Two of the most well-known quantum algorithms, those introduced by Deutsch–Jozsa and Bernstein–Vazirani, can solve promise problems with just one function query, showing an oracular separation with deterministic classical algorithms. In this work, we generalise those methods to study a family of quantum algorithms that can, with just one query, exactly solve promise problems stated over Boolean functions. We also show that these problems can be naturally ordered, inducing a partially ordered set of promise problems. We study the properties of such a poset, showing that the Deutsch–Jozsa and Bernstein–Vazirani problems are, in a certain sense, extremal problems in it, determining some of its automorphisms and proving that it is connected. We also prove that, for the problems in the poset, the corresponding classical query complexities can take any value between 1 and \(2^{n-1}+1\).

由 Deutsch–Jozsa 和 Bernstein–Vazirani 引入的两种最著名的量子算法可以仅通过一个函数查询来解决承诺问题，显示出使用确定性经典算法的预言式分离。在这项工作中，我们将这些方法推广到研究一系列量子算法，这些算法只需一个查询，就可以准确解决布尔函数上的承诺问题。我们还表明，这些问题可以自然排序，从而产生一组偏序的 Promise 问题。我们研究了这样一个偏序集的性质，表明 Deutsch-Jozsa 和 Bernstein-Vazirani 问题在某种意义上是其中的极值问题，确定了它的一些自同构并证明了它是连通的。我们还证明，对于偏序组中的问题，\(2^{n-1}+1\)。