Beyond Bell sampling: stabilizer state learning and quantum pseudorandomness lower bounds on qudits

TL;DR

This paper investigates the limitations of Bell sampling on qudits, introduces new quantum algorithms for stabiliser state learning and pseudorandomness lower bounds, and demonstrates the boundaries of these techniques for higher-dimensional quantum systems.

Contribution

It characterizes Bell sampling limitations on qudits, proposes algorithms for stabiliser state identification and pseudorandomness bounds, extending quantum learning and complexity understanding.

Findings

Bell sampling yields uniform distribution on stabiliser states, revealing no information.

Quantum algorithm identifies stabiliser states with O(n) copies and O(n^4) time for prime dimensions.

Efficiently distinguishes Haar-random states from those with stabiliser fidelity, setting pseudorandomness bounds.

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.