Quantum and classical query complexities for generalized Deutsch–Jozsa problems

Abstract

The Deutsch–Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. We generalize the Deutsch–Jozsa problem from the perspective of functional correlation, i.e., given two unknown n-bit Boolean functions f, g, the testing problem is to determine whether $$|C(f,g)|=0$$ or $$|C(f,g)|=\epsilon $$, where $$1/2^{(n-1)}\le \epsilon \le 1$$, promised that one of these is the case. Firstly, we propose two exact quantum algorithms for making distinction between $$|C(f,g)|=0$$ and $$|C(f,g)|=\epsilon $$ using the Deutsch–Jozsa algorithm and also the amplitude amplification technique with query complexity of $$O(1/\epsilon )$$, where C(f, g) denotes the correlation between two Boolean functions f, g. Secondly, we present a lower bound $${\varOmega }(1/\epsilon )$$ on the above promised problem, which proves that our quantum algorithms are optimal. Thirdly, we can accurately distinguish $$|C(f,g)|=\varepsilon $$ from $$|C(f,g)|=1$$ using the similar methods mentioned above, where $$0\le \varepsilon \le 1-1/2^{(n-1)}$$, promised that one of these is the case. We give a lower bound $${\varOmega }(1/\sqrt{1-\varepsilon })$$ and an upper bound $$O(4/\sqrt{1-\varepsilon ^{2}})$$ on this promised problem, which implies that the two bounds are almost tight. However, the query complexity of classical deterministic algorithms for two promised problems is $${\varTheta }(2^{n})$$.