Southwest Quantum Information and Technology

A theory for Trotter Error

Presenting Author: Minh Tran, University of Washington
Contributing Author(s): Nathan Wiebe, Andrew Childs, Yuan Su, Schchen Zhu

The Lie-Trotter formula and its higher-order generalizations provide direct approaches to implement the exponential of a sum of operators on classical and quantum computers. However, the error scaling of these approximations remains poorly understood despite significant effort. We address this by developing a theory of Trotter error that directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Previous work only achieves these goals for systems with geometrically local interactions and Lie-algebraic structures. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo simulation, including simulations of second-quantized plane-wave electronic structure, local Hamiltonians, power-law interactions, clustered Hamiltonians, transverse field Ising model, and quantum ferromagnets, nearly matching or outperforming the state-of-the-art results. We further show that the gate count of simulating local observables can be independent of the system size and we prove a Lieb-Robinson-type bound that nearly matches a recent bound. Our bound is provably tight for low-order formulas. For nearest-neighbor interactions and power-law interactions, our higher-order bound overestimates the complexity by only a factor of 5. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

(Session 5 : Saturday from 5:00pm - 7:00pm)