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arXiv:1202.2651

State succinctness of two-way finite automata with quantum and classical states

Shenggen Zheng, Daowen Qiu, Jozef Gruska, Lvzhou Li, Paulo Mateus

Published Feb 13, 2012• 2 citations
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Abstract

{\it Two-way quantum automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous in 2002. In this paper we study state succinctness of 2QCFA. For any $m\in {\mathbb{Z}}^+$ and any $ε<1/2$, we show that: {enumerate} there is a promise problem $A^{eq}(m)$ which can be solved by a 2QCFA with one-sided error $ε$ in a polynomial expected running time with a constant number (that depends neither on $m$ nor on $\varepsilon$) of quantum states and $\mathbf{O}(\log{\frac{1}ε)}$ classical states, whereas the sizes of the corresponding {\it deterministic finite automata} (DFA), {\it two-way nondeterministic finite automata} (2NFA) and polynomial expected running time {\it two-way probabilistic finite automata} (2PFA) are at least $2m+2$, $\sqrt{\log{m}}$, and $\sqrt[3]{(\log m)/b}$, respectively; there exists a language $L^{twin}(m)=\{wcw| w\in\{a,b\}^*\}$ over the alphabet $Σ=\{a,b,c\}$ which can be recognized by a 2QCFA with one-sided error $ε$ in an exponential expected running time with a constant number of quantum states and $\mathbf{O}(\log{\frac{1}ε)}$ classical states, whereas the sizes of the corresponding DFA, 2NFA and polynomial expected running time 2PFA are at least $2^m$, $\sqrt{m}$, and $\sqrt[3]{m/b}$, respectively; {enumerate} where $b$ is a constant.

DOI
10.48550/arxiv.1202.2651

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The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA, with a single qubit, can recognize, with bounded error, the language Leq={ambm:m∈N}L_{eq}=\{a^m b^m :m \in \mathbb{N}\}Leq​={ambm:m∈N} in expected polynomial time and the language Lpal={w∈{a,b}∗:w is a palindrome}L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}Lpal​={w∈{a,b}∗:w is a palindrome} in expected exponential time. We further demonstrate the power of 2QCFA by showing that they can recognize the word problems of many groups. In particular 2QCFA, with a single qubit and algebraic number transition amplitudes, can recognize, with bounded error, the word problem of any finitely generated virtually abelian group in expected polynomial time, as well as the word problems of a large class of linear groups in expected exponential time. This latter class (properly) includes all groups with context-free word problem. We also exhibit results for 2QCFA with any constant number of qubits. As a corollary, we obtain a direct improvement on the original Ambainis and Watrous result by showing that LeqL_{eq}Leq​ can be recognized by a 2QCFA with better parameters. As a further corollary, we show that 2QCFA can recognize certain non-context-free languages in expected polynomial time. In a companion paper, we prove matching lower bounds, thereby showing that the class of languages recognizable with bounded error by a 2QCFA in expected subexponential\mathit{subexponential}subexponential time is properly contained in the class of languages recognizable with bounded error by a 2QCFA in expected exponential\mathit{exponential}exponential time.
Two-way finite automata with quantum and classical states (2QCFA’s), introduced by Ambainis and Watrous, extend deterministic two-way finite automata with a constant-size quantum register and provide a natural hybrid model of computation. In this paper, we introduce two-way nondeterministic finite automata with quantum and classical states (2QCNFA’s), extending this framework to the nondeterministic setting. We show that 2QCNFA’s are closed under union, intersection, and marked concatenation, and that they can recognize certain comparison languages whose status for 2QCFA’s remains open. We further initiate the study of decision problems for this model by proving that 2QCNFA’s can recognize the set of invalid computations of a deterministic Turing machine, which implies that the universality problem is undecidable. In addition, we investigate descriptional complexity, showing that there is no approximation algorithm for minimization between 2QCNFA’s and classical models such as DFA’s and context-free grammars, and that the corresponding trade-offs are non-recursive.
Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and it was shown that 2QCFA have superiority over two-way probabilistic finite automata (2PFA) for recognizing some non-regular languages such as the language Leq = {anbn ∣ n ∈ N} and the palindrome language Lpal = {w ∈ {a,b}* ∣ w=wR}, where xR is x in the reverse order. It is interesting to find more languages like these that witness the superiority of 2QCFA over 2PFA. In this paper, we consider the language Lm = {xcy ∣ ∑ = {a,b,c},x,y ∈ {a,b}*, ∣x∣ = ∣y∣} that is similar to the middle language Lmiddle = {xay ∣ x,y ∈ {a,b}*, ∣x∣ = ∣y∣}. We prove that the language Lm show that Lm can be recognized by 2PFA with bounded error, but only in exponential expected time. Thus Lm is another witness of the fact that 2QCFA are more powerful than their classical counterparts.
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.
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the language Lpal={w∈{a,b}∗:w is a palindrome}L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}Lpal​={w∈{a,b}∗:w is a palindrome} with bounded error in expected time 2O(n)2^{O(n)}2O(n), on inputs of length nnn. We prove that their result essentially cannot be improved upon: a 2QCFA (of any size) cannot recognize LpalL_{pal}Lpal​ with bounded error in expected time 2o(n)2^{o(n)}2o(n). To our knowledge, this is the first example of a language that can be recognized with bounded error by a 2QCFA in exponential time but not in subexponential time. Moreover, we prove that a quantum Turing machine (QTM) running in space o(log⁡n)o(\log n)o(logn) and expected time 2n1−Ω(1)2^{n^{1-Ω(1)}}2n1−Ω(1) cannot recognize LpalL_{pal}Lpal​ with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or o(log⁡n)o(\log n)o(logn)-space QTM that recognizes any language LLL in terms of a natural "hardness measure" of LLL. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer.
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.