As a quantum-inspired, non-traditional analog solver architecture, the analog Ising machine (AIM) has emerged as a distinctive computational paradigm to address the rapidly growing demand for computational power. However, the mathematical understanding of its principles, as well as the optimization of its solution speed and accuracy, remain unclear. In this work, we for the first time systematically discuss multiple implementations of AIM and establish a unified mathematical formulation. On this basis, by treating the binarization constraint of AIM (such as injection locking) as a Lagrange multiplier in optimization theory and combining it with a Lyapunov analysis from dynamical systems theory, an analytical framework for evaluating solution speed and accuracy is constructed, and further demonstrate that conventional AIMs possess a theoretical performance upper bound. Subsequently, by elevating the binarization constraint to a control variable, we propose the controllable analog Ising machine (CAIM), which integrates control Lyapunov functions and momentum-based optimization algorithms to realize adaptive sampling-feedback control, thereby surpassing the performance limits of conventional AIMs. In a proof-of-concept CAIM demonstration implemented using an FPGA-controlled LC-oscillator Ising machine, CAIM achieves a twofold speedup and a 7\% improvement in accuracy over AIM on a 50-node all-to-all weighted MaxCut problem, validating both the effectiveness and interpretability of the proposed theoretical framework.
As a quantum-inspired, non-traditional analog solver architecture, the analog Ising machine (AIM) has emerged as a distinctive computational paradigm to address the rapidly growing demand for computational power. However, the mathematical understanding of its principles, as well as the optimization of its solution speed and accuracy, remain unclear. In this work, we for the first time systematically discuss multiple implementations of AIM and establish a unified mathematical formulation. On this basis, by treating the binarization constraint of AIM (such as injection locking) as a Lagrange multiplier in optimization theory and combining it with a Lyapunov analysis from dynamical systems theory, an analytical framework for evaluating solution speed and accuracy is constructed, and further demonstrate that conventional AIMs possess a theoretical performance upper bound. Subsequently, by elevating the binarization constraint to a control variable, we propose the controllable analog Ising machine (CAIM), which integrates control Lyapunov functions and momentum-based optimization algorithms to realize adaptive sampling-feedback control, thereby surpassing the performance limits of conventional AIMs. In a proof-of-concept CAIM demonstration implemented using an FPGA-controlled LC-oscillator Ising machine, CAIM achieves a twofold speedup and a 7% improvement in accuracy over AIM on a 50-node all-to-all weighted MaxCut problem, validating both the effectiveness and interpretability of the proposed theoretical framework.
Solving optimization problems is challenging for existing digital computers and even for future quantum hardware. The practical importance of diverse problems, from healthcare to financial optimization, has driven the emergence of specialised hardware over the past decade. However, their support for problems with only binary variables severely restricts the scope of practical problems that can be efficiently embedded. We build analog iterative machine (AIM), the first instance of an opto-electronic solver that natively implements a wider class of quadratic unconstrained mixed optimization (QUMO) problems and supports all-to-all connectivity of both continuous and binary variables.Beyond synthetic 7-bit problems at small-scale, AIM solves the financial transaction settlement problem entirely in analog domain with higher accuracy than quantum hardware and at room temperature. With compute-in-memory operation and spatial-division multiplexed representation of variables, the design of AIM paves the path to chip-scale architecture with 100 times speed-up per unit-power over the latest GPUs for solving problems with 10,000 variables. The robustness of the AIM algorithm at such scale is further demonstrated by comparing it with commercial production solvers across multiple benchmarks, where for several problems we report new best solutions. By combining the superior QUMO abstraction, sophisticated gradient descent methods inspired by machine learning, and commodity hardware, AIM introduces a novel platform with a step change in expressiveness, performance, and scalability, for optimization in the post-Moores law era.
Analog computing using bosonic computational states is a leading approach to surpassing the computational speed and energy limitations of von Neumann architectures. But the challenges of manufacturing large-scale photonic integrated circuits (PIC) has led to hybrid solutions that integrate optical analog and electronic digital components. A notable example is the coherent Ising machine (CIM), that was primarily invented for solving quadratic binary optimization problems. In this paper, we focus on a mean-field interpretation of the dynamics of optical pulses in the CIM as solutions to Langevin dynamics, a stochastic differential equation (SDE) that plays a key role in non-convex optimization and generative AI. This interpretation establishes a computational framework for understanding the system's operation, the computational role of each component, and its performance, strengths, and limitations. We then infer that the CIM is inherently a continuous state machine, capable of integrating a broad range of SDEs, in particular for solving a continuous global (or mildly constrained) optimization problems. Nevertheless, we observe that the iterative digital-to-analog and analog-to-digital conversions within the protocol create a bottleneck for the low power and high speed of optics to shine. This observation underscores the need for major advances in PIC technologies as we envision that fully analog opto-electronic realizations of such experiments can open doors for broader applications, and orders of magnitude improvements in speed and energy consumption.
