Optimal Hamiltonian Simulation by Quantum Signal Processing

TL;DR

This paper introduces an optimal quantum algorithm for Hamiltonian simulation using quantum signal processing, leveraging simple single-qubit rotations to achieve query complexity that matches theoretical lower bounds.

Contribution

It demonstrates that physical intuition and simple single-qubit rotations can lead to optimal Hamiltonian simulation algorithms, unifying quantum computation and physical system simulation.

Findings

Query complexity matches lower bounds in all parameters.

Uses quantum signal processing to transform eigenvalues efficiently.

Achieves high success probability in ancilla projection.

Abstract

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a $d$-sparse Hamiltonian $\hat{H}$ for time-interval $t$ with error $\epsilon$ is $\mathcal{O}(td\|\hat{H}\|_{\text{max}}+\frac{\log{(1/\epsilon)}}{\log{\log{(1/\epsilon)}}})$, which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (1) transducing eigenvalues of $\hat{H}$ into a single ancilla qubit, (2) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (3) projecting this ancilla with near unity success probability.