Lower Bounds for Generalized Quantum Finite Automata

We obtain several lower bounds on the language recognition power of Nayak’s [12] generalized quantum finite automata (GQFA). Techniques for proving lower bounds on Kondacs and Watrous’ one-way quantum finite automata (KWQFA) were introduced by Ambainis and Freivalds [2], and were expanded in a series of papers. We show that many of these techniques can be adapted to prove lower bounds for GQFAs. Our results imply that the class of languages recognized by GQFAs is not closed under union. Furthermore, we show that there are languages which can be recognized by GQFAs with probability p > 1/2, but not with p > 2/3. Quantum finite automata (QFA) are online, space-bounded models of quantum computation. Similar to randomized finite automata [16] where the state is a random variable over a finite set, the state of a QFA is a quantum superposition of finite dimension. The machine processes strings w ∈ Σ∗ by applying a sequence of state transformations specified by the sequence of letters in w, and the output of the machine is determined by a measurement of the machine state. A central problem is to characterize the language recognition power of QFAs. Most QFA research has been focused on the case where the transformations are limited to various combinations of unitary transformations and projective measurements. The class of languages recognized by these QFAs is a strict subset of the regular languages, so QFAs are less powerful than their classical counterparts. This is due to the fact that, unlike many other models of quantum computation such as quantum Turing machines [6] or quantum circuits [17], QFAs lack the linear space overhead which is required to convert classical computation into reversible computation [5]. However, there are languages which can be recognized by QFAs using exponentially fewer states than the smallest deterministic or randomized finite automaton [2, 7]. The simplest type of QFA is the measure-once QFA (MOQFA) model of Moore and Crutchfield [11]. These QFAs are limited to recognizing those languages whose minimal automaton is such that each letter induces a permutation on the states. Two types of generalizations of the MOQFA model have been considered. In the first type, the machine is allowed to halt before reading the entire input word. This corresponds to Kondacs and Watrous’ one-way QFAs (KWQFAs) [10]. The second type is to allow state transformations to include the application of quantum measurements, which generates some classical randomness in the system. This corresponds to Ambainis et. al’s Latvian QFAs (LQFAs) [1]. Nayak [12] investigated a model called generalized QFAs (GQFAs), which generalize both KWQFAs and LQFAs. This paper introduced new entropy-based techniques which were used to show that GQFAs cannot recognize the language Σ∗a. These techniques have since been used to obtain lower bounds on quantum random access codes [12] and quantum communication complexity [13]. However, no further lower bounds have been shown for GQFAs. In a series of papers [2, 8, 4, 3], a number of important lower bounds on the power of KWQFA were identified. The main tool used in these results was a technical lemma which is used to decompose the state space of a KWQFA into two subspaces (called the ergodic and transient subspaces) in which the state transitions have specific behaviors. In this paper, we show that this lemma can be adapted to the case of GQFA. The framework of our proof follows the basic outline of [2], however we must overcome a number of technical hurdles which arise from allowing classical randomness in the state. We use this lemma to highlight a number of relevant properties of the class of languages recognized by GQFA. Following [4], we can use the lemma to show that a certain property of the minimal automaton for L implies that L is not recognizable by a GQFA. We use this result to show that the class of languages recognized by this model is not closed under union. Furthermore, we show the existence of languages which can be recognized by GQFA with probability p = 2/3 but not p > 2/3. These results highlight the key similarities and differences between KWQFA and GQFA. The paper is organized as follows. In Section 2 we give definitions and basic properties of GQFA and we review the necessary background. In Section 3 we will state our main results. In Section 4 we prove the main technical lemma and in Section 5 we apply this lemma to prove the remaining results. In the last section we discuss open problems and future work.