Performance of Domain-Wall Encoding in Digital Ising Machine

To tackle combinatorial optimization problems using an Ising machine, the objective function and constraints must be mapped onto a quadratic unconstrained binary optimization (QUBO) model. While QUBO involves binary variables, combinatorial optimization problems frequently include integer variables, which require encoding by binary variables. This process, known as binary-integer encoding, includes various methods, one of which is domain-wall encoding - a recently proposed approach. Experiments on a quantum annealing machine have demonstrated that domain-wall encoding outperforms the commonly used one-hot encoding in terms of objective function value and the probability of obtaining the optimal solution. In a digital Ising machine, domain-wall encoding required less computation time to reach optimal solutions compared to one-hot encoding. However, its practical effectiveness in digital Ising machines remains unclear. To address this uncertainty, the performance of one-hot and domain-wall encoding methods was evaluated on a digital Ising machine using the quadratic knapsack problem (QKP). The comparison focused on the dependency of penalty coefficient and sensitivity to computation time. Domain-wall encoding demonstrated a higher feasible solution rate when relative penalty coefficients for the two constraint terms were adjusted, a strategy not commonly used in previous studies. Additionally, domain-wall encoding obtained higher performance practical evaluation metrics for QKPs with large knapsack capacities compared to one-hot encoding. Furthermore, it was observed to be more sensitive to computation time than one-hot encoding.