Implications of Quantum Automata for Contextuality

Abstract

Abstract. 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 log n)-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.

Exact Affine Counter Automata

“…In [11], some promise problems were given in order to show the superiority of two-way QFAs (2QCFAs) over two-way PFAs (2PFAs). We show that the same problem can be solved by realtime Las Vegas AfAs or AfAs with restart in linear expected time.…”

“…First we review the results given in [11]. Let PAL = {w ∈ {1, 2} * | w = w r } be the language of palindromes.…”

“…Proof. It is known that AfAs can recognize PAL with one-sided bounded-error [15,19] and so we can design a Las Vegas automaton for PAL-NPAL by using similar ideas given in [11,6].…”

“…This is the first separation result concerning AfCAs. This is a strong result since an exact one-way probabilistic one-counter automaton (PCA) is simply a oneway deterministic one-counter automaton (DCA) and it is still open whether exact one-way QCAs are more powerful than one-way DCAs and whether bounded-error one-way QCAs are more powerful than one-way bounded-error PCAs (see [14,10] for some affirmative results).In [11], it was shown that a certain promise problem can be solved by two-way QFAs (2QCFAs) exactly but in exponential time, and bounded-error two-way PFAs (2PFAs) can solve the problem only if they are allowed to use logarithmic amount of memory. We show that the same problem can be solved by realtime Las Vegas AfAs or AfAs with restart in linear expected time.…”

“…In [11], it was shown that a certain promise problem can be solved by two-way QFAs (2QCFAs) exactly but in exponential time, and bounded-error two-way PFAs (2PFAs) can solve the problem only if they are allowed to use logarithmic amount of memory. We show that the same problem can be solved by realtime Las Vegas AfAs or AfAs with restart in linear expected time.…”