Beyond Bell sampling: stabilizer state learning and quantum pseudorandomness lower bounds on qudits
<|reference_start|>Beyond Bell sampling: stabilizer state learning and quantum pseudorandomness lower bounds on qudits: 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.<|reference_end|>
@article{allcock2024beyond,
    title={Beyond Bell sampling: stabilizer state learning and quantum
  pseudorandomness lower bounds on qudits},
    author={Jonathan Allcock, Joao F. Doriguello, G'abor Ivanyos, Miklos Santha},
    journal={arXiv preprint arXiv:2405.06357},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.06357},
    primaryClass={quant-ph cs.CC cs.DS}
}