© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2105.13945 · Physical Review A

Quantum optimization of complex systems with a quantum annealer

Steven Adam Abel, Andrew Blance, Michael Spannowsky

Published Oct 10, 2022• 23 citations

Abstract

We perform an in-depth comparison of quantum annealing with several classical optimization techniques, namely, thermal annealing, Nelder-Mead, and gradient descent. The focus of our study is large quasicontinuous potentials that must be encoded using a domain wall encoding. To do this, it is important to first understand the properties of a system that is discretely encoded onto an annealer, in terms of its quantum phases, and the importance of thermal versus quantum effects. We therefore begin with a direct study of the 2D Ising model on a quantum annealer, and compare its properties directly with those of the thermal 2D Ising model. These properties include an Ising-like phase transition that can be induced by either a change in ``quantumness'' of the theory (by way of the transverse field component on the annealer), or by scaling the Ising couplings up or down. This behavior is in accord with what is expected from the physical understanding of the quantum system. We then go on to demonstrate the efficacy of the quantum annealer at minimizing several increasingly hard two-dimensional potentials. For all potentials, we find the general behavior that Nelder-Mead and gradient descent methods are very susceptible to becoming trapped in false minima, while the thermal anneal method is somewhat better at discovering the true minimum. However, and despite current limitations on its size, the quantum annealer performs a minimization very markedly better than any of these classical techniques. A quantum anneal can be designed so the system almost never gets trapped in a false minimum, and rapidly and successfully minimizes the potentials.

DOI: 10.1103/physreva.106.042607

Related papers

Effectiveness of quantum annealing for continuous-variable optimization

Shunta Arai, Hiroki Oshiyama, Hidetoshi Nishimori

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.

arXiv:2305.06631•Physical Review A•Oct 2, 2023•13 citations

Quantum versus classical annealing of Ising spin glasses

Bettina Heim, Troels F. Rønnow, Sergei V. Isakov +1 more

Quantum annealers use quantum fluctuations to escape local minima and find low-energy configurations of a physical system. Strong evidence for superiority of quantum annealing (QA) has come from comparing QA implemented through quantum Monte Carlo (QMC) simulations to classical annealing. Motivated by recent experiments, we revisit the question of when quantum speedup may be expected. Although a better scaling is seen for QA in two-dimensional Ising spin glasses, this advantage is due to time discretization artifacts and measurements that are not possible on a physical quantum annealer. Simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care must be taken when using QMC simulations to assess the potential for quantum speedup.

arXiv:1411.5693•Science•Mar 12, 2015•218 citations

Hybrid annealing: Coupling a quantum simulator to a classical computer

Tobias Graß, Maciej Lewenstein

Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.

arXiv:1611.09729•Physical Review A•May 3, 2017•11 citations

Classical criticality via quantum annealing

Pratik Sathe, Andrew D. King, Susan M. Mniszewski +3 more

Quantum annealing provides a powerful platform for simulating magnetic materials and realizing statistical physics models, presenting a compelling alternative to classical Monte Carlo methods. We demonstrate that quantum annealers can accurately reproduce phase diagrams and simulate critical phenomena without suffering from the critical slowing down that often affects classical algorithms. To illustrate this, we study the piled-up dominoes model, which interpolates between the ferromagnetic 2D Ising model and Villain's fully frustrated "odd model". We map out its phase diagram and employ finite-size scaling and Binder cumulants on a quantum annealer to study critical exponents for thermal phase transitions. Our method achieves systematic temperature control by tuning the energy scale of the Hamiltonian, eliminating the need to adjust the physical temperature of the quantum hardware. This work demonstrates how, through fine-tuning and calibration, a quantum annealer can be employed to apply sophisticated finite-size scaling techniques from statistical mechanics. Our results establish quantum annealers as robust statistical physics simulators, offering a novel pathway for studying phase transitions and critical behavior.

arXiv:2505.13625•Nature Communications•Jan 15, 2026•3 citations

Title: Quantum-Enhanced Gradient Descent for Optimization Problems in Complex Dynamical Systems

Jincheng Zhang

This paper investigates the application of quantum-enhanced gradient descent (QED) to optimize complex dynamical systems, specifically those characterized by non-linear interactions. Traditional optimization methods often struggle with the inherent complexity of these systems, requiring extensive computational resources and time. We propose a hybrid quantum-classical approach, utilizing a quantum annealer to initiate exploration and a classical optimization step to refine the solution. The core mechanism involves leveraging quantum properties like superposition and entanglement to accelerate convergence and enhance the algorithm's ability to navigate high-dimensional solution spaces. The research focuses on demonstrating the efficacy of this hybrid method through a detailed analysis of a challenging, multi-dimensional dynamical system model. The results demonstrate a significant improvement in convergence speed and solution quality compared to classical optimization techniques.

