  We benchmark the accuracy of a variational quantum eigensolver based on a
finite-depth quantum circuit encoding ground state of local Hamiltonians. We
show that in gapped phases, the accuracy improves exponentially with the depth
of the circuit. When trying to encode the ground state of conformally invariant
Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy
improves slowly with the number of layers, and a finite-size regime where it
improves again exponentially. The cross-over between the two regimes happens at
a critical number of layers whose value increases linearly with the size of the
system. We discuss the implication of these observations in the context of
comparing different variational ansatz and their effectiveness in describing
critical ground states.
