Error Scaling with System Size in Digital Quantum Simulations

In order to meaningfully compare the feasibility of implementing quantum simulation on analogue and digital platforms we must take into account the errors introduced by decomposing the simulation into gate operations for digital computation. 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 for larger systems this would imply a corresponding 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 even at the critical point and that for non-local observables away from the critical point this remains true.