Calculating response functions of coupled oscillators using quantum phase estimation

TL;DR

This paper presents a quantum algorithm for efficiently estimating the response functions of coupled harmonic oscillators, potentially enabling large speedups over classical methods by leveraging quantum phase estimation without the state preparation bottleneck.

Contribution

The authors develop a quantum algorithm for response function estimation that avoids the state preparation bottleneck and demonstrates potential exponential speedup for specific problems.

Findings

Quantum algorithm operates with logarithmic qubits in system size.

Achieves polynomial time solution for the glued-trees problem.

Potential for large quantum speedups over classical approaches.

Abstract

We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. The functional form of these response functions can be mapped to a corresponding eigenproblem of a Hermitian matrix $H$, thus suggesting the use of quantum phase estimation. Our proposed quantum algorithm operates in the standard $s$-sparse, oracle-based query access model. For a network of $N$ oscillators with maximum norm $\lVert H \rVert_{\mathrm{max}}$, and when the eigenvalue tolerance $\varepsilon$ is much smaller than the minimum eigenvalue gap, we use $\mathcal{O}(\log(N s \lVert H \rVert_{\mathrm{max}}/\varepsilon)$ algorithmic qubits and obtain a rigorous worst-case query complexity upper bound $\mathcal{O}(s \lVert H \rVert_{\mathrm{max}}/(\delta^2 \varepsilon) )$ up to logarithmic factors, where $\delta$ denotes the desired precision on the coefficients appearing in the response functions. Crucially, our proposal does not suffer from the infamous state preparation bottleneck and can as such potentially achieve large quantum speedups compared to relevant classical methods. As a proof-of-principle of exponential quantum speedup, we show that a simple adaptation of our algorithm solves the random glued-trees problem in polynomial time. We discuss practical limitations as well as potential improvements for quantifying finite size, end-to-end complexities for application to relevant instances.