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arXiv:2211.09296 · Applied Physics Express

Simulated bifurcation for higher-order cost functions

Taro Kanao, Hayato Goto

Published Dec 14, 2022• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.35848/1882-0786/acaba9

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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.
Since many combinatorial optimization problems can be mapped onto ground-state search problems of Ising models, special-purpose machines for Ising problems have attracted intense attention. Simulated bifurcation (SB) is a recently proposed algorithm to solve these Ising problems. One of the remarkable features of SB is the high-degree parallelism underlying in the algorithm, providing an opportunity to solve the Ising problems very fast by massively parallel processing. In this work, we implement the SB algorithm on FPGAs by designing massively parallel custom circuits. We then compare the FPGA-based SB machines with a state-of-the-art machine called a coherent Ising machine (CIM), a highly optimized implementation of simulated annealing (SA), and GPU-based SB machines. SB machines with spin size of 2,048/4,096 (2K/4K) on an Arria10 GX1150 FPGA have 8,192 processing elements for the matrix-vector multiplication (MM) modules (the most computationally intensive part) and achieve computation throughput of 1,873/2,027 GMAC/s for the MM modules, outperforming 2K/4K SB machines on an Nvidia Tesla V100 GPU (113/183 GMAC/s). The 2K FPGA-SB solves all-to-all connected 2000-node MAX-CUT problem 14X (/124X) faster than the CIM (/the highly-optimized SA), with much better energy efficiency (288X better than the CIM).
Developed from the Ising model, Ising machines are promising to be efficient domain-specific accelerators for solving combinatorial optimization problems (COPs). A simulated bifurcation (SB) algorithm enables an Ising machine to achieve massive parallelism in the simulation of Hamiltonian dynamics. Although SB significantly accelerates the search, more resources are required due to the use of continuous variables for the position of oscillators to obtain discrete spin states, compared to conventional simulated annealing. This article proposes ternary and multiple-value quantized SB (qSB) algorithms by discretizing the position variables used for the hardware-consuming multiply-accumulate (MAC) operations in SB. These quantization schemes do not only reduce the computational complexity, but also improve the solution quality for COPs. Specifically, the ternary qSB with dynamic threshold settings converts the MAC into addition and subtraction. To improve the precision in number representation when solving large-scale COPs, a uniform quantization scheme is applied to provide multiple-valued quantization. Alternatively, a logarithmic qSB leverages the evolution characteristics of position variables and implements multiplication by using simple shift operations. We demonstrate that using the proposed qSB improves the solution quality in a long search and accelerates energy convergence in a short search for solving COPs tackled by up to 2000 fully connected spins.
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.
Many combinatorial optimization problems can be solved by numerically simulating classical nonlinear Hamiltonian systems based on the Ising model. Solving the traveling salesman problem (TSP) using the Ising model requires a quadratically increasing number of spins with strict constraints. Unlike classical simulated annealing, simulated bifurcation (SB) can update the states of spins in parallel. This feature can potentially accelerate the convergence of Hamiltonian in the Ising model by taking advantage of modern multi-core processors. As an improved SB algorithm, the ballistic SB (bSB) algorithm is considered for solving the TSP in this paper. The TSP is converted to an Ising problem with external magnetic fields. bSB is then expanded by introducing a time-dependent factor. Experiments on benchmark datasets show that the bSB-based Ising solver offers superior performance in solution quality and convergence speed.
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.
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.