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 \({\varTheta }(2^{n})\).

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广义Deutsch-Jozsa问题的量子和经典查询复杂度

Deutsch-Jozsa算法是量子算法的第一个例子，它比任何可能的确定性经典算法都快得多。我们从函数相关的角度推广Deutsch-Jozsa问题，即给定两个未知的n位布尔函数f，  g，测试问题是确定\（| C（f，g）| = 0 \）还是\（| C（f，g）| = \ epsilon \），其中\（1/2 ^ {（n-1）} \ le \ epsilon \ le 1 \）承诺是其中一种情况。首先，我们提出了两种精确的量子算法来区分\（| C（f，g）| = 0 \）和\（| C（f，g）| = \ epsilon \）使用Deutsch-Jozsa算法以及具有查询复杂度\（O（1 / \ epsilon）\）的幅度放大技术，其中C（f，  g）表示两个布尔函数f，  g的相关性。其次，针对上述承诺问题提出了下界\（{\ varOmega}（1 / \ epsilon）\），这证明了我们的量子算法是最优的。第三，我们可以使用上述类似方法将\（| C（f，g）| = \ varepsilon \）与\（| C（f，g）| = 1 \）准确区分开，其中\（0 \ le \ varepsilon \ le 1-1 / 2 ^ {（n-1）} \），承诺情况之一。我们给下界\（{\ varOmega}（1 / \ sqrt {1- \ varepsilon}）\）和上界\（O（4 / \ sqrt {1- \ varepsilon ^ {2}}）\），这意味着两个界限几乎是紧密的。但是，经典确定性算法针对两个承诺的问题的查询复杂度为\（{\ varTheta}（2 ^ {n}）\）。