Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

TL;DR

This paper demonstrates exponential gaps between quantum and classical Ordered Binary Decision Diagrams (OBDDs), introduces a reordering technique for bounds, and establishes hierarchies for quantum and probabilistic OBDDs.

Contribution

It introduces a novel reordering method to analyze OBDD complexity, constructs explicit functions with exponential quantum-classical gaps, and extends hierarchy results for bounded error quantum and probabilistic OBDDs.

Findings

Constructed a total function with exponential classical vs. polynomial quantum OBDD complexity gap.

Established hierarchy theorems for quantum and probabilistic OBDDs of various widths.

Extended hierarchy results to read-$k$-times OBDDs with polynomial, superpolynomial, and subexponential widths.

Abstract

In this paper, we study quantum Ordered Binary Decision Diagrams($OBDD$) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function $REQ$ such that the deterministic $OBDD$ complexity of it is at least $2^{\Omega(n / \log n)}$, and the quantum $OBDD$ complexity of it is at most $O(n^2/\log n)$. It is the biggest known gap for explicit functions not representable by $OBDD$s of a linear width. Another function(shifted equality function) allows us to obtain a gap $2^{\Omega(n)}$ vs $O(n^2)$. Moreover, we prove the bounded error quantum and probabilistic $OBDD$ width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-$k$-times Ordered Binary Decision Diagrams ($k$-$OBDD$) of polynomial width, for $k = o(n / \log^3 n)$. We prove a similar hierarchy for bounded error probabilistic $k$-$OBDD$s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 -- 12, 2017