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

Understand

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.