. We present the complex vector gain-based annealer (CVGA), an analog computing platform designed to overcome energy barriers in X Y Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers using optical or photonic hardware typically represent each X Y spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CVGA addresses these limitations by using two complex fields to represent each X Y spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CVGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for X Y minimization and use them to benchmark CVGA against single-dimension X Y solvers, highlighting the benefits of higher-dimensional operation.. We present the complex vector gain-based annealer (CVGA), an analog computing platform designed to overcome energy barriers in X Y Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers using optical or photonic hardware typically represent each X Y spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CVGA addresses these limitations by using two complex fields to represent each X Y spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CVGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for X Y minimization and use them to benchmark CVGA against single-dimension X Y solvers, highlighting the benefits of higher-dimensional operation.

Mar 1, 20260 citations

Vector Ising spin annealer for minimizing Ising Hamiltonians

James Cummins, Natalia G. Berloff

Abstract Complex optimization problems can be solved via dedicated machines which encode the problem in the couplings of spin Hamiltonians. However, traditional physical minimizers often select excited states due to limitations in spin dynamics. We introduce the Vector Ising Spin Annealer (VISA), a framework in gain-based computing that leverages light-matter interactions. We show that VISA overcomes the limitations by enabling spins to operate within a three-dimensional space, thereby providing a robust solution for effectively minimizing Ising Hamiltonians. Our comparative analysis demonstrates VISA’s superior performance relative to conventional single-dimension spin optimizers, highlighting its capacity to surmount significant energy barriers in intricate landscapes. Detailed studies on cyclic and random graphs reveal VISA’s proficiency in dynamically evolving the energy landscape through time-dependent gain and penalty annealing, underscoring its potential in advancing the field of complex problem-solving in physics-inspired and physics-based computing.Abstract Complex optimization problems can be solved via dedicated machines which encode the problem in the couplings of spin Hamiltonians. However, traditional physical minimizers often select excited states due to limitations in spin dynamics. We introduce the Vector Ising Spin Annealer (VISA), a framework in gain-based computing that leverages light-matter interactions. We show that VISA overcomes the limitations by enabling spins to operate within a three-dimensional space, thereby providing a robust solution for effectively minimizing Ising Hamiltonians. Our comparative analysis demonstrates VISA’s superior performance relative to conventional single-dimension spin optimizers, highlighting its capacity to surmount significant energy barriers in intricate landscapes. Detailed studies on cyclic and random graphs reveal VISA’s proficiency in dynamically evolving the energy landscape through time-dependent gain and penalty annealing, underscoring its potential in advancing the field of complex problem-solving in physics-inspired and physics-based computing.

Communications PhysicsMay 29, 202511 citations

Ising Hamiltonian minimization: Gain-based computing with manifold reduction of soft spins vs quantum annealing

James Cummins, Hayder Salman, Natalia G. Berloff

We investigate the minimization of Ising Hamiltonians, comparing the performance of gain-based computing paradigms based on the dynamics of semiclassical soft-spin models with quantum annealing. We systematically analyze how the energy landscape for the circulant couplings of a Möbius graph evolves with increased annealing parameters. Our findings indicate that these semiclassical models face challenges due to a widening dimensionality landscape. To counteract this issue, we introduce the manifold reduction method, which restricts the soft-spin amplitudes to a defined phase space region. Concurrently, quantum annealing demonstrates a natural capability to navigate the Ising Hamiltonian's energy landscape due to its operation within the comprehensive Hilbert space. Our study indicates that physics-inspired or physics-enhanced optimizers will likely benefit from combining classical and quantum annealing techniques.We investigate the minimization of Ising Hamiltonians, comparing the performance of gain-based computing paradigms based on the dynamics of semiclassical soft-spin models with quantum annealing. We systematically analyze how the energy landscape for the circulant couplings of a Möbius graph evolves with increased annealing parameters. Our findings indicate that these semiclassical models face challenges due to a widening dimensionality landscape. To counteract this issue, we introduce the manifold reduction method, which restricts the soft-spin amplitudes to a defined phase space region. Concurrently, quantum annealing demonstrates a natural capability to navigate the Ising Hamiltonian's energy landscape due to its operation within the comprehensive Hilbert space. Our study indicates that physics-inspired or physics-enhanced optimizers will likely benefit from combining classical and quantum annealing techniques.

