Branching programs are quite popular for studying time-space lower bounds. Bera et al. recently introduced the model of generalized quantum branching program aka. GQBP that generalized two earlier models of quantum branching programs. In this work we study a restricted version of GQBP with the motivation of proving bounds on the query-space requirement of quantum-query circuits. We show the first explicit query-space lower bound for our restricted version. We prove that the well-studied OR$_n$ decision problem, given a promise that at most one position of an $n$-sized Boolean array is a 1, satisfies the bound $Q^2 s = \Omega(n^2)$, where $Q$ denotes the number of queries and $s$ denotes the width of the GQBP. We then generalize the problem to show that the same bound holds for deciding between two strings with a constant Hamming distance; this gives us query-space lower bounds on problems such as Parity and Majority. Our results produce an alternative proof of the $\Omega(\sqrt{n})$-lower bound on the query complexity of any non-constant symmetric Boolean function.

翻译：分支程序在时间空间下界研究中颇为流行。Bera等人近期提出的广义量子分支程序（GQBP）模型，统一了两种早期的量子分支程序模型。本研究以证明量子查询电路的查询空间需求下界为目标，探讨一种受限版本的GQBP。我们首次针对该受限模型给出了显式的查询空间下界证明。通过分析经典OR$_n$判定问题（承诺$n$位布尔数组中至多存在一个1），我们证明了$Q^2 s = \Omega(n^2)$的下界关系，其中$Q$表示查询次数，$s$表示GQBP的宽度。进一步将该问题推广至恒定汉明距离下的二元串判定，获得了Parity和Majority等问题的查询空间下界。我们的结论为任意非常数对称布尔函数$\Omega(\sqrt{n})$查询复杂度下界提供了替代证明。