Implications of Quantum Automata for Contextuality

We construct zero-error quantum finite automata (QFAs) for promise problems which cannot be solved by bounded-error probabilistic finite automata (PFAs). Here is a summary of our results: 1. There is a promise problem solvable by an exact two-way QFA in exponential expected time, but not by any bounded-error sublogarithmic space probabilistic Turing machine (PTM). 2. There is a promise problem solvable by an exact two-way QFA in quadratic expected time, but not by any bounded-error o(log logn)space PTMs in polynomial expected time. The same problem can be solvable by a one-way Las Vegas (or exact two-way) QFA with quantum head in linear (expected) time. 3. There is a promise problem solvable by a Las Vegas realtime QFA, but not by any bounded-error realtime PFA. The same problem can be solvable by an exact two-way QFA in linear expected time but not by any exact two-way PFA. 4. There is a family of promise problems such that each promise problem can be solvable by a two-state exact realtime QFAs, but, there is no such bound on the number of states of realtime bounded-error PFAs solving the members this family. Our results imply that there exist zero-error quantum computational devices with a single qubit of memory that cannot be simulated by any finite memory classical computational model. This provides a computational perspective on results regarding ontological theories of quantum mechanics [19], [30]. As a consequence we find that classical automata based simulation models [23], [6] are not sufficiently powerful to simulate quantum contextuality. We conclude by highlighting the interplay between results from automata models and their application to developing a general framework for quantum contextuality.