Exponential Quantum Speedup for Simulating Classical Lattice Dynamics

Simulating large scale lattice dynamics directly is computationally demanding due to the high complexity involved, yet such simulations are crucial for understanding the mechanical and thermal properties of many physical systems. In this work, we introduce a rigorous quantum framework for simulating general harmonic lattice dynamics by reformulating the classical equations as a time dependent Schr\"odinger equation governed by a sparse Hamiltonian. This transformation allows us to exploit well established quantum Hamiltonian simulation techniques, offering an exponential speedup with respect to the number of atoms $N$. The overall complexity has a logarithmic dependence on $N$, and linear dependence on both the simulation time $T$ and the Debye frequency $\omega_D$. Key to our approach is the application of the matrix valued Fej\'er Riesz theorem to the phonon dynamical matrix, which facilitates the efficient construction of the underlying Hamiltonian with translational invariance. We demonstrate the applicability of the method across a broad class of lattice models.