Abstract
We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and k-wise uniformity of probability distributions: 1) Closeness testing is the problem of distinguishing whether two n-dimensional distributions are identical or at least ε -far in ℓ1 - or ℓ2 -distance. We show that the quantum query complexities for ℓ1 - and ℓ2 -closeness testing are O(n−−√/ε) and O(1/ε) , respectively, both of which achieve optimal dependence on ε , improving the prior best results of Gilyén and Li (2019) and 2) k-wise uniformity testing is the problem of distinguishing whether a distribution over {0,1}n is uniform when restricted to any k coordinates or ε -far from any such distribution. We propose the first quantum algorithm for this problem with query complexity O(nk−−√/ε) , achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity O(nk/ε2) by O’Donnell and Zhao (2018). Moreover, when k=2 our quantum algorithm outperforms any classical one because of the classical lower bound Ω(n/ε2) . All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.
Original language | English |
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Article number | 10508394 |
Pages (from-to) | 1-12 |
Number of pages | 12 |
Journal | IEEE Transactions on Information Theory |
Volume | 70 |
Issue number | 7 |
DOIs | |
Publication status | Published - 25 Apr 2024 |
Keywords / Materials (for Non-textual outputs)
- quantum computing
- quantum algorithm
- property testing
- probability distribution