Simple models of quantum finite automata

Transcription

1 Masaryk University Faculty of Informatics Simple models of quantum finite automata Bachelor s Thesis Martin Frian Brno, Spring 2018

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3 Masaryk University Faculty of Informatics Simple models of quantum finite automata Bachelor s Thesis Martin Frian Brno, Spring 2018

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5 This is where a copy of the official signed thesis assignment and a copy of the Statement of an Author is located in the printed version of the document.

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7 Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Martin Frian Advisor: prof. RNDr. Jozef Gruska DrSc. i

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9 Acknowledgements I would like to thank my supervisor prof. RNDr. Jozef Gruska, DrSc. for his guidance and valuable advice. iii

10 Abstract We present in a uniform way several models of classical finite automata and quantum finite automata. We have studied five main models of quantum finite automata in detail. Three other models of quantum finite automata were also briefly presented. We have also presented basic results concerning the power of each model, closure properties, mutual relations, and results concerning decision problems. iv

11 Keywords finite automata, quantum finite automata, classical state, quantum state, regular languages, acceptance probability, quantum information processing, quantum mechanics v

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13 Contents Introduction 1 1 Finite automata 3 2 Basic tools of QIP 9 3 Quantum finite automata Measure-many one-way quantum finite automata Measure-once one-way qunatum finite automata Two-way quantum finite automata Enhanced quantum finite automata Quantum finite automata with open time evolution Other models of quantum finite automata Conlusion 27 Bibliography 29 vii

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15 Introduction In the classical world, each regular language can be recognized (described) by a finite automaton. Finite automata model can be seen as an abstract machine with an input tape, on which an input is written, each input symbol in a single cell, read by the reading head. Finite automaton can be in exactly one of a fixed finite set of states at any moment of discrete time. Depending on the input, this machine model can go from one state to another and this is called the transition. According to the used transition function we have the two most basic models of finite automata, deterministic finite automata (DFA) and non-deterministic finite automata (NFA). In fact, despite the different definitions of these two models, their computational power is same. Trivially, every DFA is NFA too and it was proven that for each NFA there exists a DFA recognizing the same language. However such a DFA could have exponentially larger number of states than the NFA under recognition. Another computational model of finite automata are probabilistic finite automata (PFA). It is well known, that PFA have same computational power as DFA at a certain definition of acceptance. In the quantum world, there exists also various types of computational models of finite automata. Informally, quantum finite automata (QFA) are quantum analog of PFA. However, instead of using classical states, they use quantum states. Depending on used quantum resources, QFA might be more powerful and they might have better time or space complexity than PFA or DFA. So far there are several models of QFA that use simple quantum tools but generate only a subset of regular languages. There is also a model of QFA that generates exactly the whole class of regular languages but use the whole range of quantum resources. Both of these models are not fully satisfactory. Hence we need a model of QFA, which recognises exactly the whole class of regular languages using as minimum of quantum tools as possible. Therefore the aim of this thesis is to provide an overview of the current well known main models of quantum finite automata as well as basic results concerning their power, mutual relations and decidability of basic decision problems. 1

16 This thesis is divided into four chapters. In the first chapter we present classical deterministic finite automata, their properties and several basic results concerning decidability of decision problems. We also show modifications of deterministic finite automata, namely nondeterministic finite automata, two-way deterministic finite automata and probabilistic finite autoamta. In the second chapter we provide a brief introduction to basic tools widely used in a construction of quantum finite automata. In the third chapter we present in detail current known models of quantum finite automata, namely measure-many one-way quantum finite automata, measure-once one-way quantum finite automata, two-way quantum finite automata, enhanced quantum finite automata and quantum finite automata with open time evolution. We also briefly present another models of quantum finite automata. 2

