Design QAOA by Alignment between Initial State and Mixer for Constrained Optimization*

Quantum alternating operator ansatz (QAOA) is a promising quantum algorithm for combinatorial optimization. QAOA has a strong connection to the adiabatic algorithm, which it can approximate with sufficient depth. At the same time, it is unclear to what extent the lessons from the adiabatic regime apply to QAOA as executed in practice, i.e., with small to moderate depth. In this paper, we demonstrate that the intuition from the adiabatic algorithm applies to the task of choosing the QAOA initial state. Specifically, we observe that the best performance is obtained when the initial state of QAOA is set to be the ground state of the mixing Hamiltonian, as required by the adiabatic algorithm. We provide numerical evidence using the examples of constrained portfolio optimization problems with both low (p≤3) and high (p=100) QAOA depth, suggesting that the alignment between the initial state and the ground state of the mixing Hamiltonian is beneficial in most cases. We compare many variations of Hamming-weight-preserving XY mixers, which we simulate both exactly and approximately, using different numbers of Trotter steps. We observe that lower Trotter error improves QAOA performance when the initial state is set to be the easy-to-prepare ground state of the (exact) XY model. In addition, we successfully apply QAOA with XY mixer to portfolio optimization on a trapped-ion quantum processor using 32 qubits and discuss the implications of our findings to near-term experiments.