Simulating Hamiltonian dynamics with a truncated Taylor series

TL;DR

This work addresses efficient quantum simulation of Hamiltonian dynamics by directly approximating the evolution operator with a truncated Taylor series and implementing the resulting linear combination of unitaries. The core technique combines a segmented time evolution with robust oblivious amplitude amplification to realize nonunitary Taylor terms, achieving a logarithmic dependence on the desired precision $\epsilon$ comparable to the best prior methods. The framework applies to time-independent and time-dependent Hamiltonians decomposed into a sum of unitaries (e.g., Pauli sums or sparse forms) and provides explicit circuit constructions and gate-count scalings. Overall, the approach offers a simpler, broadly applicable alternative to previous optimal algorithms while preserving the same asymptotic cost, with potential for wider use of robust LCU techniques in quantum algorithms.

Abstract

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations to directly apply the truncated Taylor series.