17 1 Finite automata Deterministic finite automata (DFA) are basic models of classical information processing devices. They consist of: 1. a finite memory of so-called states. One of the states is said to be an initial state. A subset of states is said to be a set of accepting states; 2. a set of input symbols called the alphabet; 3. a transition function, which specifies the action of the automaton during computations. For each state and each input symbol determines the next state of the automaton. Formally such an automaton is defined as follows. Definition 1. A deterministic finite automaton A (DFA) is a 5-tuple (S, Σ, δ, s 0, S a ), where S is a finite set of states Σ is an input alphabet δ : S Σ S is a transition function s 0 S is an initial state S a S is a set of accepting states Processing of an input (string) w by a DFA A is informally defined as follows. The automaton starts processing w in the initial state and reads as its input the leftmost symbol of w. The automaton then proceeds in discrete time steps as follows until it gets into an accepting state or finishes processing of the whole input string: for the current state and next symbol of the input string it comes to a new state of its memory that is determined by its transition function and moves to read next symbol of its input string. This process is formally defined via extended transition function. Definition 2. An Extended transition function is a function ˆδ : S Σ * S satisfying the following three conditions: 3

18 1. Finite automata δ(s, ɛ) = s ˆδ(s, a) = δ(s, a) ˆδ(s, aw) = ˆδ(δ(s, a), w) for any a Σ, w Σ + and s S. A language accepted by a DFA is defined as follows. Definition 3. Let A = (S, Σ, δ, s 0, S a ) be a DFA, the language accepted or recognized by A, denote as L(A), is L(A) = {w Σ * ˆδ(s 0, w) S a } One of the key results of finite automata theory is that the class of languages accepted by DFA is the class of regular languages. They are the smallest set of languages containing all finite languages and closed under the following operations: 4 union, concatenation, iteration. DFA are of large importance for several reasons: It is easy to determine in polynomial time the smallest DFA equivalent to a given A. This smallest DFA is always unique. Main decision problems, namely the emptiness problem and the membership problem are decidable in polynomial time. The emptiness problem is in addition easily reduced to the reachability problem on graphs. Equivalence problems, such as the equivalence of two states or two DFA are decidable in polynomial time too. The above model of DFA is very robust. Several of its natural modifications, presented bellow, accept again exactly the class of regular languages. Here are the main variants of the above model of DFA.

19 1. Finite automata Non-deterministic finite automata Informally, transitions in non-deterministic finite automaton are based on non-deterministic choosing of the next state. The transition function maps each pair state and input symbol to the set of potential next states. The automaton nondeterministically chooses one of them. A formal definition of NFA is as follows. Definition 4. A nondeterministic finite automaton (NFA) is a 5-tuple (S, Σ, δ, s 0, S a ), where meaning of all elements is same as in the Definition 1 except the transition function that is defined as follows δ : S Σ P(S) 1. Again, the transition function could be extended in a similar way like in the Definition 2. ˆδ : Q Σ * P(Q) satisfies the following conditions: ˆδ(s, ɛ) = {s} ˆδ(s, wa) = p ˆδ(s,w) δ(p, a) for any s S, a Σ and w Σ *. The language L(A) accepted by a NFA A is defined as follows L(A) = {w Σ * ˆδ(s 0, w) S a = } Two-way finite automata Another generalization of DFA are two-way deterministic finite automata (2DFA). The only difference between 2DFA and DFA is that in 2DFA the automaton could move in both directions while reading an input word or can stay on the same position during a transition. Now we provide a formal definition of 2DFA. Definition 5. A deterministic two-way finite automaton (2DFA) A is a 5-tuple A = (S, Σ, δ, s 0, S a ) where meaning of each element is same as in Definition 1, except the transition function defined as follows δ : S Σ S {,, } 1. Here P(S) denotes the powerset of the set S 5

