Quantum algorithms on Walsh transform and Hamming distance for Boolean functions

Abstract

Walsh spectrum or Walsh transform is an alternative description of Boolean functions. In this paper, we explore quantum algorithms to approximate the absolute value of Walsh transform $W_f$ at a single point $z_{0}$ (i.e., $|W_f(z_{0})|$) for n-variable Boolean functions with probability at least $\frac{8}{\pi ^{2}}$ using the number of $O(\frac{1}{|W_f(z_{0})|\varepsilon })$ queries, promised that the accuracy is $\varepsilon$, while the best known classical algorithm requires $O(2^{n})$ queries. The Hamming distance between Boolean functions is used to study the linearity testing and other important problems. We take advantage of Walsh transform to calculate the Hamming distance between two n-variable Boolean functions f and g using O(1) queries in some cases. Then, we exploit another quantum algorithm which converts computing Hamming distance between two Boolean functions to quantum amplitude estimation (i.e., approximate counting). If $Ham(f,g)=t\ne 0$, we can approximately compute Ham(f, g) with probability at least $\frac{2}{3}$ by combining our algorithm and ${Approx-Count(f,\varepsilon )\, algorithm}$ using the expected number of $\varTheta ( \sqrt{\frac{N}{(\lfloor \varepsilon t\rfloor +1)}}+\frac{\sqrt{t(N-t)}}{\lfloor \varepsilon t\rfloor +1})$ queries, promised that the accuracy is $\varepsilon$. Moreover, our algorithm is optimal, while the exact query complexity for the above problem is $\varTheta (N)$ and the query complexity with the accuracy $\varepsilon$ is $O(\frac{1}{\varepsilon ^{2}}N/(t+1))$ in classical algorithm, where $N=2^{n}$. Finally, we present three exact quantum query algorithms for two promise problems on Hamming distance using O(1) queries, while any classical deterministic algorithm solving the problem uses $\varOmega (2^{n})$ queries.