Product Formulas for Exponentials of Commutators

TL;DR

This paper introduces a recursive method for constructing high-order product formulas to approximate exponentials of commutators, enabling efficient quantum operations and simulations of many-body Hamiltonians.

Contribution

It presents the first high-order accurate product formulas for exponentials of commutators and demonstrates their application in quantum control and simulation.

Findings

High-order approximations for exponentials of commutators

Efficient implementation of quantum search algorithms

Applications to simulating many-body Hamiltonians

Abstract

We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating many-body interaction Hamiltonians.