  We characterize the variational power of quantum circuit tensor networks in
the representation of physical many-body ground-states. Such tensor networks
are formed by replacing the dense block unitaries and isometries in standard
tensor networks by local quantum circuits. We explore both quantum circuit
matrix product states and the quantum circuit multi-scale entanglement
renormalization ansatz, and introduce an adaptive method to optimize the
resulting circuits to high fidelity with more than $10^4$ parameters. We
benchmark their expressiveness against standard tensor networks, as well as
other common circuit architectures, for the 1D/2D Heisenberg and 1D
Fermi-Hubbard models. We find quantum circuit tensor networks to be
substantially more expressive than other quantum circuits for these problems,
and that they can even be more compact than standard tensor networks.
Extrapolating to circuit depths which can no longer be emulated classically,
this suggests a region of advantage in quantum expressiveness in the
representation of physical ground-states.