20 1. Finite automata The arrows in the transition function mean: the head should moved one cell to the left, the head should moved one cell to the right and finally don t move head. The language L(A) accepted by a 2DFA A is defined as the set of all words w, for which the automaton A starting with the head on the first symbol of w and in the state s 0 moves, after a finite number of steps, over the right end of w exactly when A comes to the one of final states. The non-deterministic modification (2NFA) of two-way finite automata can be defined in a similar way. Probabilistic finite automata The last here presented generalization of FA will be probabilistic finite automata. The following definitions of probabilistic finite automata and its components are in accordance with Rabin s [1]. A transition in probabilistic automata is defined by a function, which assigns to each pair (s, σ) S Σ a certain (transition) sequence of probabilities from the interval [0, 1]. In other words, transition function maps each pair (s, σ) to S -tuple (p 0 (s, σ),, p S 1 (s, σ)) where 0 p i (s, σ) 1. We denote the set of all these tuples as [0, 1] S. Definition 6. A probabilistic finite automaton (PFA) is a 5-tuple (S, Σ, δ, s 0, S a ), where meaning of all elements is the same as in Definition 1 except for the transition function δ. Let S = {s 0,, s n }, then δ : S Σ [0, 1] n+1, such that for (s, σ) S Σ: δ(s, σ) = (p 0 (s, σ),, p n (s, σ)), where i p i (s, σ) = 1 Suppose that a PFA A is in a state s and the head reads a symbol σ. Then A changes its state to one of the states s i S. Probability of going into state s i is the (i + 1)th element p i (s, σ) of δ(s, σ). We can again extend the transition function from symbols to words. Since transition probabilities p i (s, σ) are assumed to remain fixed and be independent of time and previous input symbols, the automaton has also definite transition probabilities for going from state s to state s i 6

21 1. Finite automata by reading word w Σ * and these probabilities can be calculated by means of products of certain stochastic matrices. Now we provide the definition of these matrices. Definition 7. For σ Σ and w Σ *, w = σ 1 σ m we define ˆδ(σ) = [p j (s i, σ)] 0 i n,0 j n ˆδ(w) = ˆδ(σ 1 ) ˆδ(σ m ) = [p j (q i, w)] 0 i n,0 j n The probability p(w), which is the probability for A of going from state s 0 into one of the final states by reading input word w is defined as follows. Definition 8. Let A = (S, Σ, δ, s 0, F), I = {i 0,, i r } and F = {s i0,, s ir }, define p A (w) = p i (s 0, w). i I Recognize exactly the whole class of regular languages by PFA so called acceptance with respect to a cut-point is used. Let c R and 0 c < 1, the language accepted by a PFA A is defined as follows: L c (A) = {w p A (w) > c}. The language L c (A) is then said to be accepted by A with respect to the cut-point c. 7

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23 2 Basic tools of QIP In this chapter, we provide a brief introduction to quantum information processing. We present quantum tools, which are used in the developing of various models of quantum finite automata. Hilbert space. Hilbert space (quantum system) is n-dimensional complex vector space denoted as H n with basis B n consisting of n- dimensional orthonormal vectors equipped with: a scalar product of vectors φ =. and ψ =. : ψ φ = α 1 α n n α i β * i i=1 where ψ denotes the conjugate transpose of vector ψ and β * i denotes conjugate transpose of the complex number β i. β 1 β n a norm of vectors: a metric: φ = φ φ = 1 dist(φ, ψ) = φ ψ Pure quantum states. A pure quantum state φ is a column vector in H n in case of a standard basis B n : α 1 φ =.. α n Such a φ can be also seen as a linear combination of basis states: φ = α 1 q α n q n. We say that φ is a superposition of the basis states q 1,, q n B n. For any i {1,, n}, α i C is called a probability amplitude. 9

24 2. Basic tools of QIP Qubits. A qubit is a pure quantum state of two dimensional Hilbert space H 2. A qubit φ is usually represented as φ = α 0 + β 1 where { 0, 1 } is usually used standard basis of H 2. A qubit is a unit of quantum information. In other words, qubits are quantum analogs of a classical bits. Quantum registers. A quantum n-qubit register is any ordered sequence of n quantum qubit systems. Quantum registers can be seen as a quantum analog of a classical register. An n-qubit register is the tensor product of n two-dimensional Hilbert spaces H 2 : n H 2 n = Unitary operators. An evolution (computation) step of a quantum system is performed by a multiplication of the state vector by a unitary operator, i.e multiplication of a unitary matrix U with a vector φ : i=1 U φ. A matrix U is unitary if for U and its adjoint matrix U it holds that: U U = UU = I. H 2 Density matrix. To each quantum state φ is assigned exactly one density matrix. The Density matrix is an n n matrix resulting from the multiplication of vectors φ and φ : 10 α 1 α 1 * α 1 α 2 * α 1 α * n φ φ = α 2 α 1 * α 2 α 2 * α 2 α * n α n α * 1 α n α * 2 α n α * n

