Quantum-Inspired Optimization through Qudit-Based Imaginary Time Evolution

Imaginary-time evolution has been shown to be a promising framework for tackling combinatorial optimization problems on quantum hardware. In this work, we propose a classical quantum-inspired strategy for solving combinatorial optimization problems with integer-valued decision variables by encoding decision variables into multi-level quantum states known as qudits. This method results in a reduced number of decision variables compared to binary formulations while inherently incorporating single-association constraints. Efficient classical simulation is enabled by constraining the system to remain in a product state throughout optimization. The qudit states are optimized by applying a sequence of unitary operators that iteratively approximate the dynamics of imaginary time evolution. Unlike previous studies, we propose a gradient-based method of adaptively choosing the Hermitian operators used to generate the state evolution at each optimization step, as a means to improve the convergence properties of the algorithm. The proposed algorithm demonstrates promising results on Min-d-Cut problem with constraints, outperforming Gurobi on penalized constraint formulation, particularly for larger values of d.