We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state.Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach.The idea is tested by the two models, the transverse Ising model and the traveling salesman problem.Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations.The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method.The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible.We also consider the traveling salesman problem.This model can be described by the Ising spin, so that we also apply the same technique to the transverse Ising model.We solve the time-dependent Schrödinger equation numerically for small size systems with various types of exchange interactions of the Ising model.Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters.We check the case of large-size systems by using the quantum Monte Carlo method.The simulation supports the results of the small-size systems, while the dynamics of the Schrödinger equation and the quantum Monte Carlo method are not the same.We find that the simulated annealing by quantum fluctuations has a better performance than the conventional method for the ground state search of the Ising-spin systems.The calculation of the traveling salesman problem is also performed as A.3 Case of Γ(t) = c/ √ t . . . . . . . . . . . . . . . . . . . . . . . .We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state.Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach.The idea is tested by the two models, the transverse Ising model and the traveling salesman problem.Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations.The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method.The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible.We also consider the traveling salesman problem.This model can be described by the Ising spin, so that we also apply the same technique to the transverse Ising model.We solve the time-dependent Schrödinger equation numerically for small size systems with various types of exchange interactions of the Ising model.Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters.We check the case of large-size systems by using the quantum Monte Carlo method.The simulation supports the results of the small-size systems, while the dynamics of the Schrödinger equation and the quantum Monte Carlo method are not the same.We find that the simulated annealing by quantum fluctuations has a better performance than the conventional method for the ground state search of the Ising-spin systems.The calculation of the traveling salesman problem is also performed as A.3 Case of Γ(t) = c/ √ t . . . . . . . . . . . . . . . . . . . . . . . .

Mar 1, 19990 citations

Study of Optimization Problems by Quantum Annealing

Tadashi Kadowaki

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schrödinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schrödinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.

arXiv:quant-ph/0205020May 5, 200239 citations

Convergence of Quantum Annealing with Real-Time Schrödinger Dynamics

Satoshi Morita, Hidetoshi Nishimori

Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.

arXiv:quant-ph/0702252Journal of the Physical Society of JapanMay 25, 200741 citations

Distinguishing thermal versus quantum annealing using probability-flux signatures across interaction networks

Yoshiaki Horiike, Yuki Kawaguchi

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 $\pm J$ 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.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 $\pm J$ 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.

cond-mat.stat-mechCross-listedarXiv:2511.16457Nov 21, 20250 citations

Quantum annealing of the random-field Ising model by transverse ferromagnetic interactions

Sei Suzuki, Hidetoshi Nishimori, Masuo Suzuki

We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to accelerate the convergence of quantum annealing of the random-field Ising model. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to accelerate the convergence of quantum annealing of the random-field Ising model. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.

arXiv:quant-ph/0702214Physical Review EMay 16, 200736 citations

Quantum annealing with antiferromagnetic fluctuations

Yuya Seki, Hidetoshi Nishimori

We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.

arXiv:1203.2418Physical Review EMay 10, 2012134 citations

\pm 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.