25 2. Basic tools of QIP Mixed state. A mixed state is a probabilistic distribution of pure quantum states. To each mixed state there is assigned exactly one density matrix. The density matrix ρ assigned to a mixed state results from the sum of density matrices assigned to pure quantum states φ i, weighted by their respective probabilities p i : where i p i = 1. ρ = i p i φ i φ i Projection measurements. A quantum state ρ H n can be measured with respect to an observable - a decomposition of H n into orthogonal subspaces. A projection measurement of a state ρ gives two outcomes: A new quantum state ρ into which the state ρ collapses. Classical information into which subspace projection of a state ρ was made. The subspace into which projection is made is chosen randomly and the corresponding probability is uniquely determined by the probability amplitudes at the representation of ρ at the basis states of the subspace. Positive-operator valued measurements. A positive-operator valued measure (POVM) is the most general observable in quantum physics. POVM is an observable specified by a set of Hermitian positive semidefinite operators {F i }, such that i F i = I and 0 F i I. Measurement of a quantum state ρ with respect to POVM gives i-th outcome with probability Tr[ρF i ]. 11

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27 3 Quantum finite automata In this chapter, we present five main models of quantum finite automata. The first three use pure quantum states and projection measurements only. The last two use mixed states and POV measurements. The most basic models of quantum automata so far are one-way quantum finite automata (1QFA) that use pure quantum states. These 1QFA can be seen as abstract machines in a similar way like classical deterministic finite automata (DFA) are. 1QFA read input similarly like a DFA but in contrast to DFA the finite memory in 1QFA is represented by basis quantum states. The transition mapping in a 1QFA is defined using quantum operators, which are required to be unitary. The states of 1QFA are represented by quantum basis states, 1QFA also make use of projection measurements. There are two main models of 1QFA with pure states, so-called measure-many and measure-one models. We introduce both of them in the following two sections. In the third section, we introduce twoway quantum finite automata with pure states. In the fourth section we present enhanced quantum finite automata. The fifth section is aimed to quantum finite automata with open time evolution. They are so far the most general models of 1QFA. In the last section we briefly present another three models of quantum finite automata, namely latvian quantum finite automata, quantum finite automata using ancilla qubits and multihead quantum finite automata. 3.1 Measure-many one-way quantum finite automata A measure-many one-way quantum finite automata (MM-1QFA) were at first introduced by Kondacs and Watrous [2]. They are defined as follows. Definition 9. A measure-many one-way quantum finite automaton (MM-1QFA) A is a 6-tuple (Q, Σ, δ, q 0, Q acc, Q rej ) where: Q is a finite set of symbols (representing basic quantum states) Σ is an input alphabet q 0 is an initial state 13

28 3. Quantum finite automata δ : Q Γ Q C [0,1] is a transition function, where Γ = Σ {#, $} is the tape alphabet of A. Symbols # and $ are said to be the left and right end markers 1, and it holds Σ {#, $} =. Q acc Q and Q rej Q are sets of accepting and rejecting states, such that Q acc Q rej =. The states in Q acc and Q rej are called halting states. The states in a Q non = Q (Q acc Q rej ) are called non-halting states. The computation (evolution) of A is performed in the Hilbert space H Q with the basis B = { q q Q}, using unitary operators U σ defined by U σ ( q ) = δ(q, σ, q 1 ) q 1 q 1 Q where σ Γ. Measurement. MM-1QFA make use of so-called computational observable O that corresponds to the orthogonal decomposition of: where E a = span{ q q Q acc } E r = span{ q q Q rej } E n = (E a E r ) H Q = E a E r E n Denote by P p, p {a, r, n} the projection operator into the subspaces E p. Computation. We follow the description of a computation of a MM- 1QFA A = (Q, Σ, δ, q 0, Q acc, Q rej ) on an input w = σ 1 σ n Γ * according to [3, p. 154]. At the beginning of a computation, A reads the leftmost input symbol σ 1 and then the corresponding operator U σ1 is applied to the initial state q 0. Onto the resulting state U σ1 q 0 is 1. These two symbols are always included to the input, in case the tape alphabet is defined. 14

