Ising machines are promising next-generation computers that aim to efficiently obtain good solutions to combinatorial optimization problems, which are converted to the Ising model. To obtain the ground state of the Ising model with high accuracy, tuning the hyperparameters in the Ising model is of great importance. Recently, a new method called Energy Landscape Transformation of Ising Problems (ELTIP) has been proposed which does not change any of the original eigenenergies at all. The previous study demonstrated that the Time-to-Solution (TTS) of Quantum Annealing (QA) decreases for the specific problems after ELTIP. We reveal the scope of the problems where ELTIP can be sufficiently beneficial. To this end, we analyze the correlation between the mean Hamming distance and the ground state probability in both the static and dynamic analysis.
Ising machines are promising next-generation computers that aim to efficiently obtain good solutions to combinatorial optimization problems, which are converted to the Ising model. To obtain the ground state of the Ising model with high accuracy, tuning the hyperparameters in the Ising model is of great importance. Recently, a new method called Energy Landscape Transformation of Ising Problems (ELTIP) has been proposed which does not change any of the original eigenenergies at all. The previous study demonstrated that the Time-to-Solution (TTS) of Quantum Annealing (QA) decreases for the specific problems after ELTIP. We reveal the scope of the problems where ELTIP can be sufficiently beneficial. To this end, we analyze the correlation between the mean Hamming distance and the ground state probability in both the static and dynamic analysis.
We have proposed the energy landscape transformation of Ising problems (ELTIP), which changes the combination of the state and eigenvalue without changing all the original eigenvalues [arXiv:2202.05927]. We study how the ELTIP affects the anti-crossing between two levels of the ground and first excited states during quantum annealing. We use a 5-spin maximum-weighted independent set for the problem to numerically investigate the anticrossing. For comparison, we introduce a non-stoquastic Hamiltonian that adds antiferromagnetic interaction to the normal transverse magnetic field. Annealing with the non-stoquastic Hamiltonian is effective for difficult problems. The non-stoquastic Hamiltonian mitigates the anti-crossing when only the energy gap between the ground state and the first excited state of the final state is small. When the ELTIP is used, the anti-crossing disappears. For the problems investigated in this paper, the ELTIP shortens the annealing time to guarantee adiabatic change more than the non-stoquastic Hamiltonian.
Ising machines are a promising approach to solve combinatorial optimization problems. They map these problems onto the Ising model and search for low-energy configurations. However, navigating the rugged energy landscapes of these systems remains difficult. To improve this navigation, classical adiabatic annealing has been proposed in the literature as a heuristic optimization method for classical Ising machines. Using this technique, the Hamiltonian of the Ising machine is gradually transformed from an easily solvable Hamiltonian to the target Hamiltonian. However, its purported effectiveness is primarily motivated by an analogy to quantum adiabatic annealing, and systematic benchmarking has remained limited.
In this work, we analyze the classical adiabatic annealing technique using continuation methods. Motivated by insights from this analysis, we propose an optimized annealing strategy we refer to as hybrid classical adiabatic annealing. We benchmark our proposed strategy using MaxCut instances with up to 800 spins and problems with external fields, for which it achieves a marginal improvement for a limited set of problems. We conclude that, although theoretically motivated and occasionally beneficial, the hybrid strategy does not offer a sufficient practical advantage over simpler, existing techniques.
Quantum annealing tends to be more difficult as the energy landscape of the problem becomes complicated with many local minima. We have found a transformation for changing the energy landscape that swaps the eigenvalues and paired states without changing the eigenvalues of the instance at all. The transformation is basically a partial recombination of the two-spin interaction coefficient Jij and the longitudinal magnetic field interaction coefficient hi. The Hamming distance corresponding to a barrier between the states changes by the transformation, which in turn affects the ground state convergence. In the quantum annealing simulation results of a small number of spin instances, the annealing time was shortened by several orders of magnitude by applying the transformation. In addition, we also obtained a result using a D-Wave quantum annealer, which also showed a big improvement in the ground state convergence.
Annealing machines such as quantum annealing machines and semiconductor-based annealing machines have been attracting attention as an efficient computing alternative for solving combinatorial optimization problems. They solve original combinatorial optimization problems by transforming them into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an annealing machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one, and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising model's bit-widths. The proposed method is composed of two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that annealing machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.
Combinatorial problems such as combinatorial optimization and constraint satisfaction problems arise in decision-making across various fields of science and technology. In real-world applications, when multiple optimal or constraint-satisfying solutions exist, enumerating all these solutions is often desirable, as it provides flexibility in decision-making. However, combinatorial problems and their enumeration versions pose significant computational challenges due to combinatorial explosion. To address these challenges, we propose enumeration algorithms for combinatorial optimization and constraint satisfaction problems using Ising machines. Ising machines are specialized devices designed to efficiently solve combinatorial problems by exploring the energy landscape of an Ising model. Ising machines typically sample lower-energy solutions with higher probability. Our enumeration algorithms repeatedly perform such sampling to collect all desirable solutions. The crux of the proposed algorithms lies in their stopping criteria for sampling-based energy landscape exploration, which are derived from probability theory. In particular, the proposed algorithms have theoretical guarantees that the failure probability of enumeration is bounded above by a user-specified value, provided that lower-cost solutions are sampled more frequently and equal-cost solutions are sampled with equal probability. Many physics-based Ising machines are expected to (approximately) satisfy these conditions. As a demonstration, we applied our algorithm using simulated annealing to maximum clique enumeration on random graphs. We found that our algorithm enumerates all maximum cliques in large, dense graphs faster than a conventional branch-and-bound algorithm specifically designed for maximum clique enumeration. These findings underscore the effectiveness and potential of our proposed approach.
Ising machines such as quantum annealing machines and semiconductor-based annealing machines can solve various combinatorial optimization problems very efficiently by transforming it into a data structure called an Ising model. At that time, the bit-widths of the coefficients of the Ising model have to be kept within the range that an Ising machine can deal with. However, by reducing the Ising-model bit-widths, its minimum energy state, or ground state, may become different from that of the original one and hence the targeted combinatorial optimization problem cannot be well solved. This paper proposes an effective method for reducing Ising-model bit-widths. The proposed method is composed of the two processes: First, given an Ising model with large coefficient bit-widths, the shift method is applied to reduce its bit-widths roughly. Second, the spin-adding method is applied to further reduce its bit-widths to those that Ising machines can deal with. Without adding too many extra spins, we efficiently reduce the coefficient bit-widths of the original Ising model. Furthermore, the ground state before and after reducing the coefficient bit-widths is not much changed in most of the practical cases. Experimental evaluations demonstrate the effectiveness of the proposed method, compared to existing methods.