of D-Wave Systems' most recent Advantage Performance Update computer, which can natively solve sparse unconstrained quadratic optimization problems with over 5,000 binary decision variables and 40,000 quadratic terms. We demonstrate that classes of contrived problems exist where this quantum annealer can provide run time benefits over a collection of established classical solution methods that represent the current state-of-the-art for benchmarking quantum annealing hardware. Although this work does not present strong evidence of an irrefutable performance benefit for this emerging optimization technology, it does exhibit encouraging progress, signaling the potential impacts on practical optimization tasks in the future.
Quantum annealing is a computational paradigm in which optimisation problems are encoded in the energy landscape of an interacting quantum system and explored through its dynamical evolution. By continuously transforming an initial Hamiltonian into one whose ground state represents the solution, the system navigates complex energy landscapes through a combination of quantum fluctuations, tunnelling processes, and dissipative dynamics. Quantum annealing is primarily designed for discrete optimisation and sampling tasks and provides a physically motivated heuristic for exploring rugged landscapes that arise across science and engineering. Modern quantum annealers realise programmable spin systems with thousands of qubits, making them among the largest controllable quantum devices currently available. Beyond optimisation, they also serve as experimental platforms for studying non-equilibrium many-body quantum dynamics in regimes that are challenging to access classically. In this review we introduce the principles of quantum annealing, describe the main hardware platforms and algorithmic techniques, and analyse the roles of tunnelling, spectral gaps, and open-system effects in determining performance. We survey applications ranging from optimisation and machine learning to quantum simulation and many-body physics, and discuss the central challenges of benchmarking, scaling, and control. These developments position quantum annealing at the interface of optimisation, stochastic sampling, and programmable quantum dynamics.