Quantum cost of simulating continuous-time Brownian motions on the unitary group

Determine the quantum gate complexity required to simulate continuous-time Brownian motion processes on the unitary group to the precision needed for applications such as generating unitary designs, and assess whether such simulations achieve linear-in-k scaling when implemented on a quantum computer.

Background

Arguments in the literature suggest that certain continuous-time Brownian motions may yield linear scaling in the number of moments matched, which is highly desirable for constructing efficient unitary designs. However, realizing such processes on a quantum computer requires analyzing the actual simulation cost, which may alter the practical efficiency.

The authors explicitly note that although such linear scaling is plausible in the continuous-time model, the complexity of simulating these processes on a quantum device has not yet been established.