  Understanding the limiting capabilities of classical methods in simulating
complex quantum systems is of paramount importance for quantum technologies.
Although many advanced approaches have been proposed and recently used to
challenge quantum advantage experiments, novel efficient methods for
approximate simulation of complex quantum systems are highly demanded. Here we
propose a scalable technique for approximate simulations of intermediate-size
quantum circuits on the basis of multi-scale entanglement renormalization
ansatz (MERA). MERA is a tensor network, whose geometry together with
orthogonality constraints imposed on its "elementary" tensors allows
approximating many-body quantum states lying beyond the area-law scaling of the
entanglement entropy. We benchmark the proposed technique for checkerboard-type
intermediate-size quantum circuits of 27 qubits with various depths. Our
approach paves a way to explore new efficient simulation techniques for quantum
many-body systems.
