Very narrow quantum OBDDs and width hierarchies for classical OBDDs

TL;DR

This paper explores the complexity differences between classical and quantum OBDDs, demonstrating quantum advantages in width efficiency and establishing new hierarchy results for OBDD widths.

Contribution

It introduces new results comparing classical and quantum OBDDs, including width bounds for specific functions and hierarchy theorems for OBDD widths.

Findings

Quantum OBDDs can compute certain functions with constant width.

Classical OBDDs require significantly larger width for the same functions.

Quantum nondeterminism can be more efficient than classical nondeterminism.

Abstract

We present several results on comparative complexity for different variants of OBDD models. - We present some results on comparative complexity of classical and quantum OBDDs. We consider a partial function depending on parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2 but any classical OBDD (deterministic or stable bounded error probabilistic) needs width 2k+1. - We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical one. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width but any classical nondeterministic OBDD for this function needs non-constant width. - We also present new hierarchies on widths of deterministic and non-deterministic OBDDs. We focus both on small and large widths.