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 vertical bar psi > on systems AB by evaluating the Renyi entropy of the reduced state rho(A) = Tr-B(vertical bar psi > and whose depth scales linearly in k*n. Here, we present a quantum algorithm requiring twice the qubit resources (2n copies of vertical bar psi >) 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.