Optimal lower bounds for quantum automata and random access codes

TL;DR

This paper establishes exponential lower bounds on the size of quantum finite automata for a specific language and improves bounds on quantum random access codes using entropy and Holevo's theorem.

Contribution

It provides the first exponential lower bounds for QFA with intermediate measurements and introduces a tighter probabilistic bound related to Holevo's theorem.

Findings

QFA for L_n require size 2^{Omega(n)}

Asymptotically optimal bounds for dense quantum codes

Tighter in-probability bounds for Holevo's theorem

Abstract

Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}. It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n, thus also improving the previous bound. The improved bound is obtained by simple entropy arguments based on Holevo's theorem. This method also allows us to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes (random access codes) introduced by Ambainis et al. We then turn to Holevo's theorem, and show that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.