Succinct quantum testers for closeness and k-wise uniformity of probability distributions

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness and
k
-wise uniformity of probability distributions. •
Closeness testing
is the problem of distinguishing whether two
n
-dimensional distributions are identical or at least ε-far in ℓ
¹
- or ℓ
²
-distance. We show that the quantum query complexities for ℓ
¹
- and ℓ
²
-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). •
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
(√
n
^k
/ε), achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexity
O
(
n
^k
/ε
²
) 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
/ε
²
). All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.