We consider the recognition problem of the Dyck Language generalized for multiple types of brackets. We provide an algorithm with quantum query complexity $O(\sqrt{n}(\log n)^{0.5k})$, where $n$ is the length of input and $k$ is the maximal nesting depth of brackets. Additionally, we show the lower bound for this problem which is $O(\sqrt{n}c^{k})$ for some constant $c$. Interestingly, classical algorithms solving the Dyck Language for multiple types of brackets substantially differ form the algorithm solving the original Dyck language. At the same time, quantum algorithms for solving both kinds of the Dyck language are of similar nature and requirements.

中文翻译：

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多类型括号 Dyck 语言的量子算法

我们考虑对多种类型的括号进行泛化的 Dyck 语言的识别问题。我们提供了一个量子查询复杂度为 $O(\sqrt{n}(\log n)^{0.5k})$ 的算法，其中 $n$ 是输入的长度，$k$ 是括号的最大嵌套深度。此外，我们还展示了这个问题的下限，即某些常量 $c$ 的 $O(\sqrt{n}c^{k})$。有趣的是，为多种类型的括号求解 Dyck 语言的经典算法与求解原始 Dyck 语言的算法大不相同。同时，解决这两种 Dyck 语言的量子算法具有相似的性质和要求。