Quantum automata cannot detect biased coins, even in the limit

TL;DR

This paper proves that quantum automata cannot distinguish biased coins from fair ones, even with infinite sequences, answering a question about their limitations in detecting bias.

Contribution

It provides a negative result showing quantum automata cannot detect bias in coins, resolving a question posed by Aaronson and Drucker (2011).

Findings

Quantum automata cannot detect biased coins.

The result holds even with infinite sequences.

It answers an open question about quantum automata capabilities.

Abstract

Aaronson and Drucker (2011) asked whether there exists a quantum finite automaton that can distinguish fair coin tosses from biased ones by spending significantly more time in accepting states, on average, given an infinite sequence of tosses. We answer this question negatively.