Classical branching programs are studied to understand the space complexity of computational problems. Prior to this work, Nakanishi and Ablayev had separately defined two different quantum versions of branching programs that we refer to as NQBP and AQBP. However, none of them, to our satisfaction, captures the intuitive idea of being able to query different variables in superposition in one step of a branching program traversal. Here we propose a quantum branching program model, referred to as GQBP, with that ability. To motivate our definition, we explicitly give examples of GQBP for n-bit Deutsch-Jozsa, n-bit Parity, and 3-bit Majority with optimal lengths. We the show several equivalences, namely, between GQBP and AQBP, GQBP and NQBP, and GQBP and query complexities (using either oracle gates and a QRAM to query input bits). In way this unifies the different results that we have for the two earlier branching programs, and also connects them to query complexity. We hope that GQBP can be used to prove space and space-time lower bounds for quantum solutions to combinatorial problems.

翻译：经典分支程序的研究旨在理解计算问题的空间复杂度。在本工作之前，Nakanishi和Ablayev分别定义了两种不同的量子分支程序版本，分别称为NQBP和AQBP。然而，这两种模型均未能令人满意地捕捉到在分支程序遍历的每一步中能够以叠加态查询不同变量的直观思想。本文提出了一种具备这一能力的量子分支程序模型，称为GQBP。为阐明这一定义，我们明确给出了n比特Deutsch-Jozsa问题、n比特Parity问题以及3比特Majority问题的最优长度GQBP实例。随后，我们证明了若干等价关系，即GQBP与AQBP之间、GQBP与NQBP之间，以及GQBP与查询复杂度（通过使用预言门和QRAM查询输入比特）之间的等价性。这在一定程度上统一了先前两种分支程序模型的不同结果，并将其与查询复杂度联系起来。我们期望GQBP能够用于证明组合问题量子解法的空间下界及时空复杂度下界。