  We investigate the problem of fast-forwarding quantum evolution, whereby the
dynamics of certain quantum systems can be simulated with gate complexity that
is sublinear in the evolution time. We provide a definition of fast-forwarding
that considers the model of quantum computation, the Hamiltonians that induce
the evolution, and the properties of the initial states. Our definition
accounts for any asymptotic complexity improvement of the general case and we
use it to demonstrate fast-forwarding in several quantum systems. In
particular, we show that some local spin systems whose Hamiltonians can be
taken into block diagonal form using an efficient quantum circuit, such as
those that are permutation-invariant, can be exponentially fast-forwarded. We
also show that certain classes of positive semidefinite local spin systems,
also known as frustration-free, can be polynomially fast-forwarded, provided
the initial state is supported on a subspace of sufficiently low energies.
Last, we show that all quadratic fermionic systems and number-conserving
quadratic bosonic systems can be exponentially fast-forwarded in a model where
quantum gates are exponentials of specific fermionic or bosonic operators,
respectively. Our results extend the classes of physical Hamiltonians that were
previously known to be fast-forwarded, while not necessarily requiring methods
that diagonalize the Hamiltonians efficiently. We further develop a connection
between fast-forwarding and precise energy measurements that also accounts for
polynomial improvements.
