New Quantum Algorithms for Computing Quantum Entropies and Distances
Qisheng Wang, Ji Guan, Junyi Liu, Zhicheng Zhang, Mingsheng Ying
TL;DR
This work develops a unified quantum framework that treats density operators as block-encodings and leverages quantum singular value transformation to compute a broad class of quantum entropies and distances. By extending block-encoding to general density operators and introducing primitives such as trace estimation, eigenvalue threshold projectors, and positive-power mappings, the authors achieve substantial speedups in the low-rank regime for von Neumann, Rényi, and Tsallis entropies, as well as for trace distance and fidelity, with complexities polynomial in rank $r$ and inverse error $1/\varepsilon$. They also establish lower bounds and hardness results (including QS ZK-based and DQC1-hardness) that delineate the limits of quantum advantage and highlight the nontrivial role of rank in complexity. The results offer practical quantum tools for analyzing quantum systems and channels, enabling efficient entropy and distance computations that were previously intractable for high-dimensional states. Overall, the paper advances a versatile, state-centric quantum algorithmic toolkit with potential impact on quantum information processing and verification tasks.
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 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 $Ω(N)$, 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.