加权矩阵和二次剩余的量子算法
Quantum Algorithms for Weighing Matrices and Quadratic Residues
摘要：
在这篇文章中，我们将研究如何利用组合对象的结构，如阿达玛矩阵和加权矩阵来设计新的量子算法。我们展示了如何使用加权矩阵的属性来构造一个量子查询复杂度明显低于经典问题的问题。它指出，该计划捕获伯恩斯坦和Vazirani的内积协议，以及格罗弗的搜索算法。在文章的第二部分，我们考虑佩利的阿达玛矩阵，它依赖于有限域上的二次特征的性质的建设。我们设计了一个查询问题，使用勒让德符号chi（这表明，如果一个有限域F_q的元素是一个二次剩余或没有）。证明了对于移位Legendre函数f_s（i）=chi（i+s），只需对f_s进行两次量子调用，就可以精确地得到F_q中的未知量s.这与任何经典的概率过程需要多于log（q）+ log（（1-e）/2）查询来解决相同问题的观察形成鲜明对比。
英文摘要：
In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is ignificantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol chi (which indicates if an element of a finite field F_q is a quadratic residue or not). It is shown how for a shifted Legendre function f_s(i)=chi(i+s), the unknown s in F_q can be obtained exactly with only two quantum calls to f_s. This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log(q) + log((1-e)/2) queries to solve the same problem.
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