The key limitation of physics-inspired hardware accelerators such as Ising machines (IM) or Hopfield neural networks (HNN) is their lack of support of higher order couplings of spins/neurons. As a result, many classes of combinatorial optimization problems featuring polynomial interactions (PUBO) require quadratic hardware embeddings, QUBO mappings. Such embeddings were shown to risk severe performance reduction due to mismatched configuration space features (energy landscape) compared to the native space formulation. Here, we propose a gradient matching (GM) algorithm which leverages existing IM or in-memory HNN architectures and approximately recovers the native energy landscape. We show an exponential scaling advantage of the GM algorithm time-to-solution. Finally, we propose a hardware design to physically realize our GM in an HNN-based solver.
The key limitation of physics-inspired hardware accelerators such as Ising machines (IM) or Hopfield neural networks (HNN) is their lack of support of higher order couplings of spins/neurons. As a result, many classes of combinatorial optimization problems featuring polynomial interactions (PUBO) require quadratic hardware embeddings, QUBO mappings. Such embeddings were shown to risk severe performance reduction due to mismatched configuration space features (energy landscape) compared to the native space formulation. Here, we propose a gradient matching (GM) algorithm which leverages existing IM or in-memory HNN architectures and approximately recovers the native energy landscape. We show an exponential scaling advantage of the GM algorithm time-to-solution. Finally, we propose a hardware design to physically realize our GM in an HNN-based solver.
Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
Abstract Ising Machines are emerging hardware architectures that efficiently solve NP-hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ∼ ~ 10000 × × performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5 − − 4 × × over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.
Physics-based Ising machines (IM) have been developed as dedicated processors for solving hard combinatorial optimization problems with higher speed and better energy efficiency. Generally, such systems employ local search heuristics to traverse energy landscapes in searching for optimal solutions. Here, we quantify and address some of the major challenges met by IMs by extending energy-landscape geometry visualization tools known as disconnectivity graphs. Using efficient sampling methods, we visually capture landscapes of problems having diverse structure and hardness manifesting as energetic and entropic barriers for IMs. We investigate energy barriers, local minima, and configuration space clustering effects caused by locality reduction methods when embedding combinatorial problems to the Ising hardware. To this end, we sample disconnectivity graphs of PUBO energy landscapes and their different QUBO mappings accounting for both local minima and saddle regions. We demonstrate that QUBO energy-landscape properties lead to the subpar performance for quadratic IMs and suggest directions for their improvement.
As a dedicated quantum device, Ising machines could solve large-scale binary optimization problems in milliseconds. There is emerging interest in utilizing Ising machines to train feedforward neural networks due to the prosperity of generative artificial intelligence. However, existing methods can only train single-layer feedforward networks because of the complex nonlinear network topology. This paper proposes an Ising learning algorithm to train quantized neural network (QNN), by incorporating two essential techinques, namely binary representation of topological network and order reduction of loss function. As far as we know, this is the first algorithm to train multi-layer feedforward networks on Ising machines, providing an alternative to gradient-based backpropagation. Firstly, training QNN is formulated as a quadratic constrained binary optimization (QCBO) problem by representing neuron connection and activation function as equality constraints. All quantized variables are encoded by binary bits based on binary encoding protocol. Secondly, QCBO is converted to a quadratic unconstrained binary optimization (QUBO) problem, that can be efficiently solved on Ising machines. The conversion leverages both penalty function and Rosenberg order reduction, who together eliminate equality constraints and reduce high-order loss function into a quadratic one. With some assumptions, theoretical analysis shows the space complexity of our algorithm is O(H2L+HLNlogH), quantifying the required number of Ising spins. Finally, the algorithm effectiveness is validated with a simulated Ising machine on MNIST dataset. After annealing 700 ms, the classification accuracy achieves 98.3%. Among 100 runs, the success probability of finding the optimal solution is 72%. Along with the increasing number of spins on Ising machine, our algorithm has the potential to train deeper neural networks.
Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output nearoptimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minorembedding of QUBO problems and the performance of Ising machines.
Ising machines, including quantum annealing machines, are promising next-generation computers for combinatorial optimization problems. Due to hardware limitations, most Ising-type hardware can only solve functions expressed in linear or quadratic terms of binary variables. Therefore, problems with higher-order terms require order reduction, which increases the number of variables and constraints and may degrade solution quality. This study evaluates the effectiveness of directly solving such problems without order reduction by using the polynomial unconstrained binary optimization (PUBO) solver implemented in Toshiba's Simulated Quantum-inspired Bifurcation Machine+ (SQBM+). We compare its performance against a conventional quadratic unconstrained binary optimization (QUBO) solver on the same hardware platform. As benchmarks, we use a vehicle routing problem with distance balancing and the low autocorrelation binary sequence (LABS) problem, both of which naturally include higher-order interactions. Results show that the PUBO solver consistently achieves superior solution quality and stability compared to its QUBO counterpart, while maintaining comparable computational time and requiring no order-reduction compilation, indicating the potential advantages of directly handling higher-order terms in practical optimization problems.