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arXiv:quant-ph/0304082 · SIAM Journal on Computing

Decidable and Undecidable Problems about Quantum Automata

Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, Natacha Portier

Published Jan 1, 2005• 76 citations
arXivHTMLPDF

Abstract

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

DOI
10.1137/s0097539703425861

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Annual Symposium on Foundations of Computer Science•Nov 22, 2002•348 citations
We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it is known that strict andnonstrict thresholds both lead to undecidable problems.
Quantum Finite Automata with Classical states (QFACs) are nondeterministic finite automata over a finite alphabet of quantum operations. We study expressiveness of this model on finite words and the corresponding emptiness problem. We show that regular languages are incomparable with those definable by Quantum Finite Automata (QFAs) and that both are strictly subsumed by QFAC-definable languages. We show that the emptiness problem for a QFAC can be reduced to the emptiness of the language intersection of a QFA and a finite automaton. This intersection is known to be decidable for strict thresholds but undecidable for non-strict cases. Furthermore, we consider the problem for flat QFACs, a restriction where the underlying automata contain no nested loops, and relate it to the higher-dimensional orbit problem, a long-standing open challenge in dynamical systems. Finally, we propose a sound and semi-complete witness searching procedure to verify the non-emptiness of one-loop QFACs, which are sufficiently expressive to represent some prominent quantum algorithms, such as Grover's search and quantum random walks.
The nondeterministic quantum finite automaton (NQFA) is the only known case where a one-way quantum finite automaton (QFA) model has been shown to be strictly superior in terms of language recognition power to its probabilistic counterpart. We give a characterization of the class of languages recognized by NQFAs, demonstrating that it is equal to the class of exclusive stochastic languages. We also characterize the class of languages that are recognized necessarily by two-sided error by QFAs. It is shown that these classes remain the same when the QFAs used in their definitions are replaced by several different model variants that have appeared in the literature. We prove several closure properties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space complexity classes are examined.
In this paper, we study some decision problems both for {\it multi-letter quantum finite automata} and {\it measure many multi-letter quantum finite automata}. We first show that given a k1k_1k1​-letter quantum finite automaton A1\mathcal{A}_1A1​ and a k2k_2k2​-letter quantum finite automaton A2\mathcal{A}_2A2​ over the same input alphabet Σ\SigmaΣ, they are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where nin_ini​, i=1,2i=1,2i=1,2, are the number of states in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. By applying a method, due to the author, used to deal with the equivalence problem of {\it measure many one-way quantum finite automata}, we also show that a k1k_1k1​-letter measure many quantum finite automaton A1\mathcal{A}_1A1​ and a k2k_2k2​-letter measure many quantum finite automaton A2\mathcal{A}_2A2​ are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where nin_ini​, i=1,2i=1,2i=1,2, are the number of states in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. Next, we study the emptiness problem of those two kinds of quantum finite automata. We show that whether the language recognized by a kkk-letter quantum finite automaton with non-strict cut-point is empty is undecidable, but we leave open the emptiness of language reorganized by a kkk-letter quantum finite automaton with strict cutpoint. We also show that whether the languages recognized by a kkk-letter measure many quantum finite automaton with both nonstrict and strict cutpoints are undecidable. And the direct consequences of the above outcomes are summarized in the paper.
Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1
Quantum finite automata were introduced by C.Moore, J.P. Crutchfield, and by A.Kondacs and J.Watrous. This notion is not a generalization of the deterministic finite automata. Moreover, it was proved that not all regular languages can be recognized by quantum finite automata. A.Ambainis and R.Freivalds proved that for some languages quantum finite automata may be exponentially more concise rather than both deterministic and probabilistic finite automata. In this paper we introduce the notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite automaton but not by a deterministic or probabilistic finite automata. This is the first result on a problem which can be solved by a quantum computer but not by a deterministic or probabilistic computer. Additionally we discover unexpected probabilistic automata recognizing complicated languages.
In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.