arXiv:2311.17359Physical Review ResearchFeb 11, 202514 citations

Adiabatic Quantum Computations and Transverse Field Models

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

Multivariable optimization problems belong to an interdisciplinary field of research connecting diverse fields like computer science, information theory and statistical physics. It involves finding an optimal solution out of the many available states or configurations, possibly satisfying a set of constraints. For physicists, the problem can be visualized as finding the global minima of the energy landscape of a given complex Hamiltonian, which is quite difficult to obtain by the obvious method of examining the energy of each state. This is due to the huge number of available states which grows exponentially with the system size (for N Ising spins, the configuration space grows as 2 N ). The search for such a global minimum becomes all the more complex when the landscape consists of several local minima separated by macroscopic barriers ( O ( N )) arising from frustrating constraints in the systems. To solve such an optimization problem in a stochastic case, the concept of simulated annealing (SA) was put forward by Kirkpatrick et al . [437]. In this method, a tunable noise is artificially induced (thermal-like fluctuations equivalent to that of a metallurgical annealing) which helps the system to escape from the local minima by hopping above the energy barriers, and staying longer in minimum energy valley(s) as the noise variable is slowly reduced to zero starting from a high value. At the same time, it is also known that in complex systems, due to many closely spaced low lying states, the annealing needs to be performed infinitely slowly to reach the true ground state; this is indeed a major drawback. We shall elaborate on this point later when we discuss the effect of system size on the minimum gap between the ground state and the first excited state of the quantum many body system under consideration. The numerical methods used in such studies are variants of Monte Carlo methods.Multivariable optimization problems belong to an interdisciplinary field of research connecting diverse fields like computer science, information theory and statistical physics. It involves finding an optimal solution out of the many available states or configurations, possibly satisfying a set of constraints. For physicists, the problem can be visualized as finding the global minima of the energy landscape of a given complex Hamiltonian, which is quite difficult to obtain by the obvious method of examining the energy of each state. This is due to the huge number of available states which grows exponentially with the system size (for N Ising spins, the configuration space grows as 2 N ). The search for such a global minimum becomes all the more complex when the landscape consists of several local minima separated by macroscopic barriers ( O ( N )) arising from frustrating constraints in the systems. To solve such an optimization problem in a stochastic case, the concept of simulated annealing (SA) was put forward by Kirkpatrick et al . [437]. In this method, a tunable noise is artificially induced (thermal-like fluctuations equivalent to that of a metallurgical annealing) which helps the system to escape from the local minima by hopping above the energy barriers, and staying longer in minimum energy valley(s) as the noise variable is slowly reduced to zero starting from a high value. At the same time, it is also known that in complex systems, due to many closely spaced low lying states, the annealing needs to be performed infinitely slowly to reach the true ground state; this is indeed a major drawback. We shall elaborate on this point later when we discuss the effect of system size on the minimum gap between the ground state and the first excited state of the quantum many body system under consideration. The numerical methods used in such studies are variants of Monte Carlo methods.

Cambridge University Press eBooksJan 28, 20150 citations

Soft vector spins with dimensional annealing for combinatorial optimization

Marvin Syed, Richard Zhipeng Wang, Natalia G. Berloff

Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.Recently, purpose-built analog hardware that can efficiently minimize the Ising energy and thereby solve a variety of combinatorial optimization problems has been receiving widespread attention. In this work, we show how multidimensional, vectorial degrees of freedom, that are either naturally present or can be artificially created in such hardware, could strengthen the capability to find optimal solutions to optimization problems. In order to achieve this, we introduce a simple model of soft vector spins that should be implementable on a variety of analog hardware platforms as well as three different dimensional annealing methods which harness the enlarged phase space of the vectorial degrees of freedom to minimize the Ising energy. We perform simulations on different benchmark problems and show that for all dimensional annealing methods we tested, vectorial degrees of freedom improve upon one-dimensional degrees of freedom when it comes to finding the ground state of the Ising model. In particular, we find that this advantage becomes most pronounced for $d \gtrsim 3$ dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.

arXiv:2604.01003Apr 1, 20260 citations

Quantum-Enhanced Differential Equation Solvers

Jincheng Zhang

Quantum-enhanced differential equation solvers represent a significant advancement in computational mathematics, offering the potential to accelerate the solution of complex mathematical models. This paper explores the feasibility of utilizing quantum algorithms, specifically quantum annealing and variational quantum eigensolver (VQE), to address computationally intensive differential equation problems. We detail the core mechanism of this approach, outlining the challenges and benefits of employing quantum computation for this domain. The paper concludes by highlighting the potential impact of this research on various fields, particularly those dealing with complex systems and high-dimensional mathematical models.Quantum-enhanced differential equation solvers represent a significant advancement in computational mathematics, offering the potential to accelerate the solution of complex mathematical models. This paper explores the feasibility of utilizing quantum algorithms, specifically quantum annealing and variational quantum eigensolver (VQE), to address computationally intensive differential equation problems. We detail the core mechanism of this approach, outlining the challenges and benefits of employing quantum computation for this domain. The paper concludes by highlighting the potential impact of this research on various fields, particularly those dealing with complex systems and high-dimensional mathematical models.

Sep 4, 20260 citations

d \gtrsim 3

dimensional degrees of freedom, with diminishing returns as the dimension is increased further. Our results could inspire new analog optimization hardware and algorithms that explicitly incorporate the advantage of vectorial degrees of freedom.