Trotter Error Scaling with System Size in Quantum Simulations

A main concern when implementing digital quantum simulation on quantum computers or simulators is the number of gate operations required. The most common implementation uses the Trotter decomposition to map an arbitrary Hamiltonian onto realisable gates. However, the proven upper bounds on the error introduced using this method grows with the system size. In order to maintain the same accuracy this would imply that the time step has to shrink with system size, leading to a sharp increase in the number of gates required. We show empirically that for local Hamiltonians the error for local observables and their correlation functions is independent of system size. This results is obtained by simulating large one- dimensional quantum Ising model at and away from the critical point using a Trotter decomposition in imaginary time . We find that the Trotter errors saturate with increasing system size even at the critical points. We finally discuss long-range models and correlations in time.