Entanglement spectroscopy with a depth-two quantum circuit

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.