  We perform a detailed resource estimate for the prospect of using deep
entanglement renormalization ansatz (DMERA) on a fault-tolerant quantum
computer, focusing on the regime in which the target system is large. For
probing a relatively large system size ($64\times 64$), we observe up to an
order of magnitude reduction in the number of qubits, compared to the
approaches based on quantum phase estimation (QPE). We discuss two
complementary strategies to measure the energy. The first approach is based on
a random sampling of the local terms of the Hamiltonian, requiring
$\mathcal{O}(1/\epsilon^2)$ invocations of quantum circuits, each of which have
depth of at most $\mathcal{O}(\log N)$, where $\epsilon$ is the relative
precision in the energy and $N$ is the system size. The second approach is
based on a coherent estimation of the expectation value of observables averaged
over space, which achieves the Heisenberg scaling while incurring only a
logarithmic cost in the system size. For estimating the energy per site of
$\epsilon$, $\mathcal{O}\left(\frac{\log N}{\epsilon} \right)$ $T$ gates and
$\mathcal{O}\left(\log N \right)$ qubits suffice. The constant factor of the
leading contribution is shown to be determined by the depth of the DMERA
circuit, the gates used in the ansatz, and the periodicity of the circuit. We
also derive tight bounds on the variance of the energy gradient, assuming the
gates are random Pauli rotations.
