与自相关谱相关的高效量子算法
Efficient Quantum Algorithms related to Autocorrelation Spectrum
摘要：
在本文中，我们提出了有效的概率算法的自相关谱的几个问题。首先，我们提出了一个量子算法，从沃尔什频谱的任何衍生物的$f()$的样本。非正式地，布尔函数$f()$在某个点$a$处的自相关系数测量值$f(x)$和$f(x \oplus a)$之间的平均相关性。布尔函数的导数是自相关到$f()$的多个值之间的相关性的扩展。沃尔什谱是很好的研究，主要是由于它的连接到量子线路的Deutsch-Jozsa问题。我们将这个思想推广到“高阶Deutsch-Jozsa”量子算法，得到对应于$f()$的某个导数的Walsh谱中的大绝对值的点.此外，我们还设计了一种算法，根据自相关系数的平方对输入点进行采样。最后，我们提供了一套不同的算法，用于估计特定系数的平方或其平方的累积和。
英文摘要：
In this paper, we propose efficient probabilistic algorithms for several problems regarding the autocorrelation spectrum. First, we present a quantum algorithm that samples from the Walsh spectrum of any derivative of $f()$. Informally, the autocorrelation coefficient of a Boolean function $f()$ at some point $a$ measures the average correlation among the values $f(x)$ and $f(x \oplus a)$. The derivative of a Boolean function is an extension of autocorrelation to correlation among multiple values of $f()$. The Walsh spectrum is well-studied primarily due to its connection to the quantum circuit for the Deutsch-Jozsa problem. We extend the idea to "Higher-order Deutsch-Jozsa" quantum algorithm to obtain points corresponding to large absolute values in the Walsh spectrum of a certain derivative of $f()$. Further, we design an algorithm to sample the input points according to squares of the autocorrelation coefficients. Finally we provide a different set of algorithms for estimating the square of a particular coefficient or cumulative sum of their squares.
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