  We propose a variational quantum eigensolver (VQE) for the simulation of
strongly-correlated quantum matter based on a multi-scale entanglement
renormalization ansatz (MERA) and gradient-based optimization. This MERA
quantum eigensolver can have substantially lower computation costs than
corresponding classical algorithms. Due to its narrow causal cone, the
algorithm can be implemented on noisy intermediate-scale quantum (NISQ) devices
and still describe large systems. It is particularly attractive for ion-trap
devices with ion-shuttling capabilities. The number of required qubits is
system-size independent, and increases only to a logarithmic scaling when using
quantum amplitude estimation to speed up gradient evaluations. Translation
invariance can be used to make computation costs square-logarithmic in the
system size and describe the thermodynamic limit. We demonstrate the approach
numerically for a MERA with Trotterized disentanglers and isometries. With a
few Trotter steps, one recovers the accuracy of the full MERA.
