Quantum algorithm for estimating Renyi entropies of quantum states

摘要: We describe a quantum algorithm to estimate the $\alpha$-Renyi entropy of an unknown density matrix $\rho\in\mathcal{C}^{d\times d}$ for $\alpha\neq 1$ by combining recent technique singular value transformations with method estimating normalised traces in one clean qubit model. consider oracular input model where state is prepared via oracle that outputs purified version state, assumed be non-singular. Our additive precision $\epsilon$, using expected total number $O\left(\frac{1}{(x\epsilon)^2}\right)$ independent applications circuit which coherently queries unitary $O\left(\frac{1}{\delta}\log \frac{d}{\epsilon}\right)$ times, each case measuring single output qubit. Here $\delta$ lower cutoff on smallest eigenvalue $\rho$ and $x=\frac{1}{d}\!\mathop{Tr}{\rho^\alpha}$. The measurements made this can compared results sample complexity generally require $\Theta(d^2/\epsilon^2)$ samples. Furthermore, we also show multiplicative approximations obtained iteratively approximations, overhead logarithmic dimension $d$.