Hamiltonian simulation for low-energy states with optimal time dependence

Abstract

We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. Assuming access to a block-encoding of $H'=(H-E)/\lambda$ for some $E \in \mathbb R$, the goal is to implement an $\epsilon$-approximation to $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+\Delta/\lambda]$ of $H'$. We present a quantum algorithm that uses $O(t\sqrt{\lambda\Gamma} + \sqrt{\lambda/\Gamma}\log(1/\epsilon))$ queries to the block-encoding for any $\Gamma$ such that $\Delta \leq \Gamma \leq \lambda$. When $\log(1/\epsilon) = o(t\lambda)$ and $\Delta/\lambda = o(1)$, this result improves over generic methods with query complexity $\Omega(t\lambda)$. Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform. Using standard access models for $H$, we show that the ability to efficiently block-encode $H'$ is equivalent to $H$ being what we refer to as a "gap-amplifiable" Hamiltonian. This includes physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low-energy simulation algorithms. We also provide lower bounds for low-energy simulation. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to $t$, $\Delta$, and $\lambda$. In the practically relevant regime where $\log (1/\epsilon) = o(t\Delta)$ and $\Delta/\lambda = o(1)$, we also prove a matching lower bound in gate complexity (up to log factors). To establish the query lower bounds, we consider $\mathrm{PARITY}\circ\mathrm{OR}$ and degree bounds on trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction acting on a low-energy state.