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Vacuum and Surface Science

Principle and Applications of Simulated Bifurcation Machine

Hayato Goto

Published Mar 9, 2020• 0 citations
Publisher

Abstract

Recently, we have proposed a new heuristic algorithm for combinatorial optimization inspired by our proposed quantum computer. We named the algorithm “simulated bifurcation (SB) algorithm,” because it is based on the numerical simulation of nonlinear Hamiltonian systems exhibiting bifurcations. Here we present the principle, performance, and applications of our SB machine.

DOI
10.1380/vss.63.129

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Recently, we have proposed a new heuristic algorithm for combinatorial optimization inspired by our proposed quantum computer. We named the algorithm “simulated bifurcation (SB) algorithm,” because it is based on the numerical simulation of nonlinear Hamiltonian systems exhibiting bifurcations. Here we present the principle, performance, and applications of our SB machine.
Combinatorial optimization problems are ubiquitous but difficult to solve. Hardware devices for these problems have recently been developed by various approaches, including quantum computers. Inspired by recently proposed quantum adiabatic optimization using a nonlinear oscillator network, we propose a new optimization algorithm simulating adiabatic evolutions of classical nonlinear Hamiltonian systems exhibiting bifurcation phenomena, which we call simulated bifurcation (SB). SB is based on adiabatic and chaotic (ergodic) evolutions of nonlinear Hamiltonian systems. SB is also suitable for parallel computing because of its simultaneous updating. Implementing SB with a field-programmable gate array, we demonstrate that the SB machine can obtain good approximate solutions of an all-to-all connected 2000-node MAX-CUT problem in 0.5 ms, which is about 10 times faster than a state-of-the-art laser-based machine called a coherent Ising machine. SB will accelerate large-scale combinatorial optimization harnessing digital computer technologies and also offer a new application of computational and mathematical physics.
Simulated Bifurcation (SB) algorithms, inspired by quantum annealing, can efficiently solve large-scale combinatorial optimization problems on classical hardware, often outperforming traditional approaches such as simulated annealing. However, their tendency to be trapped in local optima limits global solution quality. In this work, we introduce Tabu-Enhanced Simulated Bifurcation (TESB), an improved SB variant that incorporates a Tabu Search-inspired mechanism. By leveraging a dynamic penalty guided by early search history, TESB can naturally avoid revisiting suboptimal regions. On Max-Cut benchmarks, TESB achieves up to a three-order-of-magnitude reduction in Time-to-Solution compared to standard SB. When applied to particle track reconstruction in high-energy physics, TESB identifies lower-energy configurations on problems exceeding 100,000 spin variables, demonstrating enhanced scalability and performance across a wide range of combinatorial tasks. Combinatorial optimization is a challenging yet crucial class of problems, and Simulated Bifurcation (SB) is a promising approximate solver developed in recent years. The authors enhance SB with ideas from Tabu Search to improve its performance, demonstrating clear gains on Max-Cut and particle track reconstruction problems.
Abstract High-performance Ising machines for solving combinatorial optimization problems have been developed with digital processors implementing heuristic algorithms such as simulated bifurcation (SB). Although Ising machines have been designed for second-order cost functions, there are practical problems expressed naturally by higher-order cost functions. In this work, we extend SB to such higher-order cost functions. By solving a problem having third-order cost functions, we show that the higher-order SB can outperform not only the second-order SB with additional spin variables, but also simulated annealing applied directly to the third-order cost functions. This result suggests that the higher-order SB can be practically useful.
Combinatorial optimization problems are economically valuable but computationally hard to solve. Many practical combinatorial optimizations can be converted to the ground-state search problems of Ising spin models. Simulated bifurcation (SB) is a quantum-inspired algorithm to solve these Ising problems. One of the remarkable features of SB is the high-degree parallelism, providing an opportunity for quickly solving those problems by massively parallel processing. In this article, we review our recent works on the design and implementation of high-performance FPGA-based accelerators for SB and their applications toward innovative real-time systems that make optimal responses to ever-changing situations. An example of such an application is an ultrafast financial transaction machine that detects the most profitable cross-currency arbitrage opportunities at microsecond speeds. Also, we show a scale-out architecture for SB-based Ising machines with all-to-all spin-spin couplings that allows continued scaling of both machine size and computational throughput by connecting multiple chips, rather than scaling up a single chip.
We introduce Simulated Bifurcation Quantum Annealing (SBQA), a quantum-inspired optimization algorithm that extends simulated bifurcation by incorporating inter-replica interactions to mimic quantum tunneling. SBQA retains the efficiency and parallelism of simulated bifurcation while improving performance on sparse and rugged energy landscapes. We derive its equations of motion, analyze parameter dependence, and propose a lightweight auto-tuning strategy. A comprehensive benchmarking study on both large-scale problems and smaller instances relevant for current quantum hardware shows that SBQA systematically improves on SBM in the sparse and rugged regimes where SBM is known to struggle, while remaining competitive and versatile across a diverse set of tested problem families. These results position SBQA as a practical quantum-inspired optimization heuristic and a stronger classical baseline for the sparse and rugged regimes studied here.
Combinatorial optimization problems are economically valuable but computationally hard to solve. Many practical combinatorial optimizations can be converted to the ground-state search problems of Ising spin models. Simulated bifurcation (SB) is a quantum-inspired algorithm to solve these Ising problems. One of the remarkable features of SB is the high-degree parallelism, providing an opportunity for quickly solving those problems by massively parallel processing. In this article, we review our recent works on the design and implementation of high-performance FPGA-based accelerators for SB and their applications toward innovative real-time systems that make optimal responses to ever-changing situations. An example of such an application is an ultrafast financial transaction machine that detects the most profitable cross-currency arbitrage opportunities at microsecond speeds. Also, we show a scale-out architecture for SB-based Ising machines with all-to-all spin-spin couplings that allows continued scaling of both machine size and computational throughput by connecting multiple chips, rather than scaling up a single chip.