A Generalized Quantum Branching Program

TL;DR

This paper introduces a new quantum branching program model, GQBP, that allows querying multiple variables in superposition, unifying previous models and linking them to quantum query complexity for analyzing computational problems.

Contribution

The paper proposes GQBP, a generalized quantum branching program model that captures superposition queries, unifies existing models, and connects to quantum query complexity.

Findings

GQBP can efficiently implement Deutsch-Jozsa, Parity, and Majority functions.

GQBP is equivalent to existing quantum branching programs and query complexity models.

GQBP provides a framework for proving space and space-time lower bounds in quantum computing.

Abstract

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.