Quantum vs. Classical Read-once Branching Programs

TL;DR

This paper establishes new bounds for quantum read-once branching programs, showing their computational power can be incomparable to classical counterparts, with specific functions requiring exponential size in the classical case but manageable quantumly.

Contribution

It introduces the first nontrivial bounds for quantum read-once branching programs and demonstrates their incomparability with classical models through explicit functions and complexity bounds.

Findings

Quantum read-once branching programs can compute certain functions efficiently.

Classical randomized read-once branching programs require exponential size for these functions.

Quantum models reading each input variable once have exponential lower bounds for set-disjointness.

Abstract

The paper presents the first nontrivial upper and lower bounds for (non-oblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2^{\Omega(n)} for computing the set-disjointness function DISJ_n from communication complexity theory with two-sided error bounded by a constant smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multi-partition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented.