Quantum Query-Space Lower Bounds Using Branching Programs

TL;DR

This paper establishes the first explicit quantum query-space lower bounds for a restricted class of generalized quantum branching programs, impacting understanding of quantum query complexity for specific decision problems.

Contribution

It introduces the first explicit query-space lower bounds for a restricted GQBP model, advancing quantum complexity theory for decision problems.

Findings

Proves $Q^2 s = ext{Omega}(n^2)$ lower bound for OR$_n$ under a promise.

Extends bounds to problems like Parity and Majority with constant Hamming distance.

Provides an alternative proof for the $ ext{Omega}(\sqrt{n})$ lower bound on symmetric Boolean functions.

Abstract

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.