  Using the path integral associated to a cMERA tensor network, we provide an
operational definition for the complexity of a cMERA circuit/state which is
relevant to investigate the complexity of states in quantum field theory. In
this framework, it is possible to explicitly establish the correspondence
(Minimal) Complexity $=$ (Least) Action. Remarkably, it is also shown how the
cMERA complexity action functional can be seen as the action of a Liouville
field theory, thus establishing a connection with two dimensional quantum
gravity. Concretely, the Liouville mode is identified with the variational
parameter defining the cMERA circuit. The rate of complexity growth along the
cMERA renormalization group flow is obtained and shown to saturate limits which
are in close resemblance to the fundamental bounds to the speed of evolution in
unitary quantum dynamics, known as quantum speed limits. We also show that the
complexity of a cMERA circuit measured through these complexity functionals,
can be cast in terms of the variationally-optimized amount of left-right
entanglement created along the cMERA renormalization flow. Our results suggest
that the patterns of entanglement in states of a QFT could determine their dual
gravitational descriptions through a principle of least complexity.
