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Calculating response functions of coupled oscillators using quantum phase estimation

Sven Danz1,2,*, Mario Berta3,4, Stefan Schröder5, Pascal Kienast5, Frank K. Wilhelm1,2, and Alessandro Ciani1

  • *Contact author: s.danz@fz-juelich.de
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Phys. Rev. Research 7, 023264 – Published 13 June, 2025

DOI: https://doi.org/10.1103/5c6d-r1fb

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 ∥H∥max, and when the eigenvalue tolerance ɛ is much smaller than the minimum eigenvalue gap, we use O(log(Ns∥H∥max/ɛ) algorithmic qubits and obtain a rigorous worst-case query complexity upper bound O(s∥H∥max/(δ2ɛ)) up to logarithmic factors, where δ 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.

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Phys. Rev. Research 7, 023264– Published 13 June, 2025
  • Received 18 June 2024
  • Accepted 23 May 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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