  The multi-scale entanglement renormalization ansatz (MERA) is a hierarchical
class of tensor network states motivated by the real-space renormalization
group, and it is used to simulate strongly correlated quantum many-body
systems. For prominent MERA structures in one and two spatial dimensions, we
determine the optimal scaling of contraction costs as well as corresponding
contraction sequences and algorithmic phase diagrams. This is motivated by
recent efforts to employ MERA in hybrid quantum-classical algorithms, where the
MERA tensors are Trotterized, i.e., chosen as brickwall circuits of two-qubit
gates, and observables as well as energy gradients are evaluated by sampling
causal-cone states. We investigate whether tensor Trotterization and/or
variational Monte Carlo sampling can lead to quantum-inspired classical MERA
algorithms that perform better than the traditional alternating least-squares
optimization of full MERA. Algorithmic phase diagrams indicate the best MERA
method depending on the scaling of the energy accuracy and the number of
Trotter steps with the bond dimension.
