  We present a quantum algorithm to compute the entanglement spectrum of
arbitrary quantum states. The interesting universal part of the entanglement
spectrum is typically contained in the largest eigenvalues of the density
matrix which can be obtained from the lower Renyi entropies through the
Newton-Girard method. Obtaining the $p$ largest eigenvalues
($\lambda_1>\lambda_2\ldots>\lambda_p$) requires a parallel circuit depth of
$\mathcal{O}(p(\lambda_1/\lambda_p)^p)$ and $\mathcal{O}(p\log(N))$ qubits
where up to $p$ copies of the quantum state defined on a Hilbert space of size
$N$ are needed as the input. We validate this procedure for the entanglement
spectrum of the topologically-ordered Laughlin wave function corresponding to
the quantum Hall state at filling factor $\nu=1/3$. Our scaling analysis
exposes the tradeoffs between time and number of qubits for obtaining the
entanglement spectrum in the thermodynamic limit using finite-size digital
quantum computers. We also illustrate the utility of the second Renyi entropy
in predicting a topological phase transition and in extracting the localization
length in a many-body localized system.
