Hamiltonian simulation with nearly optimal dependence on all parameters

TL;DR

This paper introduces a quantum algorithm for sparse Hamiltonian simulation that achieves nearly optimal complexity across all parameters, combining quantum walk and fractional-query methods for improved efficiency.

Contribution

It combines quantum walk and fractional-query techniques to achieve nearly optimal dependence on all parameters in Hamiltonian simulation.

Findings

Complexity is logarithmic in inverse error.

Complexity is nearly linear in the product of evolution time, sparsity, and Hamiltonian magnitude.

Proves a new lower bound on dependence on evolution time.

Abstract

We present an algorithm for sparse Hamiltonian simulation whose complexity is optimal (up to log factors) as a function of all parameters of interest. Previous algorithms had optimal or near-optimal scaling in some parameters at the cost of poor scaling in others. Hamiltonian simulation via a quantum walk has optimal dependence on the sparsity at the expense of poor scaling in the allowed error. In contrast, an approach based on fractional-query simulation provides optimal scaling in the error at the expense of poor scaling in the sparsity. Here we combine the two approaches, achieving the best features of both. By implementing a linear combination of quantum walk steps with coefficients given by Bessel functions, our algorithm's complexity (as measured by the number of queries and 2-qubit gates) is logarithmic in the inverse error, and nearly linear in the product $\tau$ of the evolution time, the sparsity, and the magnitude of the largest entry of the Hamiltonian. Our dependence on the error is optimal, and we prove a new lower bound showing that no algorithm can have sublinear dependence on $\tau$.