Hamiltonian simulation for low-energy states with optimal time dependence

Understand

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 $\Omega(t\lambda)$.