Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas

Product formula (PF), which approximates the time evolution under a many-body Hamiltonian by the product of local time evolution operators, is one of the central approaches for simulating quantum dynamics by quantum computers. It has been of great interest whether PFs have a bound of the error from the exact time evolution, which is expressed by commutators among local terms (called commutator scaling), since it brings the substantial suppression of the computational cost in the system size. Although recent studies have revealed the presence and the explicit formulas of the PF error bounds for time-independent systems, those for time-dependent Hamiltonians remain to be a difficult problem except for low-order PFs. In this paper, we derive an explicit error bound of generic PFs for smooth time-dependent Hamiltonians, which is expressed by commutators among local terms and their time derivatives. This error bound can also host the substantial suppression in the system size for generic local Hamiltonians with finite-, short-, and long-ranged interactions, thereby giving a much better estimate of gate counts. Our derivation employs Floquet theory; Embedding generic smooth time-dependent Hamiltonians into time-periodic ones, we map the time-dependent PF error to the time-independent one defined on an infinite-dimensional space. This approach allows to obtain the error bounds not only for the ordinary time-dependent PF but also for its various family. In particular, we also clarify the explicit error bound of a time-dependent multi-product formula, with which we can achieve much smaller error by a linear combination of time-dependent PFs. Our results will shed light on various applications of quantum computers, ranging from quantum simulation of nonequilibrium materials to faster algorithms exploiting time-dependent Hamiltonians like adiabatic state preparation.