Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1211.3178 · Physical Review A

Non-Hermitian quantum annealing in the ferromagnetic Ising model

Alexander I. Nesterov, Juan Carlos Beas Zepeda, G. P. Berman

Published Apr 25, 2013• 10 citations
arXivHTMLPDF

Abstract

We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to $\mathrm{ln}N$, where $N$ is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NP-complete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

DOI
10.1103/physreva.87.042332

Related papers

Non-Hermitian Quantum Annealing in the Transverse Ising Model

Alexander I. Nesterov, Juan Carlos Beas Zepeda, G. P. Berman

We create a non-Hermitian quantum optimization algorithm to find the ground state of an Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytic and numerical results demonstrate that the total annealing time is proportional to ln N, where N is the number of spins. This encouraging result is important for the rapid solution of NP-complete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

•Nov 14, 2012•0 citations

Quantum search using non-Hermitian adiabatic evolution

Alexander I. Nesterov, G. P. Berman

We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover's problem [L. K. Grover, Phys. Rev. Lett. 79, 325 (1997)] of finding a marked item inside of an unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters and is not exponentially small. This allows a significant reduction of the searching time, which is proportional to the number of qubits. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment, and we discuss alternate ways to solve $NP$-complete problems.

arXiv:1208.4642•Physical Review A•Nov 12, 2012•10 citations

Analytical solution for ground state estimation by a nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Bin Yan, Nikolai A. Sinitsyn

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 $H_I$ by implementing time evolution with a Hamiltonian $H(t)=H_I+g(t) H_t$. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling $g(t)$ and a specific mixing Hamiltonian $H_t$ that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of $H_I$, 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 $H_I$ annealing schedule. This proves the possibility for spin glasses of a faster than $\sim 1/\log^{\beta} T$ energy relaxation.

•Oct 24, 2021•0 citations

Ising formulation of integer optimization problems for utilizing quantum annealing in iterative improvement strategy

Shuntaro Okada, Masayuki Ohzeki

Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinatorial optimization problems. However, quantum annealing cannot be used to improve a candidate solution obtained by other algorithms because it starts from an initial state where all spin configurations are found with a uniform probability. In this study, we propose an Ising formulation of integer optimization problems to utilize quantum annealing in the iterative improvement strategy. Our formulation exploits the biased sampling of degenerated ground states in transverse magnetic field quantum annealing. We also analytically show that a first-order phase transition is successfully avoided for a fully connected ferromagnetic Potts model if the overlap between a ground state and a candidate solution exceeds a threshold. The proposed formulation is applicable to a wide range of integer optimization problems and enables us to hybridize quantum annealing with other optimization algorithms.

arXiv:2211.03957•Nov 8, 2022•0 citations

Effectiveness of Hybrid Optimization Method for Quantum Annealing Machines

Shuta Kikuchi, Nozomu Togawa, Shu Tanaka

To enhance the performance of quantum annealing machines, several methods have been proposed to reduce the number of spins by fixing spin values through preprocessing. We proposed a hybrid optimization method that combines a simulated annealing (SA)-based non-quantum-type Ising machine with a quantum annealing machine. However, its applicability remains unclear. Therefore, we evaluated the performance of the hybrid method on large-size Ising models and analyzed its characteristics. The results indicate that the hybrid method improves upon solutions obtained by the preprocessing SA, even if the Ising models cannot be embedded in the quantum annealing machine. We analyzed the method from three perspectives: preprocessing, spin-fixed sub-Ising model generation method, and the accuracy of the quantum annealing machine. From the viewpoint of the minimum energy gap, we found that solving the sub-Ising model with a quantum annealing machine results in a higher solution accuracy than solving the original Ising model. Additionally, we demonstrated that the number of fixed spins and the accuracy of the quantum annealing machine affect the dependency of the solution accuracy on the sub-Ising model size.

arXiv:2507.15544•Jul 21, 2025•0 citations

Variational optimization of the quantum annealing schedule for the Lechner-Hauke-Zoller scheme

Yuki Susa, Hidetoshi Nishimori

The annealing schedule is optimized for a parameter in the Lechner-Hauke-Zoller (LHZ) scheme for quantum annealing designed for the all-to-all-interacting Ising model representing generic combinatorial optimization problems. We adapt the variational approach proposed by Matsuura et al. (arXiv:2003.09913) to the annealing schedule of a term representing a constraint for variables intrinsic to the LHZ scheme with the annealing schedule of other terms kept intact. Numerical results for a simple ferromagnetic model and the spin-glass problem show that nonmonotonic annealing schedules optimize the performance measured by the residual energy and the final ground-state fidelity. This improvement does not accompany a notable increase in the instantaneous energy gap, which suggests the importance of a dynamical viewpoint in addition to static analyses in the study of practically relevant diabatic processes in quantum annealing.

