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+log{N})) 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+log{N})) 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.