29 3. Quantum finite automata then applied measurement by the observable O. This measurement projects U σ1 q 0 to a vector ψ with the probability equal to the square of the norm of ψ. Vector ψ is of one of the subspaces E a, E r, E n. In case that ψ E a or ψ E r, the automaton accepts or rejects an input word. If ψ E n, then ψ is normalized and the automaton continues processing next symbol σ 1. This process goes on if and only if the projection into E n occurs. This computation can be seen as an application of the composed operator P n U σn P n U σn 1 P n U σ1 q 0. To define formally the overall probability of accepting or rejecting the input by A we define the so-called set of total states V A = H Q C C. The total state of A can be interpreted as A is at any time of the computation in a total state (ψ, p a, p r ), that is the probability p a of accepting the input, probability p r of rejecting the input and neither with the probability ψ 2 = 1 p a p r. For each symbol σ Σ, the evolution of A as symbol σ is read can be described by an operator T σ : V A V A, which is defined as follows: T σ : (ψ, p a, p r ) (P n U σ ψ, p a + P a U σ ψ 2, p r + P r U σ ψ 2 ). Acceptance of a language. The acceptance of a language L by a MM- 1QFA A, with a cut-point λ is formally defined as follows. L is said to be accepted by a MM-1QFA A with bounded-error probability if A accepts any x L with probability at least λ + ɛ, for some ɛ > 0 and rejects any x / L with probability at least λ + ɛ. L is said to be accepted with unbounded-error probability if A accepts any x L with probability at least λ and rejects any x / L with probability at least λ. A cut-point λ is usually equal to 1 2 unless otherwise stated. Brodsky and Pipenger [4] introduced the partitioning of languages accepted by a MM-1QFA to the several natural classes. Denote by RMM ɛ the class of languages accepted by a MM-QFA with margin ɛ. Next let us define the class RMM= RMM ɛ as a class of languages ɛ>0 15

30 3. Quantum finite automata accepted by a MM-1QFA with bounded-error probability. The class of languages accepted by a MM-QFA with unbounded-error probability is denoted as UMM. Languages accepted by MM-1QFA. The class of languages accepted by MM-1QFA with bounded-error probability is a proper subset of regular languages. For example, it is known that the language {a, b} * b cannot be accepted by any MM-1QFA. This is due to fact that linear operators doing transitions in a MM-1QFA are required to be unitary (and therefore reversible). Ambainis, K, ikusts, and Valdats [5] formulated several necessary and some sufficient conditions for a language to be recognized by an MM-1QFA. They also formulated conditions which are necessary and sufficient for a subclass of regular languages to be accepted by MM-QFA. Ambainis and Freivalds [4] showed that MM-1QFA could accept a language L with probability more than 7 9 if and only if L could be accepted by an AF-reversible automaton 2. Moreover, Ambainis and K, ikusts [6] proved that any language not accepted by an AF-reversible automaton can be accepted by MM-1QFA with probability less than They also determined maximum probabilities for several languages accepted by MM-1QFA. Ambainis et al. [7] constructed an infinite hierarchy of regular languages. Each language in the hierarchy can be accepted by some MM-1QFA with a probability smaller than is the probability of acceptance of the preceding language in the hierarchy. It was shown, that these probabilities converge to 1 2. Closure properties. The classes RMM and UMM of languages accepted by MM-1QFA are closed under: complement inverse homomorphism word quotient 2. AF-reversible automaton is a variation of a DFA where for any s S and any σ Σ there is at most one s 1 S such that δ(s 1, σ) = s. 16