arXiv:2012.01694•Physical Review A•Feb 26, 2021•27 citations
We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to lnN\mathrm{ln}NlnN, where NNN is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NP-complete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.
We create a non-Hermitian quantum optimization algorithm to find the ground state of an Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytic and numerical results demonstrate that the total annealing time is proportional to ln N, where N is the number of spins. This encouraging result is important for the rapid solution of NP-complete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.
We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover's problem [L. K. Grover, Phys. Rev. Lett. 79, 325 (1997)] of finding a marked item inside of an unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters and is not exponentially small. This allows a significant reduction of the searching time, which is proportional to the number of qubits. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment, and we discuss alternate ways to solve NPNPNP-complete problems.
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 HIH_IHI​ by implementing time evolution with a Hamiltonian H(t)=HI+g(t)HtH(t)=H_I+g(t) H_tH(t)=HI​+g(t)Ht​. We will assume that the nonadiabatic annealing protocol is defined by a specific decaying coupling g(t)g(t)g(t) and a specific mixing Hamiltonian HtH_tHt​ that make the model analytically solvable arbitrarily far from the adiabatic regime. In specific cases of HIH_IHI​, 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 TTT and uncorrelated from the HIH_IHI​ annealing schedule. This proves the possibility for spin glasses of a faster than ∼1/log⁡βT\sim 1/\log^{\beta} T∼1/logβT energy relaxation.
Quantum annealing is a heuristic algorithm for searching the ground state of an Ising model. Heuristic algorithms aim to obtain near-optimal solutions with a reasonable computation time. Accordingly, many algorithms have so far been proposed. In general, the performance of heuristic algorithms strongly depends on the instance of the combinatorial optimization problem to be solved because they escape the local minima in different ways. Therefore, combining several algorithms to exploit their complementary strength is effective for obtaining highly accurate solutions for a wide range of combinatorial optimization problems. However, quantum annealing cannot be used to improve a candidate solution obtained by other algorithms because it starts from an initial state where all spin configurations are found with a uniform probability. In this study, we propose an Ising formulation of integer optimization problems to utilize quantum annealing in the iterative improvement strategy. Our formulation exploits the biased sampling of degenerated ground states in transverse magnetic field quantum annealing. We also analytically show that a first-order phase transition is successfully avoided for a fully connected ferromagnetic Potts model if the overlap between a ground state and a candidate solution exceeds a threshold. The proposed formulation is applicable to a wide range of integer optimization problems and enables us to hybridize quantum annealing with other optimization algorithms.
To enhance the performance of quantum annealing machines, several methods have been proposed to reduce the number of spins by fixing spin values through preprocessing. We proposed a hybrid optimization method that combines a simulated annealing (SA)-based non-quantum-type Ising machine with a quantum annealing machine. However, its applicability remains unclear. Therefore, we evaluated the performance of the hybrid method on large-size Ising models and analyzed its characteristics. The results indicate that the hybrid method improves upon solutions obtained by the preprocessing SA, even if the Ising models cannot be embedded in the quantum annealing machine. We analyzed the method from three perspectives: preprocessing, spin-fixed sub-Ising model generation method, and the accuracy of the quantum annealing machine. From the viewpoint of the minimum energy gap, we found that solving the sub-Ising model with a quantum annealing machine results in a higher solution accuracy than solving the original Ising model. Additionally, we demonstrated that the number of fixed spins and the accuracy of the quantum annealing machine affect the dependency of the solution accuracy on the sub-Ising model size.
The annealing schedule is optimized for a parameter in the Lechner-Hauke-Zoller (LHZ) scheme for quantum annealing designed for the all-to-all-interacting Ising model representing generic combinatorial optimization problems. We adapt the variational approach proposed by Matsuura et al. (arXiv:2003.09913) to the annealing schedule of a term representing a constraint for variables intrinsic to the LHZ scheme with the annealing schedule of other terms kept intact. Numerical results for a simple ferromagnetic model and the spin-glass problem show that nonmonotonic annealing schedules optimize the performance measured by the residual energy and the final ground-state fidelity. This improvement does not accompany a notable increase in the instantaneous energy gap, which suggests the importance of a dynamical viewpoint in addition to static analyses in the study of practically relevant diabatic processes in quantum annealing.