Theory of Trotter Error with Commutator Scaling

Time-dependent unbounded Hamiltonian simulation with vector norm scaling

“…Therefore in the context of time-dependent simulations, the use of the generalized Trotter method could reduce the simulation cost. Our proof of the operator norm error bounds mainly follow the procedure proposed in [17], and our results generalize the first and second order time-independent results in [17] in the sense that, when the scalar functions f 1 and f 2 are constant functions, both time-dependent standard Trotter formula and time-dependent generalized Trotter formula degenerate to the same time-independent Trotter formula, and the corresponding operator norm error bound is of commutator scaling.…”

“…In particular, for a d-sparse Hamiltonian with bounded H max (the largest element of H in absolute value), the complexity of the quantum signal processing (QSP) method [36] is O (T d H max + log(1/ )/ log log(1/ )), which matches complexity lower bounds in all parameters. Meanwhile the error bound of the high order Trotter-Suzuki scheme has also been significantly improved [17], which yields near-best asymptotic complexities for simulating problems such as k-local Hamiltonians, and improves previous error bounds for Hamiltonians with long range interactions. On the other hand, simulation with time-dependent Hamiltonians appears ubiquitously, such as in the context of quantum controls [57,38,42,21,43,39], non-adiabatic quantum dynamics [48,18], and adiabatic quantum computation [22,47,2], to name a few.…”

“…Given the improved error commutator scaling estimates for time-independent simulations [17], it is natural to ask whether the commutator scaling of the operator norm still holds for time-dependent simulations. The second contribution of this paper is to reveal that for time-dependent simulations, the error of standard Trotter method does not exhibit commutator scalings, while the commutator scaling holds for the generalized Trotter method (Theorem 2).…”

“…For the time-independent Trotter formula, the work of [17,20] prove a commutator type of error of any order by writing down an explicit error representation via variation of parameters formula. Here we follow the procedure in [17] to write down the corresponding error representations for standard and generalized time-dependent Trotter formulae, which turn out to be the starting point for proving both the operator norm and the vector norm error bounds. Although we only present the error representation on the interval [0, h], this is just for notation simplicity and with minor modifications the results naturally hold on [t, t + h] for any t. The proofs are given in Appendix B.…”

“…If H(t) ≡ H is time-independent, then the solution can be expressed in closed form as |ψ(T ) = exp(−iT H) |ψ 0 . The task of creating such a universal quantum simulator was first conceptualized by Lloyd [34], and the past few years have witnessed significant progresses in the development of new quantum algorithms as well as the improvement of theoretical error bounds of existing quantum algorithms for time-independent Hamiltonian simulation [3,4,9,6,7,36,14,37,15,11,35,16,17,12,49]. In particular, for a d-sparse Hamiltonian with bounded H max (the largest element of H in absolute value), the complexity of the quantum signal processing (QSP) method [36] is O (T d H max + log(1/ )/ log log(1/ )), which matches complexity lower bounds in all parameters.…”