Practical Quantum Circuit Implementation for Simulating Coupled Classical Oscillators

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 $$\mathcal{O}\bigl(\log_2² N\,\log_2(1/\varepsilon)\bigr)$$, 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.