We study the deep multi-scale entanglement renormalization ansatz (DMERA) on quantum hardware and the causal cone of a subset of the qubits which make up the ansatz. This causal cone spans O(M+logN) physical qubits on a quantum device, where M and N are the subset size and the total number qubits in the ansatz respectively. This allows for the determination of the von Neumann entanglement entropy of the N qubit wave-function using O(M+logN) qubits by diagonalization of the reduced density matrix (RDM). We show this by randomly initializing a 16-qubit DMERA and diagonalizing the resulting RDM of the M-qubit subsystem using density matrix simulation. As an example of practical interest, we also encode the variational ground state of the quantum critical long-range transverse field Ising model (LRTIM) on 8 spins using DMERA. We perform density matrix simulation with and without noise to obtain entanglement entropies in separate experiments using only 4 qubits. Finally we repeat the experiment on the IBM Kyoto backend reproducing simulation results.