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 $\mathcal{O}({\log }_2^{2}N{\log }_2(1/\epsilon ))$, where $N$ is the number of oscillators and $\epsilon$ is the target accuracy of the approximation. For more general, heterogeneous spring-mass systems, the total gate complexity is $\mathcal{O}(N{\log }_{2}N{\log }_2(1/\epsilon ))$. 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.

26 pages, 16 figures