Classical Algorithms for Hamiltonian Dynamics Mean Value and Guided Local Hamiltonian Problem

The efficient simulation of quantum dynamics and ground states is a central challenge in physics and a key frontier for quantum advantage. While short-time evolution in one-dimensional systems can often be simulated classically, extending this to higher dimensions remains difficult. Here, we introduce an efficient classical algorithm for simulating the short-time dynamics of arbitrary local quantum systems. For any local Hamiltonian $H$ and constant evolution time $t$, our method estimates expectation values of the form $\langle\psi|e^{iHt}Oe^{-iHt}|\psi\rangle$ for global Pauli observables $O$ and stabilizer states $|\psi\rangle$, with high precision and exponentially small failure probability. Furthermore, we present a classical dequantization of a tailored quantum algorithm that efficiently solves the guided local Hamiltonian (GLH) problem to constant additive error - previously considered classically hard and hence a promising candidate for quantum computational advantage. These results reveal unexpected classical tractability in constant-time quantum dynamics and fundamental connections between Hamiltonian dynamics mean value and the GLH problem. Our work refines the boundary between classical and quantum computational power, identifying sharper criteria for regimes where quantum advantage may genuinely emerge.