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Ising machines are specialized computers for finding the lowest energy states of Ising spin models, onto which many practical combinatorial optimization problems can be mapped. Simulated bifurcation (SB) is a quantum-inspired parallelizable algorithm for Ising problems that enables scalable multi-chip implementations of Ising machines. However, the computational performance of a previously proposed multi-chip architecture tends to saturate as the number of chips increases for a given problem size because computation and communication are exclusive in the time domain. In this paper, we propose a streaming architecture for multi-chip implementations of SB-based Ising machines with full spin-to-spin connectivity. The data flow in in-chip computation is harmonized with the data flow in inter-chip communication, enabling the computation and communication to overlap and the communication time to be hidden. Systematic experiments demonstrate linear strong scaling of performance up to the vicinity of the ideal communication limit determined only by the latency of chip-to-chip communication. Our eight-FPGA (field-programmable gate array) cluster can compute a 32,768-spin problem with a high pipeline efficiency of 97.9%. The performance of a 79-FPGA cluster for a 100,000-spin problem, projected using a theoretical performance model validated on smaller experimental clusters, is comparable to that of a state-of-the-art 100,000-spin optical Ising machine.

Correlation-diversified portfolios can be constructed by finding the maximum independent sets (MISs) in market graphs with edges corresponding to correlations between two stocks. The computational complexity of finding the MIS increases exponentially as the size of the market graph increases, making the MIS selection in a large-scale market graph difficult. Here we construct a diversified portfolio through solving the MIS problem for a large-scale market graph with a combinatorial optimization solver (an Ising machine) based on a quantum-inspired algorithm called simulated bifurcation (SB) and investigate the investment performance of the constructed portfolio using long-term historical market data at the Tokyo Stock Exchange. Comparisons using stock universes of various sizes (TOPIX 100, Nikkei 225, TOPIX 1000, and TOPIX which includes approximately 2,000 constituents) show that the SB-based solver outperforms conventional MIS solvers in terms of computation-time and solution-accuracy. By using the SB-based solver, we optimized the parameters of a MIS portfolio strategy through iteration of the backcast simulation that calculates the performance of the MIS portfolio strategy based on a large-scale universe covering more than 1,700 Japanese stocks for a long period of 10 years. It has been found that the best MIS portfolio strategy (Sharpe ratio = 1.16, annualized return/risk = 16.3%/14.0%) outperforms the major indices such as TOPIX (0.66, 10.0%/15.2%) and MSCI Japan Minimum Volatility Index (0.64, 7.7%/12.1%) for the period from 2013 to 2023. Factor analyses reveal that the selection of small-capitalization and low-correlation stocks results in the portfolio performance with not only the relatively low risk (the diversification effect as expected) but also the relatively high return.

Financial portfolio construction problems are often formulated as quadratic and discrete (combinatorial) optimization that belong to the nondeterministic polynomial time (NP)-hard class in computational complexity theory. Ising machines are hardware devices that work in quantum-mechanical/quantum-inspired principles for quickly solving NP-hard optimization problems, which potentially enable making trading decisions based on NP-hard optimization in the time constraints for high-speed trading strategies. Here we report a real-time stock trading system that determines long(buying)/short(selling) positions through NP-hard portfolio optimization for improving the Sharpe ratio using an embedded Ising machine based on a quantum-inspired algorithm called simulated bifurcation. The Ising machine selects a balanced (delta-neutral) group of stocks from an $N$ -stock universe according to an objective function involving maximizing instantaneous expected returns defined as deviations from volume-weighted average prices and minimizing the summation of statistical correlation factors (for diversification). It has been demonstrated in the Tokyo Stock Exchange that the trading strategy based on NP-hard portfolio optimization for $N=128$ is executable with the FPGA (field-programmable gate array)-based trading system with a response latency of $164 \mu \text{s}$ .

Pairs-trading is a trading strategy that involves matching a long position with a short position in two stocks aiming at market-neutral profits. While a typical pairs-trading system monitors the prices of two statistically correlated stocks for detecting a temporary divergence, monitoring and analyzing the prices of more stocks would potentially lead to finding more trading opportunities. Here we report a stock pairs-trading system that finds trading opportunities for any two stocks in an $N$ -stock universe using a combinatorial optimization accelerator based on a quantum-inspired algorithm called simulated bifurcation. The trading opportunities are detected through solving an optimal path search problem in an $N$ -node directed graph with edge weights corresponding to the products of instantaneous price differences and statistical correlation factors between two stocks. The accelerator is one of Ising machines and operates consecutively to find multiple opportunities in a market situation with avoiding duplicate detections by a tabu search technique. It has been demonstrated in the Tokyo Stock Exchange that the FPGA (field-programmable gate array)-based trading system has a sufficiently low latency ( $33 \mu \text{s}$ for $N$ =15 or 210 pairs) to execute the pairs-trading strategy based on optimal path search in market graphs.