New Quantum Algorithms for Computing Quantum Entropies and Distances
Abstract
We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the best known (and even quantum) ones in the lowrank case, some of which achieve exponential speedups. In particular, for $N$dimensional quantum states of rank $r$, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error $\varepsilon$ have time complexity of $\tilde O(r^2/\varepsilon^2)$, $\tilde O(r^5/\varepsilon^6)$ and $\tilde O(r^{6.5}/\varepsilon^{7.5})$, respectively. In contrast, the known algorithms for the von Neumann entropy and trace distance require quantum time complexity of $\Omega(N)$ [AISW19,GL20,GHS21], and the best known one for fidelity requires $\tilde O(r^{21.5}/\varepsilon^{23.5})$ [WZC+21]. The key idea of our quantum algorithms is to extend blockencoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. In particular, we introduce a novel technique for eigenvalue transformation of density operators and their (noninteger) positive powers, based on the powerful quantum singular value transformation (QSVT) [GSLW19]. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound of the minimal nonzero eigenvalue of density operators. In addition, we provide some techniques of independent interest for trace estimation, linear combinations, and eigenvalue threshold projectors of (subnormalized) density operators, which will be, we believe, useful in other quantum algorithms.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 arXiv:
 arXiv:2203.13522
 Bibcode:
 2022arXiv220313522W
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 51 pages