检验分布性质的量子算法
Quantum algorithms for testing properties of distributions
摘要：
假设一个人可以访问从某个N元集合上的两个未知概率分布P和Q生成样本的预言机。在L_1范数下，需要多少样本来检验两个分布是接近还是远离？在过去的几年里，这个问题和相关的问题在性能测试领域得到了广泛的研究。本文研究了检验分布性质的量子算法。结果表明，在量子环境下，P和Q之间的L_1-距离可以用近似N^{1/2}次查询以恒定精度估计，而经典计算机需要Ω（N）次查询.我们还描述了量子算法测试的均匀性和一致性与查询复杂性O（N^{1/3}）。这些问题的经典查询复杂度已知为Ω（N^{1/2}）。
英文摘要：
Suppose one has access to oracles generating samples from two unknown probability distributions P and Q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L_1-norm ? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L_1-distance between P and Q can be estimated with a constant precision using approximately N^{1/2} queries in the quantum settings, whereas classical computers need Ω(N) queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity O(N^{1/3}). The classical query complexity of these problems is known to be Ω(N^{1/2}).
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