Beyond Bell sampling: stabilizer state learning and quantum pseudorandomness lower bounds on qudits
Jonathan Allcock, Joao F. Doriguello, Gábor Ivanyos, Miklos Santha
TL;DR
Bell sampling provides limited information for qudits with $d>2$, so this work develops a formal framework to study qudit Bell difference sampling and its limitations. It introduces an involuted Weyl distribution and proves that stabiliser information concentrates on a subspace $\mathscr{M}+J(\mathscr{M})$, not the full stabiliser group, highlighting why Bell-based learning fails on qudits. The authors then present two stabiliser-state learning strategies: (i) Bell sampling with conjugates to efficiently identify the stabiliser with $O(n)$ copies and $O(n^3)$–$O(n^4)$ time, and (ii) a quadratic-function method that recovers $\mathbf{V},\mathbf{W}$ and phases from $O(n)$ copies in $O(n^3\operatorname{rank}(\mathbf{W}))$ time. They also develop a Haar-vs-stabiliser-distinguishing algorithm using a stabiliser tester to prove pseudorandomness lower bounds for doped Clifford circuits on qudits, with corollaries extending to general dimension and tight bounds on non-Clifford resources. Together, these results advance understanding of stabiliser-state learning and pseudorandomness on qudits, showing how to overcome Bell-based limitations and exposing fundamental limits for Clifford-based pseudorandomness with limited non-Clifford resources.
Abstract
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.