Practical Quantum Circuit Implementation for Simulating Coupled Classical Oscillators
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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. (2023), 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 mass–spring setting, our circuit construction requires a gate complexity of <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}\bigl (\log _{2}^{2} N\,\log _{2}(1/\varepsilon )\bigr )$ </tex-math></inline-formula>, where N is the number of oscillators and <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> is the target accuracy of the approximation. For more general, heterogeneous mass–spring systems, the total gate complexity is <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}\bigl (N\log _{2} N\,\log _{2}(1/\varepsilon )\bigr )$ </tex-math></inline-formula>. Both settings require <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(\log _{2} N)$ </tex-math></inline-formula> 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.