XY-mixer ansatz assisted by counterdiabatic driving for combinational optimization

XY mixer are a type of mixing Hamiltonians used in the quantum alternating operator ansatzs (QAOA) framework for solving combinatorial optimization problems where the feasible subspace consists of states with the same Hamming weight. In this paper, we propose an extension of the XY-mixer ansatzes to solve optimization problems that do not adhere to this specific subspace structure. For problems, such as maximal independent set, one can employ positive semidefinite programming and a greedy algorithm to obtain upper and lower bounds for the optimal solution, then utilize the XY-mixer ansatzes in the separated subspaces between the upper and lower bounds to address the problem at hand. Additionally, we find suitable counterdiabatic (CD) driving terms that complement the XY-mixer ansatzes. These driving terms accelerate the QAOA evolution towards the target state and also confine the evolution within the subspace. By combining XY-mixer ansatzes with counterdiabatic driving (XY-CD mixers), we present a generalized QAOA-based scheme for finding higher-quality approximate solutions to combinatorial problems.