Beyond Bell sampling: stabilizer state learning and quantum pseudorandomness lower bounds on qudits
CoRR(2024)
摘要
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 qudits 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
(ℂ^p)^⊗ 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(logn/logd) auxiliary non-Clifford single-qudit
gates cannot prepare computationally pseudorandom quantum states.
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