Optimal Hamiltonian simulation for time-periodic systems

Implementing time-evolution operators under many-body Hamiltonians, called Hamiltonian simulation, is one of the most important tasks for potential applications of quantum computers toward condensed matter physics and quantum chemistry. Recently, the qubitization has achieved a comparably good way to organize time-evolution under time-independent Hamiltonians, with an arbitrarily small error . The number of gates required for the qubitization has a theoretically optimal scaling both in time and inverse error . In contrast, for time-dependent Hamiltonians, the existing algorithms largely rely on the truncated Dyson-series expansion. Due to the difficulty of handling time-dependency, we need much resources compared to the qubitization. It is a nontrivial and important question whether there exists an optimal algorithm for time-dependent cases whose resource is as large as that for time-independent cases.



In our study, we focus on time-dependent Hamiltonians with periodicity, and obtain the answer to the above question. To be precise, we organize the optimal simulation algorithm for time-periodic Hamitlonians, with the help of Floquet theory. Despite the existence of time-dependency, the scaling of the computational cost is the same as or slightly larger than the qubitization, and much smaller than conventional algorithms for time-dependent systems. Our algorithm has broad and promising applications to nonequilibrium phenomena in condensed matter physics and quantum chemistry.