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First-Order Trotter Error from a Second-Order Perspective
David Layden
TL;DR
Known Trotter error bounds can overestimate true error, while numerical studies report unexpected scaling even for simple first-order product formulas. For Hamiltonians split into two realizable terms, the paper relates first- and second-order formulas to refine the bound and identify the source of its terms. Numerical tests show the bound gives a good quantitative and qualitative description of the true error.
Problem
Known bounds can overestimate true error, and numerical studies report unexpected error scaling even for simple first-order product formulas used in near-term quantum simulation.
Source: p. 1Method
For Hamiltonians decomposed into two realizable terms, the authors relate first- and second-order product formulas to refine the error bound and identify the source of each term.
Source: pp. 2, 4Results
The bound gives a good quantitative and qualitative description of the true Trotter error in numerical tests.
Source: p. 4Takeaways & Limitations
For L = 2, error interference occurs primarily between adjacent timesteps, while circuit-end discrepancies introduce the O(t/r) term.
Source: p. 4For observables measured in an H1 or H2 eigenbasis from a corresponding eigenstate, PF1 and PF2 are equivalent and the O(t/r) error term is absent.
Source: p. 4
Abstract
from arXiv · showhide
Simulating quantum dynamics beyond the reach of classical computers is one of the main envisioned applications of quantum computers. The most promising quantum algorithms to this end in the near-term are the simplest, which use the Trotter formula and its higher-order variants to approximate the dynamics of interest. The approximation error of these algorithms is often poorly understood, even in the most basic cases, which are particularly relevant for experiments. Recent studies have reported anomalously low approximation error with unexpected scaling in such cases, which they attribute to quantum interference between the errors from different steps of the algorithm. Here we provide a simpler picture of these effects by relating the Trotter formula to its second-order variant. Our method generalizes state-of-the-art error bounds without the technical caveats of prior studies, and elucidates how each part of the total error arises from the underlying quantum circuit. We compare our bound to the true error numerically, and find a close match over many orders of magnitude in the simulation parameters. Our findings reduce the required circuit depth for the most basic quantum simulation algorithms, and illustrate a useful method for bounding simulation error more broadly.