Maxime Dion

The quality of an Trotterized quantum evolution is often characterized using the spectral norm between the exact and Trotter evolutions. This worst-case scenario approach overestimate the error in most cases, which translate in an overestimation of the resources required to perform the evolution with a target accuracy. In this work, we introduce an approach to characterize the quality of a Trotterized evolution for finite-size quantum systems based on the infidelity between the states prepared by the exact evolution and the Trotterized evolution. We relate the fidelity to a phase operator, which is central to this approach and can be constructed using the multivariate Baker-Campbell-Hausdorff formula. We demonstrate the utility of that operator by using its eigenbasis to construct the state which maximizes the Trotter infidelity. That infidelity acts as an exact bound for the infidelity of any state evolved using a Trotterized evolution. We compare the Trotter infidelity to the operator norm distance to get insights on why the later generally overestimate the Trotter error in practice. We then study the infidelity bound in the limits of short evolutions and of a large number of Trotter steps. While we focus on the first-order Trotterization, some results and conclusions directly apply to second order or higher.