Quantum and classical algorithms for nonlinear unitary dynamics

TL;DR

This paper develops a quantum algorithm for nonlinear differential equations that matches known exponential lower bounds and introduces classical algorithms with comparable scaling under certain conditions.

Contribution

It presents a quantum algorithm for nonlinear unitary dynamics that attains optimal query complexity and introduces classical methods with similar scaling in specific scenarios.

Findings

Quantum algorithm matches exponential lower bounds for nonlinear differential equations.

Classical Euler-based algorithm scales similarly to the quantum algorithm in restricted cases.

A randomized classical path integration algorithm acts as a true analogue to the quantum approach when sign problems are absent.

Abstract

Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending this to a nonlinear problem has proven challenging, with exponential lower bounds having been demonstrated for the time scaling. We provide a quantum algorithm matching these bounds. Specifically, we find that for a non-linear differential equation of the form $\frac{d|u\rangle}{dt} = A|u\rangle + B|u\rangle^{\otimes2}$ for evolution of time $T$, error tolerance $\epsilon$ and $c$ dependent on the strength of the nonlinearity, the number of queries to the differential operators that approaches the scaling of the quantum lower bound of $e^{o(T\|B\|)}$ queries in the limit of strong non-linearity. Finally, we introduce a classical algorithm based on the Euler method allowing comparably scaling to the quantum algorithm in a restricted case, as well as a randomized classical algorithm based on path integration that acts as a true analogue to the quantum algorithm in that it scales comparably to the quantum algorithm in cases where sign problems are absent.