We solve a model that has basic features that are desired for quantum annealing computations: entanglement in the ground state, controllable annealing speed, ground state energy separated by a gap during the whole evolution, and a programmable computational problem that is encoded by parameters of the Ising part of the spin Hamiltonian. Our solution enables exact nonperturbative characterization of final nonadiabatic excitations, including a scaling of their number with the annealing rate and the system size. We prove that quantum correlations can accelerate computations and, at the end of the annealing protocol, lead to the perfect Gibbs distribution of all microstates.
We solve a model that has basic features that are desired for quantum annealing computations: entanglement in the ground state, controllable annealing speed, ground state energy separated by a gap during the whole evolution, and a programmable computational problem that is encoded by parameters of the Ising part of the spin Hamiltonian. Our solution enables exact nonperturbative characterization of final nonadiabatic excitations, including a scaling of their number with the annealing rate and the system size. We prove that quantum correlations can accelerate computations and, at the end of the annealing protocol, lead to the perfect Gibbs distribution of all microstates.
We point to the existence of an analytical solution to a general quantum annealing (QA) problem of finding low energy states of an arbitrary Ising spin Hamiltonian HI by implementing time evolution with a Hamiltonian H(t)=HI+g(t)Ht. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t) and a specific mixing Hamiltonian Ht that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HI, the solution shows the possibility of a considerable quantum speedup of finding the Ising ground state. We then compare predictions of our solution to results of numerical simulations and argue that the solvable QA protocol produces the optimal performance in the limit of maximal complexity of the computational problem. Our solution demonstrates for the most complex spin glasses a power-law energy relaxation with the annealing time T and uncorrelated from the HI annealing schedule. This proves the possibility for spin glasses of a faster than ∼1/logβT energy relaxation.
Abstract Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, H I . Here, we point to the existence of an analytical solution for such a problem for an arbitrary H I beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of H I , the solution also shows a considerable quantum speedup in computations.
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.
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.
The Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. This model provides insight into a range of phenomena in correlated electron systems. We wish to examine the paradigm of quantum algorithms for solving such many-body problems. The focus of our current work is on the one-dimensional model that is integrable, meaning that there exist analytical results for determining its ground state. In particular, we demonstrate how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian. We perform simulations for systems with up to 40 qubits to study the scaling of required annealing time for obtaining the ground state. We find that for the half-filled cases considered, there is a polynomial quantum speedup over algorithms based on the Bethe-ansatz equations.
Simulated annealing provides a heuristic solution to combinatorial optimization problems. The cost function of a problem is mapped onto the energy function of a physical many-body system, and, by using thermal or quantum fluctuations, the system explores the state space to find the ground state, which corresponds to the optimal solution of the problem. Studies have highlighted both the similarities and differences between thermal and quantum fluctuations. Nevertheless, fundamental understanding of thermal and quantum annealing remains incomplete, making it unclear how quantum annealing outperforms thermal annealing in which problem instances. Here, we investigate the many-body dynamics of thermal and quantum annealing by examining all possible interaction networks of ±J Ising spin systems up to seven spins. Our comprehensive investigation reveals that differences between thermal and quantum annealing emerge for particular interaction networks, indicating that the structure of the energy landscape distinguishes the two dynamics. We identify the microscopic origin of these differences through probability fluxes in state space, finding that the two dynamics are broadly similar, but that quantum tunnelling produces qualitative differences. Our results provide insight into how thermal and quantum fluctuations navigate a system toward the ground state in simulated annealing, and are experimentally verifiable in atomic, molecular, and optical systems. Furthermore, these insights may improve mappings of optimization problems to Ising spin systems, yielding more accurate solutions in faster simulated annealing and thus benefiting real-world applications in industry. Our comprehensive survey of interaction networks and visualization of probability flux can help to understand, predict, and control quantum advantage in quantum annealing.