Theory of Trotter Error with Commutator Scaling

Open Access

Theory of Trotter Error with Commutator Scaling

Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu

Phys. Rev. X 11, 011020 – Published 1 February 2021

Abstract

The Lie-Trotter formula, together with its higher-order 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 Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic 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 second-quantized plane-wave electronic structure, $k$ -local Hamiltonians, rapidly decaying power-law 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 power-law interacting systems, which implies a Lieb-Robinson bound as a by-product. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and it is close to tight for power-law interactions and other orderings of terms. This result suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.

Received 21 January 2020
Revised 7 July 2020
Accepted 4 November 2020

DOI:https://doi.org/10.1103/PhysRevX.11.011020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum algorithms Quantum simulation

Quantum Information, Science & Technology

Authors & Affiliations

Andrew M. Childs ^1,2,3, Yuan Su ^1,2,3, Minh C. Tran ^3,4, Nathan Wiebe^5,6,7, and Shuchen Zhu⁸

¹Department of Computer Science, University of Maryland, College Park, Maryland 20742, USA
²Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742, USA
³Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
⁴Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
⁵Department of Physics, University of Washington, Seattle, Washington 98195, USA
⁶Pacific Northwest National Laboratory, Richland, Washington 99354, USA
⁷Google Inc., Venice, California 90291, USA
⁸Department of Computer Science, Georgetown University, Washington, DC 20057, USA

Popular Summary

Quantum computers hold the promise of simulating quantum systems much more efficiently than their classical counterparts, enabling potential development of new pharmaceuticals, catalysts, and materials. Among all known quantum simulation algorithms, product formulas provide the simplest approach, making them popular for experimental realizations. Furthermore, evidence suggests that product formulas perform well in practice. Here, we develop a theory for bounding the error of product formulas.

Until now, the error scaling of product formulas has been poorly understood, leaving a dramatic gap between their provable performance and actual behavior. This gap has made it hard to identify the fastest simulation algorithm and to find optimized implementations for near-term applications of quantum computers.

Our theory allows us to provide a host of improved algorithms for quantum simulation, including various systems of practical relevance. This analysis puts the product-formula algorithm on a rigorous foundation, in many cases justifying its superiority over the best previous algorithms. We also connect our result to classical simulation of quantum systems and to the speed of information propagation in physical systems.

This work represents significant progress toward a precise characterization of the product-formula error. Whereas recent studies have favored more advanced simulation algorithms that are easier to analyze but harder to implement, our result demonstrates that product formulas can outperform these sophisticated algorithms. Our findings thus reemphasize the role of product formulas in quantum simulation, which we hope will bridge the gap between theoretical investigation and near-term experimental realization of quantum computing.

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Vol. 11, Iss. 1 — January - March 2021

Subject Areas

Quantum Information

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Images

Figure 1
Demonstration of the second-order product formula for simulating the evolution of an observable $B$ supported on $S (B) = Δ B_{0}$ . Each rectangle represents a unitary supported on the sites covered by the width of the rectangle. The evolution unitary $e^{- i t H}$ is decomposed using the second-order product formula into $ϒ = 2$ stages. Each stage is a sequence of $Γ = 3$ matrix exponentials generated by Hamiltonian terms supported on parts of the system. Some of these unitaries (red shaded rectangles) can subsequently be commuted through $B$ in the expression $𝒮_{trunc}^{†} (t) B 𝒮_{trunc} (t)$ to cancel out with their Hermitian conjugate. As a result, the time-evolved version of $B$ can be effectively described by the remaining unitaries (light-gray rectangles).
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Figure 2
Construction of the Hamiltonian $H_{lc}$ within the light cone such that $e^{i t H} A e^{- i t H} \approx e^{i t H_{lc}} A e^{- i t H_{lc}}$ . Each rectangle represents a unitary supported on the sites covered by the width of the rectangle. Specifically, the rectangle in the top panel represents the evolution $e^{- i t H}$ that we want to decompose. We divide the evolution into $r$ steps. Within each step, we truncate the Hamiltonian to $H_{trunc}$ and decompose its evolution using a product formula $𝒮_{trunc}$ . We commute certain matrix exponentials in $𝒮_{trunc}$ (represented by red shaded rectangles) through the observable to cancel their counterpart, obtaining $𝒮_{reduce}$ in the second panel. We reverse this procedure within the light cone to construct $𝒮_{lc}$ in the third panel, which approximates $e^{- i t H_{lc}}$ as illustrated in the bottom panel.
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Figure 3
Comparison of $r$ for the Heisenberg model using the analytic bound (see Proposition F.4 in Ref. [31]), the commutator bound (see Theorem F.11 in Ref. [31]), locality-based bound (see Supplementary Material IV B in Ref. [35]), and our new bounds, Proposition M.1 and Proposition M.2. Error bars are omitted as they are negligibly small on the plot. Straight lines show power-law fits to the data. Note that the exponent for the empirical data is based on brute-force simulations of small systems and thus may not precisely capture the true asymptotic scaling due to finite-size effects.
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Figure 4
Comparison of $r$ for the power-law Heisenberg model using the analytic bound (see Proposition F.4 of Ref. [31]), 1-norm bound (Lemma 1), a bound from counting argument (58), and our bound (Proposition M.2). Error bars are omitted as they are negligibly small on the plot. Straight lines show power-law fits to the data. Note that the exponent for the empirical data is based on brute-force simulations of small systems and thus may not precisely capture the true asymptotic scaling due to finite-size effects.
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Figure 5
Comparison of the empirical scaling exponents of the Trotter numbers (purple squares) against our bound (green squares) as functions of the system size at different values of the power-law exponent $α$ . The error bars of the empirical values represent the standard deviation of the fitted exponents (see Fig. 4) over five instances of the random field $h_{j}$ . The bound is derived from the counting argument in Eq. (53) and therefore has no standard deviation. We attribute the systematic difference between our bound and the empirical values to the fact that we only compute the empirical Trotter numbers up to $n = 11$ , which may not capture the precise asymptotic scaling in the large- $n$ limit. We also include the scaling exponent of the analytic bound in Ref. [31] (red dash-dotted line) as well as the theoretical exponent in the limit $α \to \infty$ , i.e., nearest-neighbor interactions (blue dashed line), for references.
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