  Noisy intermediate-scale quantum (NISQ) computers have gate errors and
decoherence, limiting the depth of circuits that can be implemented on them. A
strategy for NISQ algorithms is to reduce the circuit depth at the expense of
increasing the qubit count. Here, we exploit this trade-off for an application
called entanglement spectroscopy, where one computes the entanglement of a
state $| \psi \rangle$ on systems $AB$ by evaluating the R\'enyi entropy of the
reduced state $\rho_A = {\rm Tr}_B(| \psi \rangle \langle \psi |)$. For a
$k$-qubit state $\rho(k)$, the R\'enyi entropy of order $n$ is computed via
${\rm Tr}(\rho(k)^{n})$, with the complexity growing exponentially in $k$ for
classical computers. Johri, Steiger, and Troyer [PRB 96, 195136 (2017)]
introduced a quantum algorithm that requires $n$ copies of $| \psi \rangle$ and
whose depth scales linearly in $k*n$. Here, we present a quantum algorithm
requiring twice the qubit resources ($2n$ copies of $| \psi \rangle$) but with
a depth that is independent of both $k$ and $n$. Surprisingly this depth is
only two gates. Our numerical simulations show that this short depth leads to
an increased robustness to noise.
