A Generalized Quantum Branching Program

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.