Quantum Advantage in Learning Quantum Dynamics via Fourier coefficient extraction

TL;DR

This paper demonstrates that quantum algorithms can exponentially outperform classical ones in learning unknown quantum Hamiltonian dynamics by extracting Fourier coefficients, introducing a new subroutine method for parametrized circuits.

Contribution

The paper introduces a novel quantum learning algorithm based on Fourier coefficient extraction and a new subroutine method, providing provable exponential advantages in learning quantum dynamics.

Findings

Quantum algorithms outperform classical in learning Hamiltonian dynamics

Introduction of a new subroutine method for parametrized circuits

Limitations of generalizing the method are discussed

Abstract

One of the key challenges in quantum machine learning is finding relevant machine learning tasks with a provable quantum advantage. A natural candidate for this is learning unknown Hamiltonian dynamics. Here, we tackle the supervised learning version of this problem, where we are given random examples of the inputs to the dynamics as classical data, paired with the expectation values of some observable after the time evolution, as corresponding output labels. The task is to replicate the corresponding input-output function. We prove that this task can yield provable exponential classical-quantum learning advantages under common complexity assumptions in natural settings. To design our quantum learning algorithms, we introduce a new method, which we term \textit{\subroutine}~algorithm for parametrized circuit functions, and which may be of independent interest. Furthermore, we discuss the limitations of generalizing our method to arbitrary quantum dynamics while maintaining provable guarantees. We explain that significant generalizations are impossible under certain complexity-theoretic assumptions, but nonetheless, we provide a heuristic kernel method, where we trade-off provable correctness for broader applicability.