Decidable and Undecidable Problems about Quantum Automata

Abstract

We study the following decision problem: is the language recognized by a quantum finite automaton empty or non-empty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or non-strict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata for which it is known that strict and non-strict thresholds both lead to undecidable problems.

(Un)decidable Problems about Reachability of Quantum Systems

“…Undecidability of Gf (globally reachable), Uf (ultimately forever reachable) and If (infinitely often reachable) comes from a straightforward reduction from the emptiness problem for quantum automata [7]. However, undecidability of Ff (eventually reachable) requires a careful reduction from the halting problem for 2-counter Minsky machines [18].…”

Decidability of the Membership Problem for 2 × 2 integer matrices

“…Matrices and matrix products play a crucial role in the representation and analysis of various computational processes, i.e., linear recurrent sequences [19,28,29], arithmetic circuits [16], hybrid and dynamical systems [27,3], probabilistic and quantum automata [8], stochastic games, broadcast protocols [15], optical systems [17], etc. Unfortunately, many simply formulated and elementary problems for matrices are inherently difficult to solve even in dimension two, and most of these problems become undecidable in general starting from dimension three or four.…”

An overview of quantum computation models: quantum automata

“…Quantum automata involve various types, as classical automata [10], and include quantum finite automata (QFA) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], quantum sequential machines (QSM) [29][30][31], quantum pushdown automata (QPDA) [26,32,33], quantum Turing machines (QTM) [6,[34][35][36][37][38][39], quantum multi-counter machines (QMCM) [38], quantum multi-stack machines (QMSM) [38], quantum cellular automata (QCA) [1,40,41], and automata theory based on quantum logic, called orthomodular latticevalued finite automata (l-VFA) [42][43][44][45][46][47][48][49][50] as well as the other models. In this paper, we only review QFA, QSM, QPDA, QTM, and l-VFA.…”