Improved Time-independent Hamiltonian Simulation

TL;DR

This paper introduces a novel Hamiltonian simulation method leveraging quantum singular value transformation, enabling efficient, logarithmic scaling of approximation error for time-independent Hamiltonians decomposed into efficiently simulatable parts.

Contribution

It presents a new simulation technique that surpasses product formula methods by achieving polylogarithmic error scaling using quantum singular value transformation.

Findings

Achieves polylogarithmic scaling of approximation error.

Enables efficient simulation of decomposed Hamiltonians.

Improves over traditional product formula approaches.

Abstract

We describe a simple method for simulating time-independent Hamiltonian $H$ that could be decomposed as $H = \sum_{i=1}^m H_i$ where each $H_i$ can be efficiently simulated. Approaches relying on product formula generally work by splitting the evolution time into segments, and approximate the evolution in each segment by the evolution of composing Hamiltonian $H_i$. This key step incur a constraint, that prohibits a (poly)logarithmic scaling on approximation error. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate $H_i$ in an alternative way, which then allows us to construct and simulate the main Hamiltonian $H$ with polylogarithmical scaling on the inverse of desired error, which is a major improvement with respect to product formula approaches.