Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

中文翻译：

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精确的 Quantum 1 查询算法和复杂性

Deutsch-Jozsa 问题 (D-J) 具有精确的量子 1 查询复杂性（“精确”意味着没有错误），但需要超指数查询才能获得最优经典确定性决策树。D-J 问题等价于一个对称的偏布尔函数，实际上，所有具有精确量子 1-查询复杂度的对称偏布尔函数都已经被找出，并且这些函数都可以通过 D-J 算法来计算。一个特殊情况是所有具有精确量子 1 查询复杂度的对称布尔函数都直接跟随，并且这些函数显然也是具有精确量子 1 查询复杂度的全布尔函数。然后存在关于具有精确量子 1 查询复杂性的部分布尔函数的未决问题，并且已经发现了新结果，但仍有一些问题悬而未决。在本文中，我们回顾了这些关于精确量子 1 查询复杂度的结果，特别是，我们还获得了一个新结果，即构造了一个具有精确量子 1 查询复杂度的部分布尔函数，它不能通过 D-J 算法计算。指出进一步的问题，以供今后研究。