  A highly anticipated use of quantum computers is the simulation of complex
quantum systems including molecules and other many-body systems. One promising
method involves directly applying a linear combination of unitaries (LCU) to
approximate a Taylor series by truncating after some order. Here we present an
adaptation of that method, optimized for Hamiltonians with terms of widely
varying magnitude, as is commonly the case in electronic structure
calculations. We show that it is more efficient to apply LCU using a truncation
that retains larger magnitude terms as determined by an iterative procedure. We
obtain bounds on the simulation error for this generalized truncated Taylor
method, and for a range of molecular simulations, we report these bounds as
well as exact numerical results. We find that our adaptive method can typically
improve the simulation accuracy by an order of magnitude, for a given circuit
depth.