•Sep 7, 2026•0 citations

Title: Quantum Gradient Descent for Optimization

Jincheng Zhang

Quantum Gradient Descent for Optimization Quantum optimization is a rapidly growing field that seeks to utilize quantum mechanical phenomena to improve the efficiency of optimization algorithms. This paper investigates the application of quantum gradient descent to complex systems, proposing a hybrid approach that combines the strengths of both quantum and classical computation. We explore the potential of quantum annealing and quantum error correction to enhance gradient calculations, thereby accelerating the search for optimal solutions. The core mechanism involves leveraging quantum properties to perform gradient computations with significantly reduced computational complexity compared to traditional classical methods. This research aims to provide a novel algorithm that demonstrates substantial performance gains for challenging optimization problems, particularly those exhibiting inherent quantum behavior. The paper details the design of the algorithm, its implementation within a simulated environment, and preliminary results demonstrating improved convergence speed and solution quality. Further work will focus on refining the quantum-enhanced gradient calculation to maximize efficiency and scalability.

•Sep 3, 2026•0 citations

We perform an in-depth comparison of quantum annealing with several classical optimization techniques, namely, thermal annealing, Nelder-Mead, and gradient descent. The focus of our study is large quasicontinuous potentials that must be encoded using a domain wall encoding. To do this, it is important to first understand the properties of a system that is discretely encoded onto an annealer, in terms of its quantum phases, and the importance of thermal versus quantum effects. We therefore begin with a direct study of the 2D Ising model on a quantum annealer, and compare its properties directly with those of the thermal 2D Ising model. These properties include an Ising-like phase transition that can be induced by either a change in ``quantumness'' of the theory (by way of the transverse field component on the annealer), or by scaling the Ising couplings up or down. This behavior is in accord with what is expected from the physical understanding of the quantum system. We then go on to demonstrate the efficacy of the quantum annealer at minimizing several increasingly hard two-dimensional potentials. For all potentials, we find the general behavior that Nelder-Mead and gradient descent methods are very susceptible to becoming trapped in false minima, while the thermal anneal method is somewhat better at discovering the true minimum. However, and despite current limitations on its size, the quantum annealer performs a minimization very markedly better than any of these classical techniques. A quantum anneal can be designed so the system almost never gets trapped in a false minimum, and rapidly and successfully minimizes the potentials.

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.

Quantum annealers use quantum fluctuations to escape local minima and find low-energy configurations of a physical system. Strong evidence for superiority of quantum annealing (QA) has come from comparing QA implemented through quantum Monte Carlo (QMC) simulations to classical annealing. Motivated by recent experiments, we revisit the question of when quantum speedup may be expected. Although a better scaling is seen for QA in two-dimensional Ising spin glasses, this advantage is due to time discretization artifacts and measurements that are not possible on a physical quantum annealer. Simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care must be taken when using QMC simulations to assess the potential for quantum speedup.

Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.

Quantum annealing provides a powerful platform for simulating magnetic materials and realizing statistical physics models, presenting a compelling alternative to classical Monte Carlo methods. We demonstrate that quantum annealers can accurately reproduce phase diagrams and simulate critical phenomena without suffering from the critical slowing down that often affects classical algorithms. To illustrate this, we study the piled-up dominoes model, which interpolates between the ferromagnetic 2D Ising model and Villain's fully frustrated "odd model". We map out its phase diagram and employ finite-size scaling and Binder cumulants on a quantum annealer to study critical exponents for thermal phase transitions. Our method achieves systematic temperature control by tuning the energy scale of the Hamiltonian, eliminating the need to adjust the physical temperature of the quantum hardware. This work demonstrates how, through fine-tuning and calibration, a quantum annealer can be employed to apply sophisticated finite-size scaling techniques from statistical mechanics. Our results establish quantum annealers as robust statistical physics simulators, offering a novel pathway for studying phase transitions and critical behavior.

This paper investigates the application of quantum-enhanced gradient descent (QED) to optimize complex dynamical systems, specifically those characterized by non-linear interactions. Traditional optimization methods often struggle with the inherent complexity of these systems, requiring extensive computational resources and time. We propose a hybrid quantum-classical approach, utilizing a quantum annealer to initiate exploration and a classical optimization step to refine the solution. The core mechanism involves leveraging quantum properties like superposition and entanglement to accelerate convergence and enhance the algorithm's ability to navigate high-dimensional solution spaces. The research focuses on demonstrating the efficacy of this hybrid method through a detailed analysis of a challenging, multi-dimensional dynamical system model. The results demonstrate a significant improvement in convergence speed and solution quality compared to classical optimization techniques.

Quantum Gradient Descent for Optimization Quantum optimization is a rapidly growing field that seeks to utilize quantum mechanical phenomena to improve the efficiency of optimization algorithms. This paper investigates the application of quantum gradient descent to complex systems, proposing a hybrid approach that combines the strengths of both quantum and classical computation. We explore the potential of quantum annealing and quantum error correction to enhance gradient calculations, thereby accelerating the search for optimal solutions. The core mechanism involves leveraging quantum properties to perform gradient computations with significantly reduced computational complexity compared to traditional classical methods. This research aims to provide a novel algorithm that demonstrates substantial performance gains for challenging optimization problems, particularly those exhibiting inherent quantum behavior. The paper details the design of the algorithm, its implementation within a simulated environment, and preliminary results demonstrating improved convergence speed and solution quality. Further work will focus on refining the quantum-enhanced gradient calculation to maximize efficiency and scalability.