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 byproduct. 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 is close to tight for power-law 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.
Highlights
Product formulas provide a convenient approach to decomposing the evolution of a sum of operators
While many previous works focused on the performance of specific formulas, the theory we develop holds for any formula; we use the term “product formula” to emphasize this generality
Product formulas and their generalizations [26,27,28,29] can perform significantly better when the operator summands commute or nearly commute—a unique feature that does not seem to hold for other quantum simulation algorithms
Summary
Product formulas provide a convenient approach to decomposing the evolution of a sum of operators. This paper considers the application of product formulas to simulating quantum systems. It has been known for over two decades that these formulas are useful for digital quantum simulation and quantum Monte Carlo methods. Their error scaling is poorly understood, and existing bounds can be several orders of magnitude larger than those observed in practice, even for simulating relatively small systems. Despite the sophistication of these “postTrotter methods” and their optimality in certain general models, our work shows that they can be provably outperformed by product formulas for simulating many quantum systems
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