Approximating Hamiltonian dynamics with the Nystr\"om method

TL;DR

This paper introduces a classical randomized algorithm using the Nyström method to efficiently approximate Hamiltonian dynamics, potentially enabling classical simulation of certain quantum evolutions under specific conditions.

Contribution

It develops a novel Nyström-based randomized algorithm for approximating Hermitian matrix exponentials in quantum simulation, with polynomial runtime in system size and Hamiltonian norm.

Findings

Classical algorithms can simulate certain quantum evolutions efficiently under structural assumptions.

The proposed method achieves polynomial runtime in the number of qubits and Hamiltonian Frobenius norm.

Strong sampling assumptions may allow classical poly-logarithmic time simulation of some quantum computations.

Abstract

Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nystr\"om method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.