We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum R\'{e}nyi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the prior best (and even quantum) ones in the low-rank 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 $\tilde O(r/\varepsilon^2)$, $\tilde O(r^5/\varepsilon^6)$ and $\tilde O(r^{6.5}/\varepsilon^{7.5})$, respectively. By contrast, prior quantum algorithms for the von Neumann entropy and trace distance usually have time complexity $\Omega(N)$, and the prior best one for fidelity has time complexity $\tilde O(r^{12.5}/\varepsilon^{13.5})$. The key idea of our quantum algorithms is to extend block-encoding 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. 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 on the minimal non-zero eigenvalue of density operators.

翻译：我们提出了一系列用于计算多种量子熵与距离的量子算法，包括冯·诺依曼熵、量子Rényi熵、迹距离以及保真度。在低秩情形下，所提算法显著超越了先前最优（甚至量子）算法，其中部分算法实现了指数级加速。具体而言，对于秩为r的N维量子态，我们提出的计算冯·诺依曼熵、迹距离和保真度（允许加性误差ε）的量子算法时间复杂度分别为$\tilde O(r/\varepsilon^2)$、$\tilde O(r^5/\varepsilon^6)$和$\tilde O(r^{6.5}/\varepsilon^{7.5})$。相比之下，先前计算冯·诺依曼熵与迹距离的量子算法通常具有$\Omega(N)$的时间复杂度，而先前最优保真度算法的时间复杂度为$\tilde O(r^{12.5}/\varepsilon^{13.5})$。我们量子算法的核心思想是将先前工作中针对酉算子的块编码技术扩展至量子态（即密度算子）。这一目标通过开发多种操控量子态并从中提取信息的便捷技术得以实现。相较于现有方法，我们技术的优势在于无需对密度算子施加任何限制；与此形成鲜明对比的是，先前方法通常要求密度算子的最小非零特征值存在下界。