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Ising machines are expected to solve combinatorial optimization problems efficiently by representing them as Ising models or equivalent quadratic unconstrained binary optimization (QUBO) models . However, upper bound exists on the computable problem size due to the hardware limitations of Ising machines. This paper propose a new hybrid annealing method based on partial QUBO extraction, called subQUBO model extraction, with multiple solution instances. For a given QUBO model, the proposed method obtains <inline-formula><tex-math notation="LaTeX">$N_I$</tex-math><alternatives><mml:math><mml:msub><mml:mi>N</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq1-3138629.gif"/></alternatives></inline-formula> quasi-optimal solutions (quasi-ground-state solutions) in some way using a classical computer. The solutions giving these quasi-optimal solutions are called <italic>solution instances</italic>. We extract a size-limited subQUBO model as follows based on a strong theoretical background: we randomly select <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math><alternatives><mml:math><mml:msub><mml:mi>N</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq2-3138629.gif"/></alternatives></inline-formula> <inline-formula><tex-math notation="LaTeX">$(N_S&#x003C;N_I)$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>S</mml:mi></mml:msub><mml:mo>&#x003C;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>I</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="atobe-ieq3-3138629.gif"/></alternatives></inline-formula> solution instances among them and focus on a particular binary variable <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math><alternatives><mml:math><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq4-3138629.gif"/></alternatives></inline-formula> in the <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math><alternatives><mml:math><mml:msub><mml:mi>N</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq5-3138629.gif"/></alternatives></inline-formula> solution instances. If <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math><alternatives><mml:math><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq6-3138629.gif"/></alternatives></inline-formula> value is much <italic>varied</italic> over <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math><alternatives><mml:math><mml:msub><mml:mi>N</mml:mi><mml:mi>S</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="atobe-ieq7-3138629.gif"/></alternatives></inline-formula> solution instances, it is included in the subQUBO model; otherwise, it is not. We find a (quasi-)ground-state solution of the extracted subQUBO model using an Ising machine and add it as a new solution instance. By repeating this process, we can finally obtain a (quasi-)ground-state solution of the original QUBO model. Experimental evaluations confirm that the proposed method can obtain better quasi-ground-state solution than existing methods for large-sized QUBO models.

Hybrid Annealing Method Based on subQUBO Model Extraction With Multiple Solution Instances