We investigate the minimization of Ising Hamiltonians, comparing the performance of gain-based computing paradigms based on the dynamics of semiclassical soft-spin models with quantum annealing. We systematically analyze how the energy landscape for the circulant couplings of a Möbius graph evolves with increased annealing parameters. Our findings indicate that these semiclassical models face challenges due to a widening dimensionality landscape. To counteract this issue, we introduce the manifold reduction method, which restricts the soft-spin amplitudes to a defined phase space region. Concurrently, quantum annealing demonstrates a natural capability to navigate the Ising Hamiltonian's energy landscape due to its operation within the comprehensive Hilbert space. Our study indicates that physics-inspired or physics-enhanced optimizers will likely benefit from combining classical and quantum annealing techniques.
We investigate the minimization of Ising Hamiltonians, comparing the performance of gain-based computing paradigms based on the dynamics of semiclassical soft-spin models with quantum annealing. We systematically analyze how the energy landscape for the circulant couplings of a Möbius graph evolves with increased annealing parameters. Our findings indicate that these semiclassical models face challenges due to a widening dimensionality landscape. To counteract this issue, we introduce the manifold reduction method, which restricts the soft-spin amplitudes to a defined phase space region. Concurrently, quantum annealing demonstrates a natural capability to navigate the Ising Hamiltonian's energy landscape due to its operation within the comprehensive Hilbert space. Our study indicates that physics-inspired or physics-enhanced optimizers will likely benefit from combining classical and quantum annealing techniques.
Abstract Complex optimization problems can be solved via dedicated machines which encode the problem in the couplings of spin Hamiltonians. However, traditional physical minimizers often select excited states due to limitations in spin dynamics. We introduce the Vector Ising Spin Annealer (VISA), a framework in gain-based computing that leverages light-matter interactions. We show that VISA overcomes the limitations by enabling spins to operate within a three-dimensional space, thereby providing a robust solution for effectively minimizing Ising Hamiltonians. Our comparative analysis demonstrates VISA’s superior performance relative to conventional single-dimension spin optimizers, highlighting its capacity to surmount significant energy barriers in intricate landscapes. Detailed studies on cyclic and random graphs reveal VISA’s proficiency in dynamically evolving the energy landscape through time-dependent gain and penalty annealing, underscoring its potential in advancing the field of complex problem-solving in physics-inspired and physics-based computing.
Ising machines are a promising approach to solve combinatorial optimization problems. They map these problems onto the Ising model and search for low-energy configurations. However, navigating the rugged energy landscapes of these systems remains difficult. To improve this navigation, classical adiabatic annealing has been proposed in the literature as a heuristic optimization method for classical Ising machines. Using this technique, the Hamiltonian of the Ising machine is gradually transformed from an easily solvable Hamiltonian to the target Hamiltonian. However, its purported effectiveness is primarily motivated by an analogy to quantum adiabatic annealing, and systematic benchmarking has remained limited.
In this work, we analyze the classical adiabatic annealing technique using continuation methods. Motivated by insights from this analysis, we propose an optimized annealing strategy we refer to as hybrid classical adiabatic annealing. We benchmark our proposed strategy using MaxCut instances with up to 800 spins and problems with external fields, for which it achieves a marginal improvement for a limited set of problems. We conclude that, although theoretically motivated and occasionally beneficial, the hybrid strategy does not offer a sufficient practical advantage over simpler, existing techniques.
Optimization problems, which involve finding the best solution from a set of possible solutions, are ubiquitous in various fields, from finance to engineering. Traditional algorithms sometimes struggle with these problems, especially when the solution space is vast, or the landscape is filled with numerous local minima. Quantum-inspired computing, which emulates quantum mechanical principles on classical hardware, emerges as a promising paradigm to address these challenges. This paper delves into two notable approaches: coherent Ising machines (CIM) and graphics processing unit (GPU)-accelerated simulated annealing. In essence, both methods offer innovative strategies to navigate the solution landscape, potentially bypassing the pitfalls of local optima and ensuring more efficient convergence to solutions. By harnessing the strengths of these quantum-inspired techniques, we can pave the way for enhanced computational capabilities in tackling complex optimization problems, even without a fault-tolerant quantum computer.
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.
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.
Classical and quantum systems are used to simulate the Ising Hamiltonian, an essential component in large-scale optimization and machine learning. However, as the system size increases, devices like quantum annealers and coherent Ising machines face an exponential drop in their success rate. Here, we introduce a novel approach involving high-dimensional embeddings of the Ising Hamiltonian and a technique called "dimensional annealing" to counteract the decrease in performance. This approach leads to an exponential improvement in the success rate and other performance metrics, slowing down the decline in performance as the system size grows. A thorough examination of convergence dynamics in high-performance computing validates the new methodology. Additionally, we suggest practical implementations using technologies like coherent Ising machines, all-optical systems, and hybrid digital systems. The proposed hyperscaling heuristics can also be applied to other quantum or classical Ising devices by adjusting parameters such as nonlinear gain, loss, and nonlocal couplings.