Calculating response functions of coupled oscillators using quantum phase estimation

TL;DR

The paper investigates quantum-speedups for computing frequency-response functions of networks of $N$ coupled harmonic oscillators by recasting the problem as an eigenproblem of a Hermitian matrix $H$ and applying quantum phase estimation in a sparse, oracle-access setting. The local response function $G_{uu}( u)$ is expressed in terms of eigenvalues $ u_j$ and weights $W_{u j}^2$, enabling a QPE-based algorithm that reads these quantities from phase information without requiring full eigenstate preparation. The authors derive a detailed complexity framework, showing polylogarithmic qubit counts and problem-dependent query complexities, with worst-case behavior tied to eigenvalue gaps; they also demonstrate an exponential-speedup-worthy instance via the random glued-trees problem and discuss limitations, optimizations (eigenvalue filtering, amplitude estimation), and practical considerations for finite-size applications. The work connects quantum-eigenproblem techniques to mechanical and structural simulations and suggests concrete pathways to quantify end-to-end performance for relevant industrial and physical scenarios.

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 $\mathcal{O}(s \lVert H \rVert_{\mathrm{max}}/(δ^2 \varepsilon) )$ 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.

Figures (8)

• Figure 1: In-process-workpiece (IPW) of a single blade demonstrator (left) and the detailed representation of the finite element mesh. The cutting tool is shown in blue. With the cutter location-dependent IPW, material properties (density, Young's modulus, Poisson's ratio), and boundary conditions as input, the finite element modal analysis can be conducted and the frequency response function (FRF) calculated by solving the corresponding eigenproblem. Typical solvers used for this task are based on the QR algorithm and have complexity $\mathcal{O}(N^3)$trefethen97press1992arbenzNotes.
• Figure 2: A representation of coupled oscillators. Point-like masses $m_u$ in blue are connected by springs with spring constant $\kappa_{wu}$. The force $f_v$ applied to mass $v$ yields a displacement $x_u$ at mass $u$.
• Figure 3: QPE circuit for the computation of the local response function. The state preparation is realized by a series of NOT gates. QPE is performed using the walk operator $V$ defined in \ref{['eq:qw_op']} followed by the inverse quantum Fourier transform ($\mathrm{QFT}^{\dagger}$) NielsenChuang. Sampling from the phase register and repeating many times allows us to estimate the relevant eigenvalues $\lambda_j$ and the weights $W_{uj}^2$ in \ref{['eq:diagresponse']}.
• Figure 4: Example of random glued-trees with $n_c=3$.
• Figure 5: Circuit for the state transformation $U_T$. It consists of three parts. The preparation of the superposition with $O_P$, the matrix encoding in the amplitude and the multiplication with the complex sign. The order of the registers in here is chosen for a compact circuit layout. The first and fourth register combine into $\ket{\psi_u}$ of \ref{['eq:ut1']}. The fifth register is $\ket{u}$ in \ref{['eq:ut1']}. The second, third and last registers (in blue) are of auxiliary nature and will be reinitialized after the full block-encoding of $H$. In the second register we store the intermediate angle (\ref{['eq:anglehuv']}). The third register is used for the binary sign $\Theta(-H_{uv})$ and the last for $\Theta(\mathrm{sgn}(u-v))$.

Theorems & Definitions (1)

• Theorem 1