A Theory of Trotter Error
Abstract
The LieTrotter formula, together with its higherorder generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the BakerCampbellHausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real and imaginarytime evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Liealgebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of secondquantized planewave electronic structure, $k$local Hamiltonians, rapidly decaying powerlaw interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for powerlaw interacting systems, which implies a LiebRobinson bound as a byproduct. Our analysis reproduces known tight bounds for first and secondorder formulas. Our higherorder bound overestimates the complexity of simulating a onedimensional Heisenberg model with an evenodd ordering of terms by only a factor of $5$, and is close to tight for powerlaw interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.08854
 Bibcode:
 2019arXiv191208854C
 Keywords:

 Quantum Physics;
 Condensed Matter  Strongly Correlated Electrons;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Numerical Analysis;
 Physics  Chemical Physics
 EPrint:
 82 pages, 5 figures. Enhanced version of the article published in Physical Review X at http://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.011020