  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.