31 3. Quantum finite automata These classes of languages are not closed under other Boolean operations and homomorphism. The power of MM-1QFA As mentioned above, MM-1QFA accept only a proper subset of regular languages. This implies that MM-1QFA are less powerful than DFA and therefore any modification of DFA presented in Chapter 1 is more powerful than MM-1QFA. Size-space efficiency of a MM-1QFA. Ambainis and Freivalds [8] showed, that there exist MM-1QFA which are exponentially more space efficient than DFA recognizing the same language. For example language L p = {a i p i, where p is prime} could be accepted by MM- 1QFA with probability arbitrarily close to 1, with exponentially fewer states than any PFA recognizing L p. However, sometimes classical finite automata could be more space efficient than corresponding MM-1QFA. Ambainis et al. showed that minimal DFA recognizing L m = {w1 w {0, 1} *, w m} is exponentially more space efficient than any MM-1QFA recognizing L m. Equivalence problem The problem whether two MM-1QFA are equivalent is formulated as follows. Two MM-1QFA A 1 and A 2 are said to be equivalent if for any input w A 1 and A 2 accept with equal probability. Li and Qiu [9] showed, that this problem could be solved with a polynomial time algorithm. In [10] they improved the criterion in their original theorem. At this improvement the theorem says that two MM-1QFA A 1 and A 2 are equivalent if and only if they are n n2 2 1 equivalent, where n 1 and n 2 are number of states of A 1 and A 2. In such a case there is polynomial time algorithm deciding whether A 1 and A 2 are equivalent or not. Minimization problem Mateus and Qiu [11] proved that the problem of minimization of states of MM-1QFA is decidable in EXPSPACE. 3.2 Measure-once one-way qunatum finite automata Measure-once one-way quantum finite automata (MO-1QFA) can be seen as abstract machines in a similar way like MM-1QFA, with two 17

32 3. Quantum finite automata main differences. The first one is in the performing of projective measurements. The projection measurement in the MO-1QFA is performed only once and only at the right endmarker. The second difference is that MO-1QFA don t have the set Q rej of rejecting states. MO-1QFA were introduced by Moore and Crutchfield [12]. According to their work, the MO-1QFA are defined as follows. Definition 10. A measure-once one-way quantum finite automaton A (MO-1QFA) is a 5-tuple (Q, Σ, δ, q 0, Q acc ) where meaning of all elements is same as in the Definition 9 except for the transition function δ. Here δ : Q Σ Q C [0,1] and δ is required to be unitary, thus for all q 1, q 2 Q and σ Γ δ must satisfy the following condition { δ(q 1, σ, q)δ(q 2, σ, q) = q Q 1 q 1 = q 2 0 q 1 = q 2 The transition function δ is again (like in MM-1QFA) represented by a set {U σ } σ Γ of unitary operators. Computation. The computation of a MO-1QFA A = (Q, Σ, δ, q 0, Q acc ) on an input w = σ 1 σ n proceeds as follows. At the beginning of a computation, the automaton is in the state q 0. A reads the first input symbol σ 1 and then unitary operator, U σ1, corresponding to symbol σ 1 is applied. This process goes on, until the right-most symbol, σ n is reached. The projective measurement is performed after the last unitary operator is applied. The whole computation may be seen as follows: δ(w) = PU σn U σ1 q 0 Acceptance of a language. Both, bounded and unbounded, error acceptance are defined in a similar way like in MM-1QFA, section 3.1. The same holds for classes RMO, the class of languages accepted by MO-1QFA with bounded-error probability and UMO, the class of languages accepted by MO-1QFA with unbounded error-probability. 18

