超越奇偶校验的精确量子查询算法--超越对称函数
Exact Quantum Query Algorithms Outperforming Parity -- Beyond The Symmetric functions
摘要：
在精确量子查询模型中，几乎所有存在非平凡查询算法的布尔函数本质上都是对称的。在这个领域中最著名的技术利用奇偶校验决策树，其中两个比特的奇偶校验可以通过单个查询获得。因此，精确的量子查询算法优于奇偶校验决策树是罕见的。在本文中，我们首先获得最佳精确量子查询算法（$Q_{algo}(f)$）的直接和为基础的类$Ω\left( 2^{\frac{\sqrt{n}}{2}} \right)$非对称函数。我们构造这些算法，通过分析的代数范式与一个新的解开策略。接下来，我们得到的广义奇偶校验决策树复杂度（$D_{\oplus}(f)$）分析沃尔什频谱。最后，我们表明，$Q_{algo}$的查询复杂性是$\lceil \frac{3n}{4} \rceil$，而$D_{\oplus}(f)$不同的类之间的$n-1$和$\lceil \frac{3n}{4} \rceil+1$，强调在许多情况下，这两个措施之间的线性分离。据我们所知，这是第一个家庭的算法超越广义奇偶校验（因此奇偶校验）的一大类非对称函数。我们还实现了这些技术的一个更大的（双指数在$\frac{n}{4}$）类的Maiorana-McFarland型函数，但只能获得部分结果使用类似的算法技术。
英文摘要：
In Exact Quantum Query model, almost all of the Boolean functions for which non-trivial query algorithms exist are symmetric in nature. The most well known techniques in this domain exploit parity decision trees, in which the parity of two bits can be obtained by a single query. Thus, exact quantum query algorithms outperforming parity decision trees are rare. In this paper we first obtain optimal exact quantum query algorithms ($Q_{algo}(f)$) for a direct sum based class of $Ω\left( 2^{\frac{\sqrt{n}}{2}} \right)$ non-symmetric functions. We construct these algorithms by analyzing the algebraic normal form together with a novel untangling strategy. Next we obtain the generalized parity decision tree complexity ($D_{\oplus}(f)$) analysing the Walsh Spectrum. Finally, we show that query complexity of $Q_{algo}$ is $\lceil \frac{3n}{4} \rceil$ whereas $D_{\oplus}(f)$ varies between $n-1$ and $\lceil \frac{3n}{4} \rceil+1$ for different classes, underlining linear separation between the two measures in many cases. To the best of our knowledge, this is the first family of algorithms beyond generalized parity (and thus parity) for a large class of non-symmetric functions. We also implement these techniques for a larger (doubly exponential in $\frac{n}{4}$) class of Maiorana-McFarland type functions, but could only obtain partial results using similar algorithmic techniques.
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