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 $\langleψ|e^{iHt}Oe^{-iHt}|ψ\rangle$ for global Pauli observables $O$ and stabilizer states $|ψ\rangle$, 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.
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 ⟨ψ∣eiHtOe−iHt∣ψ⟩ for global Pauli observables O and stabilizer states ∣ψ⟩, 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.
The local Hamiltonian (LH) problem, the quantum analog of the classical constraint satisfaction problem, is a cornerstone of quantum computation and complexity theory. It is known to be QMA-complete, indicating that it is challenging even for quantum computers. Interestingly, the guided local Hamiltonian (GLH) problem -- an LH problem with a guiding state that has a non-trivial overlap with the ground state -- can be efficiently solved on a quantum computer and is proved to be BQP-complete. This makes the GLH problem a valuable framework for exploring the fundamental separation between classical and quantum computation. Remarkably, the quantum algorithm for solving the GLH problem can be `dequantized' (i.e., made classically simulatable) under certain conditions, such as when only constant accuracy is required and when the Hamiltonian satisfies an unrealistic constant operator norm constraint. In this work, we relieve these restrictions by introducing a dequantized classical algorithm for a randomized quantum imaginary-time evolution quantum algorithm. We demonstrate that it achieves either limited or arbitrary constant accuracy, depending on whether the guiding state's overlap is general or exceeds a certain threshold. Crucially, our approach eliminates the constant operator norm constraint on the Hamiltonian, opening its applicability to realistic problems. Our results advance the classical solution of the GLH problem in practical settings and provide new insights into the boundary between classical and quantum computational power.
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good "guiding state" is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly (1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up to a polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2^O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.
We construct classical algorithms computing an approximation of the ground state energy of an arbitrary k-local Hamiltonian acting on n qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in poly(1/χ,n) time and poly(n) space, where χ denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in 2O(n) time and poly(n) space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.
Recent progress in the development of quantum technologies has enabled the direct investigation of the dynamics of increasingly complex quantum many-body systems.This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage.Here, we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian.For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error.In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb-Robinson bound.Our results use cluster-expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence.This has important physical consequences besides our efficient algorithms.In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.
We establish general conditions for the rapid convergence of propagation algorithms for computing expectation values of local observables in quantum many-body systems, covering both spin and fermionic systems. Our results apply to evolutions that are sufficiently contractive, providing a general mechanism by which contractivity controls the growth generated by local interactions and enables efficient classical simulation. As a first application, we consider noisy time evolution generated by local Hamiltonians on arbitrary interaction graphs. We show that the dynamics can be efficiently simulated when the noise strength λ is sufficiently large compared to the degree of the interaction graph. More generally, for systems with an interaction strength u, our bounds determine a time horizon tmax as a function of the interaction strength u, the noise strength λ, and the interaction structure below which the algorithm is efficient. In the noiseless limit, λ=0, our analysis extends the efficiently simulable time scale from log(1/u) proved in Facelli, Fawzi, and Fawzi (2026) to tmax∼1/u matching the recent improvement by Zhao, Marvian and Tong (2026). As a second application, we apply the same framework to the computation of local observables in Gibbs states of local Hamiltonians. We consider a pseudo-Lindbladian whose steady state is the Gibbs state and show that, at sufficiently high temperature, its propagation algorithm converges rapidly. In particular, for weakly-interacting systems with interaction strength u, our result applies up to inverse temperatures β∼log(1/u).
We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable A at time t. By analyzing the convergence of the perturbative expansion in the interaction strength λ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime λ∣t∣2D+1=O(1), when the Hamiltonian is geometrically local on a D-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to λ∣t∣=O(1) with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to λ∣t∣=O(1), modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.