  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.
