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arXiv:2411.19604 · Physical review. E

Continuous approximation of the Ising Hamiltonian: Exact ground states and applications to fidelity assessment in Ising machines

Amirhossein Rezaei, Mahmood Hasani, Alireza Rezaei, Seyyed M. H. Halataei

Published Jul 14, 2025• 1 citation
arXivHTMLPDF

Abstract

In this study, we present an alternative analytical approach to solving large-scale Ising problems by reformulating the discrete Ising Hamiltonian into a continuous framework. This transformation enables us to derive exact solutions for a nontrivial class of fully connected Ising models. To validate our method, we conducted numerical experiments comparing our analytical solutions with those obtained from a quantum-inspired Ising algorithm and a quantum Ising machine. The results demonstrate that the quantum-inspired algorithm and brute-force method successfully align with our solutions, while the quantum Ising machine exhibits notable deviations. Our method offers promising avenues for analytically solving diverse Ising problem instances, while the class of Ising problems addressed here provides a robust framework for assessing the fidelity of Ising machines.

DOI
10.1103/w88d-7rqz

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In this study, we present an alternative analytical approach to solving large-scale Ising problems by reformulating the discrete Ising Hamiltonian into a continuous framework. This transformation enables us to derive exact solutions for a nontrivial class of fully connected Ising models. To validate our method, we conducted numerical experiments comparing our analytical solutions with those obtained from a quantum-inspired Ising algorithm and a quantum Ising machine. The results demonstrate that the quantum-inspired algorithm and brute-force method successfully align with our solutions, while the quantum Ising machine exhibits notable deviations. Our method offers promising avenues for analytically solving diverse Ising problem instances, while the class of Ising problems addressed here provides a robust framework for assessing the fidelity of Ising machines.
Various combinatorial optimization NP-hard problems can be reduced to finding the minimizer of an Ising model, which is a discrete mathematical model. It is an intellectual challenge to develop some mathematical tools or algorithms for solving the Ising model. Over the past decades, some continuous approaches or algorithms have been proposed from physical, mathematical or computational views for optimizing the Ising model such as quantum annealing, the coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. However, the mathematical principle of these algorithms is far from being understood. In this paper, we reveal the mathematical mechanism of dynamical system algorithms for the Ising model by Morse theory and variational methods. We prove that the dynamical system algorithms can be designed to minimize a continuous function whose local minimum points give all the candidates of the Ising model and the global minimum gives the minimizer of Ising problem. Using this mathematical mechanism, we can easily understand several dynamical system algorithms of the Ising model such as the coherent Ising machine, the Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm. Furthermore, motivated by the works of C. Conley, we study transit and capture properties of the simulated bifurcation algorithm to explain its convergence by the low energy transit and capture in celestial mechanics. A detailed discussion on 222-spin and 333-spin Ising models is presented as application.
Many combinatorial optimization problems can be reformulated as finding the ground state of the Ising model. Existing Ising solvers are mostly inspired by simulated annealing. Although annealing techniques offer scalability, they lack convergence guarantees and are sensitive to the cooling schedule. We propose solving the Ising problem by relaxing the binary spins to continuous variables and introducing an attraction potential that steers the solution toward binary spin configurations. A key property of this potential is that its combination with the Ising energy produces a Hamiltonian that can be written as a difference of convex polynomials. This enables us to design efficient iterative algorithms that require a single matrix-vector multiplication per iteration and provide convergence guarantees. We implement our Ising solver on a wide range of GPU platforms, from edge devices to high-performance computing clusters, and demonstrate that it consistently outperforms existing solvers across problem sizes ranging from small (10310^3103 spins) to ultra-large (10810^8108 spins).
Unconventional computers based on superconducting qubits [1] and analog/digital hardware accelerators [2]–[5] have been developed and solved hard combinatorial optimization problems (COPs) with higher efficiency than classical computers by integrating the Ising models in hardware and observing their natural convergence behavior towards the lower Ising Hamiltonian (or energy). Ouantum annealer based on superconducting qubits [1] has demonstrated its capability of finding solutions to COPs using the quantum tunneling effect. However, the quantum computer requires operating at ultra-low temperatures (15-20mK), consumes excessive power (>10kW), and is not scalable yet to solve real-world problems. Recently, ASIC hardware accelerators [2]–[5] using low-cost CMOS technologies have implemented the Ising model by integrating many physical spins (i.e., computing units of the Ising model) to accelerate the finding of solutions to hard COPs. The hardware accelerators, so-called Ising machines, solved hard COPs much faster than classical computers and consumed several orders of magnitude lower energy. Nevertheless, they still have to overcome the fundamental scalability issues for solving large-scale COPs in the real world. The performance of existing discrete-time Ising machines has been limited by the exponentially increasing number of operation cycles when the problem size increases for the large-scale COPs. in addition, prior works have not been able to integrate massive random number generators (RNG), the essential building blocks of the Ising machines for finding better solutions, that could occupy a significant hardware footprint. Prior works have implemented them off-chip, or a few have been integrated on-chip and shared by many spins. This paper proposes a novel continuous-time analog Ising machine based on coupled inveiter chain circuits to address the scalability issues of the existing discrete-time Ising machines. Besides the benefits of the continuous-time operation, which significantly lowers the computing energy and latency, we introduce an equalization method for finding better solutions without RNGs.
Hard combinatorial optimization problems, often mapped to Ising models, promise potential solutions with quantum advantage but are constrained by limited qubit counts in near-term devices. We present an innovative quantum-inspired framework that dynamically compresses large Ising models to fit available quantum hardware of different sizes. Thus, we aim to bridge the gap between large-scale optimization and current hardware capabilities. Our method leverages a physics-inspired GNN architecture to capture complex interactions in Ising models and accurately predict alignments among neighboring spins (aka qubits) at ground states. By progressively merging such aligned spins, we can reduce the model size while preserving the underlying optimization structure. It also provides a natural trade-off between the solution quality and size reduction, meeting different hardware constraints of quantum computing devices. Extensive numerical studies on Ising instances of diverse topologies show that our method can reduce instance size at multiple levels with virtually no losses in solution quality on the latest D-Wave quantum annealers. Additionally, we demonstrate that GNN architectures can address the parameter initialization challenge in variational quantum algorithms, specifically QAOA for MaxCut problems, by learning structure-to-parameter mappings that achieve performance comparable with physics-based heuristics when trained on sufficiently large datasets.
Obtaining exact solutions to the Schrödinger equation for atoms, molecules, and extended systems continues to be a “Holy Grail” problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, “Can these machines be used to perform electronic structure calculations?” Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H 2, He 2, HeH +, and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. This is an early step in developing generalized methods on such devices for chemical physics.
Ground-state fidelity (GSF) and quantum renormalization group theory (QRG) have proven useful tools in the study of quantum critical systems. Here we lay out a general, unified formalism of GSF and QRG; specifically, we propose a method to calculate GSF through QRG, obviating the need for calculating or approximating ground states. This method thus enhances characterization of quantum criticality as well as scaling analysis of relevant properties with system size. We illustrate the formalism in the one-dimensional Ising model in a transverse field and the anisotropic spin-1/2 Heisenberg model.