33 3. Quantum finite automata Languages accepted by a MO-1QFA. Bertoni and Cerpentieri [13] proved, that the class RMO is exactly the class of languages accepted by AF-reversible automata, i.e the class of reversible regular languages. Brodsky and Pippenger [4] showed, that the class RMO is exactly the set of languages recognized by group automata 3. Therefore the class RMO is a proper subset of regular languages. In addition, they also showed that UMO is a subclass of stochastic languages, moreover in [14] they showed that UMO doesn t contain any finite language and therefore this class of languages is a proper subset of stochastic languages. Pumping lemma. Moore and Crutchfield [12] showed the following variation of the Pumping lemma for MO-1QFA. Suppose A be an MO-1QFA. Let p(w) be the probability of A accepting a word w. Then for any word x, any ɛ > 0 and any words u, v, there exists an integer k such that p(uw k v) p(uv) ɛ. Moreover, there is a constant c such that k (cɛ) n, if A s Hilbert space is n-dimensional. Closure properties. Moore and Crutchefield [12] proved some closure properties. For example, they proved closure under (weighted) addition and multiplication, the complex-valued analogs of OR and AND operations. Space-size efficiency. Again like MM-1QFA, MO-1QFA are sometimes more space efficient than its classical counterparts. Several papers studied the space efficiency of MO-1QFA, namely [15, 16, 17, 18, 19, 20]. Equivalence problem. The equivalence problem for MO-1QFA was firstly studied by Brodsky and Pippenger [4]. They designed a polynomial-time algorithm solving this problem. Later, based on their idea, Koshiba [21] developed a polynomial-time algorithm solving this problem. Using a different approach, Li and Qiu [22] solved the equivalence problem too. The above results can be summarize as follows. 3. A group automaton is a variant of DFA, where for each s S and any σ Σ there is a unique p Q, such that δ(p, σ) = s. 19

34 3. Quantum finite automata Two MO-1QFA A 1 and A 2 are equivalent if and only if they are (n 1 + n 2 ) 2 -equivalent, where n 1 is the number of states of the automaton A 1 and n 2 is the number of states of the automaton A 2. Moreover, there is a polynomial-time algorithm in n 1, n 2 solving this problem. Minimization problem. The minimization problem of MO-1QFA was solved by Mateus, Qiu and Li [11]. They proved that state minimization problem of MO-1QFA is in EXPSPACE. They also proposed a method for state minimization, which can be applied for all computational models whose equivalence is known to be decidable. Emptiness problem. Blondel et al. [23] studied the emptiness problem for MO-1QFA. They showed, that this problem is decidable for MO-1QFA accepting language with bounded error probability. For acceptance with unbounded-error probability, the emptiness problem is undecidable. This problem was further studied by Hirvensalo [24] and he improved some results. The power of MO-1QFA. MO-1QFA is less powerful than MM- 1QFA because MO-1QFA cannot recognize finite languages, while MM-1QFA recognize all finite languages. In fact, MM-1QFA is generalization of MO-1QFA. 3.3 Two-way quantum finite automata Two-way quantum finite automata (2QFA) can be seen as quantum variants of classical 2FA. 2QFA were first introduced by Kondacs and Watrous [2]. According to them 2QFA are defined as follows. Definition 11. A two-way quantum finite automaton (2QFA) A is a 6-tuple (Σ, Q, δ, Q acc, Q rej ) where meaning of each element is same as in Definition 9 except the transition function. Here δ : Q Γ Q {,, } C [0,1]. Arrows in the transition function mean: denotes the head should move one cell to the left, denotes the head should move one cell to the right and finally denotes the head should stay on the same cell. 20

35 3. Quantum finite automata The transition function is required to satisfy the following four conditions 4 for any q 1, q 2 Q, σ, σ 1, σ 2 Γ, d {,, } : 1. Local probability and orthogonality condition: { 1 q δ(q 1, σ, p, d)δ(q 2, σ, p, d) = 1 = q 2 p,d 0 q 1 = q 2 2. Separability condition I: 3. Separability condition II: 4. Separability condition III: δ(q 1, σ, p, )δ(q 2, σ, p, ) = 0 p δ(q 1, σ, p, )δ(q 2, σ, p, ) = 0 p δ(q 1, σ, p, )δ(q 2, σ, p, ) = 0 p Acceptance of language. The language acceptance can be defined in a similar way like in section 3.1. Another acceptance often used is acceptance with one-side error. A language L is said to be accepted by a 2QFA A with one-side error if for all w L, the probability of accepting w by A is 1 and for w / L the probability of rejecting w by A is at least 1 ɛ, for 0 ɛ 2 1. Languages accepted by 2QFA. 2QFA accept the whole class of regular languages. Moreover, 2QFA can also accept some context-free and non-context free languages with one-sided error probability. Kondacs and Watrous [2] showed that languages L eq = {a n b n n N} and L = {a n b n c n n N} can be accepted by 2QFA in linear time. Yakaryılmaz and Cem Say [25] constructed more efficient 2QFA accepting the language L eq. 4. These four conditions are equivalent to the requirement that evolution of A is unitary. The proof can be found for example in [3, p. 159] 21

