Simulating large-scale lattice dynamics remains a long-standing challenge in condensed matter and materials science, where mechanical and thermal behaviors arise from coupled vibrational modes. We introduce a quantum algorithm that reformulates general harmonic lattice dynamics as a time-dependent Schrödinger equation governed by a sparse, Hermitian Hamiltonian. This enables the use of Hamiltonian simulation techniques on quantum devices, offering exponential speedup in the number of atoms $N$. Our approach applies to arbitrary harmonic lattices with vector-valued dynamics. A key ingredient is a matrix-valued Fejér-Riesz factorization of the phonon dynamical matrix, which preserves translational symmetry and enables efficient assembly of the Hamiltonian operator. We demonstrate the method’s applicability across a broad class of lattice models.

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• Lattice dynamics
• Quantum computation
• Quantum simulation