  We determine the computational power of isometric tensor network states
(isoTNS), a variational ansatz originally developed to numerically find and
compute properties of gapped ground states and topological states in two
dimensions. By mapping 2D isoTNS to 1+1D unitary quantum circuits, we find that
computing local expectation values in isoTNS is $\textsf{BQP}$-complete. We
then introduce injective isoTNS, which are those isoTNS that are the unique
ground states of frustration-free Hamiltonians, and which are characterized by
an injectivity parameter $\delta\in(0,1/D]$, where $D$ is the bond dimension of
the isoTNS. We show that injectivity necessarily adds depolarizing noise to the
circuit at a rate $\eta=\delta^2D^2$. We show that weakly injective isoTNS
(small $\delta$) are still $\textsf{BQP}$-complete, but that there exists an
efficient classical algorithm to compute local expectation values in strongly
injective isoTNS ($\eta\geq0.41$). Sampling from isoTNS corresponds to
monitored quantum dynamics and we exhibit a family of isoTNS that undergo a
phase transition from a hard regime to an easy phase where the monitored
circuit can be sampled efficiently. Our results can be used to design provable
algorithms to contract isoTNS. Our mapping between ground states of certain
frustration-free Hamiltonians to open circuit dynamics in one dimension fewer
may be of independent interest.
