Practical Quantum Circuit Implementation for Simulating Coupled Classical Oscillators

TL;DR

This paper develops a quantum circuit method for simulating large coupled classical oscillators, reducing computational costs and enabling larger-scale studies with quantum hardware.

Contribution

It presents a detailed quantum circuit construction for simulating 1D spring-mass systems, incorporating key quantum algorithms for efficient Hamiltonian simulation.

Findings

Circuit complexity scales logarithmically with system size and accuracy.

Numerical simulations match classical results across configurations.

Method reduces computational costs for large-scale oscillator systems.

Abstract

Simulating large-scale coupled-oscillator systems presents substantial computational challenges for classical algorithms, particularly when pursuing first-principles analyses in the thermodynamic limit. Motivated by the quantum algorithm framework proposed by Babbush et al., we present and implement a detailed quantum circuit construction for simulating one-dimensional spring-mass systems. Our approach incorporates key quantum subroutines, including block encoding, quantum singular value transformation (QSVT), and amplitude amplification, to realize the unitary time-evolution operator associated with simulating classical oscillators dynamics. In the uniform spring-mass setting, our circuit construction requires a gate complexity of $\mathcal{O}\bigl(\log_2^2 N\,\log_2(1/\varepsilon)\bigr)$, where $N$ is the number of oscillators and $\varepsilon$ is the target accuracy of the approximation. For more general, heterogeneous spring-mass systems, the total gate complexity is $\mathcal{O}\bigl(N\log_2 N\,\log_2(1/\varepsilon)\bigr)$. Both settings require $\mathcal{O}(\log_2 N)$ qubits. Numerical simulations agree with classical solvers across all tested configurations, indicating that this circuit-based Hamiltonian simulation approach can substantially reduce computational costs and potentially enable larger-scale many-body studies on future quantum hardware.