Dense Quantum Coding and a Lower Bound for 1-way Quantum Automata

TL;DR

This paper explores the limits of quantum encoding efficiency for classical information and establishes lower bounds on quantum automata size, revealing both potential and constraints of quantum data compression.

Contribution

It introduces non-trivial quantum encodings surpassing classical ones and provides a lower bound on quantum encoding succinctness and automata size.

Findings

Quantum encodings can outperform classical encodings in certain cases.

There is a lower bound on the succinctness of quantum encodings.

Exponential lower bound on 1-way quantum automata size for specific languages.

Abstract

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.