Abstract Solving large-scale Quadratic Unconstrained Binary Optimization (QUBO) problems is critical in various fields, including finance, physics, biology and engineering. However, these problems remain intractable on conventional computing architectures. As a result, alternative solvers—such as Ising Machines (IMs) based on networks of coupled electronic, mechanical or photonic parametric oscillators (POs)—have recently been developed. PO-based Ising Machines (IMs) aim to find the ground state of an Ising Hamiltonian, which encodes the solution to a QUBO problem. However, they rely on an energy minimization process based on gradient descent, making them inherently susceptible to getting trapped in local minima and, as a result, to identifying inaccurate solutions. In this work, we introduce and validate a QUBO solver—the Analog Floquet Solver (AFS)—which enhances the dynamics of PO-based Ising machines (IMs) by leveraging Floquet states that emerge spontaneously in POs coupled to high quality-factor resonances. These states enable the AFS to embed periodic time modulation into its energy minimization process, allowing it to escape local minima during the search for QUBO problems’ solutions. As a result, the AFS significantly increases the likelihood of identifying accurate solutions compared to conventional PO-based IMs. More generally, this work establishes a new paradigm in analog computing that can be physically implemented using existing technologies across a variety of physical domains.
With the growing demand for efficient computing to tackle complex real-world challenges like combinatorial optimization, both academia and industry are exploring alternative computing paradigms. The Ising machine (IM) is one such approach, solving optimization problems by mapping their cost functions onto the Ising model. In this framework, finding the optimal solution equates to determining the Ising spin configuration corresponding to the ground state of the Ising Hamiltonian. Analog IMs, relying on artificial spins that can take analogue rather than binary values, have shown great potential in reaching this ground at high speeds. Optical implementations, such as the opto-electronic IM developed within our research group, are particularly promising due to their higher bandwidth, energy efficiency, and lower latency compared to electronic IMs. However, scaling remains a challenge for large problems. To overcome this, we propose a novel analog optical IM utilizing a spatial light modulator (SLM). Our design differs from other IMs relying on SLMs by employing analog spin variables instead of binary ones and leveraging gradient descent-based feedback rather than a computer-driven Metropolis–Hastings sampling process. By exploiting the inherent parallelism of light, this SLM-based IM is expected to efficiently handle optimization problems with millions of spins while still maintaining high energy efficiency. At the conference, we will also offer more insight into the performance of these analogue IMs and will delve deeper in the bit resolution required for the different optical components such as the employed SLM. Based on the dynamical evolution of the IM, we will discuss strategies to further improve the problem-solving capacity and speed of these optical IMs.
Solving large-scale quadratic unconstrained binary optimization (QUBO) problems is critical in various fields, including physics, finance, and engineering. However, these problems remain intractable on conventional computing architectures. Alternative solvers, such as Ising machines (IMs) based on networks of coupled electronic, mechanical, or photonic parametric oscillators (POs), have recently been developed. PO-based IMs aim to find the ground state of an Ising Hamiltonian, which encodes the solution to a QUBO problem. However, their analog nature and their energy-minimization process based on gradient descent make PO-based IMs inherently susceptible to identifying inaccurate solutions. In this work, we introduce and validate a QUBO solver—the analog Floquet solver (AFS)—which enhances the dynamics of PO-based IMs by leveraging Floquet states that emerge spontaneously in POs coupled to high-quality-factor resonances. These states enable the AFS to embed periodic time modulation into its energy-minimization process, allowing it to escape local minima during the search for QUBO problem solutions. As a result, the AFS significantly increases the likelihood of identifying accurate solutions compared to conventional PO-based IMs. More generally, this work defines a new paradigm in analog computing—spanning both classical and quantum realms—that can be physically realized with existing technologies across diverse physical domains.
Simulated Adiabatic Bifurcation (aSB) is a quantum-inspired algorithm that provides approximate solutions for large-scale optimization problems using the Ising model. It emulates the quantum adiabatic evolution of a network of non-linear Kerr oscillators on classical platforms. These oscillators undergo bifurcation, where each branch corresponds to a spin state, with the network creating an energy imbalance to deter-mine the optimal solution. This approach is highly parallelizable, making it suitable for implementation on GPUs and FPGAs. However, classical emulation introduces analog errors, potentially compromising performance. To address this, alternative methods like ballistic (bSB) and discrete (dSB) evolutions were developed, with the ballistic approach further enhanced by thermal fluctuations (HbSB). Our comprehensive analysis of bSB, dSB, and HbSB using benchmark Max-Cut and knapsack problems aimed to identify the best balance of speed, area, and accuracy. We found that fixed-point number representation allows more algorithm iterations within the same timeframe as floating-point representation. The proposed FPGA architecture implements dSB and its heated version, leveraging high parallelizability and efficient memory or-ganization. This design supports larger coefficients and improves problem-solving efficiency without multipliers for matrix-vector evaluation. Challenges include mitigating data dependency between matrix-vector multiplication and time evolution, and developing a preconditioning method for adapting problem coefficients, making the architecture suitable for real-world applications and potential ASIC release.