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arXiv:2606.26233

A fidelity metric for quantum annealing benchmarked by extreme scaling quantum Monte-Carlo simulations

Gabriel Gouraud, Miha Srdinsek, Xavier Waintal

quant-phcond-mat.str-elSubmitted Jun 24, 2026• Announced Jun 26, 2026• 0 citations• Revised Jul 11, 2026 (v2)
v1 · Jun 24, 2026v2 · Jul 11, 2026
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Abstract

Quantum annealers are supposed to follow adiabatically the ground state of a system as its Hamiltonian slowly interpolates between a trivial phase and a non-trivial one; the non-trivial ground state being the solution to an optimization problem. Overwhelmingly, their performances are measured in terms of how well or fast the optimization problem is solved. While pragmatic, this approach is inherently brittle as it strongly depends on the problem considered and the classical algorithm used as the reference benchmark. Here, we propose a quantity that not only measures the end result but also the quality of the actual quantum annealing process itself. Our metric is the quantum annealing counterpart of the fidelity-per gate of gate-based quantum computers. It takes the form of an accuracy $ε$ for the equation of state of the annealer. We calculate benchmark values of $ε$ using two variants of the simulated quantum annealing technique for Rydberg atoms systems. Our first approach uses variational quantum Monte-Carlo with an ansatz inspired by thermal annealing. It suggests that within $ε\sim 10^{-2}-10^{-3}$, a quantum annealer is indistinguishable from its thermal classical counterpart. Critically, we could reach this precision up to $100,000,000$ atoms on a single CPU. Our second approach (based on Green function quantum Monte-Carlo) reaches accuracies around $ε\sim 10^{-4}$ and we have run it up to $100,000$ atoms. These results outperform current Rydberg atom quantum annealing experimental platforms in both precision and size by orders of magnitude and put severe constraints for future hardware.

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Quantum annealers are supposed to follow adiabatically the ground state of a system as its Hamiltonian slowly interpolates between a trivial phase and a non-trivial one; the non-trivial ground state being the solution to an optimization problem. Overwhelmingly, their performances are measured in terms of how well or fast the optimization problem is solved. While pragmatic, this approach is inherently brittle as it strongly depends on the problem considered and the classical algorithm used as the reference benchmark. Here, we propose a quantity that not only measures the end result but also the quality of the actual quantum annealing process itself. Our metric is the quantum annealing counterpart of the fidelity-per gate of gate-based quantum computers. It takes the form of an accuracy εεε for the equation of state of the annealer. We calculate benchmark values of εεε using two variants of the simulated quantum annealing technique for Rydberg atoms systems. Our first approach uses variational quantum Monte-Carlo with an ansatz inspired by thermal annealing. It suggests that within ε∼10−2−10−3ε\sim 10^{-2}-10^{-3}ε∼10−2−10−3, a quantum annealer is indistinguishable from its thermal classical counterpart. Critically, we could reach this precision up to 100,000,000100,000,000100,000,000 atoms on a single CPU. Our second approach (based on Green function quantum Monte-Carlo) reaches accuracies around ε∼10−4ε\sim 10^{-4}ε∼10−4 and we have run it up to 100,000100,000100,000 atoms. These results outperform current Rydberg atom quantum annealing experimental platforms in both precision and size by orders of magnitude and put severe constraints for future hardware.
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 (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.
Combinatorial optimization is anticipated to be one of the primary use cases for quantum computation in the coming years. The Quantum Approximate Optimization Algorithm and Quantum Annealing can potentially demonstrate significant run-time performance benefits over current state-of-the-art solutions. Inspired by existing methods to characterize classical optimization algorithms, we analyze the solution quality obtained by solving Max-cut problems using gate-model quantum devices and a quantum annealing device. This is used to guide the development of an advanced benchmarking framework for quantum computers designed to evaluate the trade-off between run-time execution performance and the solution quality for iterative hybrid quantum-classical applications. The framework generates performance profiles through compelling visualizations that show performance progression as a function of time for various problem sizes and illustrates algorithm limitations uncovered by the benchmarking approach. As an illustration, we explore the factors that influence quantum computing system throughput, using results obtained through execution on various quantum simulators and quantum hardware systems.
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 review here some recent work in the field of quantum annealing, alias adiabatic quantum computation. The idea of quantum annealing is to perform optimization by a quantum adiabatic evolution which tracks the ground state of a suitable time-dependent Hamiltonian, where 'ℏ' is slowly switched off. We illustrate several applications of quantum annealing strategies, starting from textbook toy-models—double-well potentials and other one-dimensional examples, with and without disorder. These examples display in a clear way the crucial differences between classical and quantum annealing. We then discuss applications of quantum annealing to challenging hard optimization problems, such as the random Ising model, the travelling salesman problem and Boolean satisfiability problems. The techniques used to implement quantum annealing are either deterministic Schrödinger's evolutions, for the toy models, or path-integral Monte Carlo and Green's function Monte Carlo approaches, for the hard optimization problems. The crucial role played by disorder and the associated non-trivial Landau–Zener tunnelling phenomena is discussed and emphasized.
Abstract Quantum Monte Carlo (QMC) techniques are widely used in a variety of scientific problems and much work has been dedicated to developing optimized algorithms that can accelerate QMC on standard processors (CPU). With the advent of various special purpose devices and domain specific hardware, it has become increasingly important to establish clear benchmarks of what improvements these technologies offer compared to existing technologies. In this paper, we demonstrate 2 to 3 orders of magnitude acceleration of a standard QMC algorithm using a specially designed digital processor, and a further 2 to 3 orders of magnitude by mapping it to a clockless analog processor. Our demonstration provides a roadmap for 5 to 6 orders of magnitude acceleration for a transverse field Ising model (TFIM) and could possibly be extended to other QMC models as well. The clockless analog hardware can be viewed as the classical counterpart of the quantum annealer and provides performance within a factor of < 10 of the latter. The convergence time for the clockless analog hardware scales with the number of qubits as ∼ N, improving the ∼ N2 scaling for CPU implementations, but appears worse than that reported for quantum annealers by D-Wave.