Bell sampling is a simple yet powerful measurement primitive that has recently attracted a lot of attention, and has proven to be a valuable tool in studying stabiliser states. Unfortunately, however, it is known that Bell sampling fails when used on qu\emph{d}its of dimension $d>2$. In this paper, we explore and quantify the limitations of Bell sampling on qudits, and propose new quantum algorithms to circumvent the use of Bell sampling in solving two important problems: learning stabiliser states and providing pseudorandomness lower bounds on qudits. More specifically, as our first result, we characterise the output distribution corresponding to Bell sampling on copies of a stabiliser state and show that the output can be uniformly random, and hence reveal no information. As our second result, for $d=p$ prime we devise a quantum algorithm to identify an unknown stabiliser state in $(\mathbb{C}^p)^{\otimes n}$ that uses $O(n)$ copies of the input state and runs in time $O(n^4)$. As our third result, we provide a quantum algorithm that efficiently distinguishes a Haar-random state from a state with non-negligible stabiliser fidelity. As a corollary, any Clifford circuit on qudits of dimension $d$ using $O(\log{n}/\log{d})$ auxiliary non-Clifford single-qudit gates cannot prepare computationally pseudorandom quantum states.

翻译：贝尔采样是一种简单而强大的测量原语，近期引起了广泛关注，已被证明是研究稳定子态的重要工具。然而，已知当将其用于维度$d>2$的qu\emph{d}it系统时，该采样方法会失效。本文探索并量化了贝尔采样在qudit系统中的局限性，并提出新的量子算法以规避其在解决两个重要问题时的使用：学习稳定子态与建立qudit系统伪随机性下界。具体而言，作为首个结果，我们刻画了稳定子态副本上贝尔采样对应的输出分布，并证明该输出可能呈现均匀随机性，因此不泄露任何信息。作为第二个结果，针对素数维$d=p$的情况，我们设计了一种量子算法来识别$(\mathbb{C}^p)^{\otimes n}$中的未知稳定子态，该算法仅需使用$O(n)$份输入态副本，且运行时间为$O(n^4)$。作为第三个结果，我们提出了一种能有效区分Haar随机态与具有非可忽略稳定子保真度的量子态的算法。由此推论，任意使用$O(\log{n}/\log{d})$个辅助非Clifford单qudit门、维度为$d$的qudit系统Clifford电路，均无法制备计算上伪随机的量子态。