36 3. Quantum finite automata The power of 2QFA. The fact that 2QFA can accept also non-regular languages implies that 2QFA are more powerful than any classical finite automata, MM-1QFA and MO-1QFA. 3.4 Enhanced quantum finite automata So far, we have considered only quantum finite automata that work only with pure quantum states, and on which only unitary operators and projective measurements are performed. Enhanced quantum finite automata (EQFA) allow any orthogonal measurement as a valid intermediate computational step. EQFA also use mixed quantum states and the transitions aren t required to be unitary, so the evolution in EQFA is no longer reversible. EQFA were introduced by Nayak [26]. They are formally defined as follows. Definition 12. An enhanced quantum finite automaton (EQFA) is a 7-tuple (Σ, Q, {U σ }, Q acc, Q rej, Q non, q 0 ) where Σ is a finite set of input symbols. This set of symbols, is extended to Γ = Σ {$, #}, the so-called tape alphabet. Symbols $ and # are again left and right end-markers and it holds that Σ {$, #} =. Q is a finite set of symbols representing basis states {U σ } is a finite set of a superoperators which are given by a composition of a finite sequence of unitary transformations and orthogonal measurements on the space C Q, for each σ Γ Q acc, Q rej, Q non are finite sets of symbols representing accepting, rejecting and non-halting states. It holds that Q acc Q, Q rej Q and Q non = Q (Q acc Q rej ) and Q acc Q rej = q 0 denotes an initial state Computation. The computation of a EQFA A on an input word w proceeds as follows. In the beginning, the automaton A is in the state q 0 q 0. Then for each σ Γ of an input word the transition proceeds as follows: 22

37 3. Quantum finite automata At first, the corresponding transformation U σ is applied on the current state ρ to obtain a new state ρ Then new state ρ is then measured with respect to the observable E acc E rej E non, where E acc = span{ q Q acc }, E rej = span{ q Q rej }, E non = span{ q Q non }. The probability of observing E i is equal to Tr(P i ρ ) 5 where P i is the orthogonal projection onto E i. In case that the outcome from E acc or E rej is observed, the input word is accepted or rejected. Otherwise the computation continues with the automaton in the state P non ρ P non. Acceptance of a language. Language L is said to be accepted by EQFA A with probability p > 1 2 if A at the end of a computation over its input accepts each w L with probability at least 1 2 and rejects each w / L with probability at least 2 1. Languages accepted by EQFA. Despite the fact that EQFA use such powerful quantum tools, they accept only proper subset of regular languages. Nayak [26] showed that the language {0, 1} * 0 cannot be accepted by any EQFA with probability bounded away from 1 2. Space-size efficiency. EQFA can be sometimes more space efficient than QFA using pure states (MM-QFA, MO-QFA). For example, Nayak [26] also proved that EQFA recognizing language L n = {w0 w {0, 1} *, w n} are exponentially more space efficient than any MM- QFA recognizing this language. The power of EQFA. It is clear that EQFA are less powerful than classical finite automata (DFA). However EQFA are more powerful than MM-QFA and MO-QFA. 5. Tr denotes the {em trace}. 23

38 3. Quantum finite automata 3.5 Quantum finite automata with open time evolution So far quantum finite automata with open time evolution (OT-QFA) are the most general cases of one-way quantum finite automata. OT-QFA were introduced by Hirvensalo [27] and are defined as follows. Definition 13. A quantum finite automaton with open time evolution (OT-QFA) is a 5-tuple where Σ is a finite alphabet A = (Σ, Q, δ, q 0, F) Q is a finite set of symbols representing states δ : Σ l + Tr (l(h Q )) 6 is a transition function q 0 is a symbol representing an initial state F Q is the set of symbols representing final states The transition function can be extended from symbols to words in a similar way like in Definition 2. Here we extend the domain of δ into whole Σ + and then δ must satisfies these two conditions: δ(1) = I (the identity mapping) δ(aw) = δ(w)δ(a) for a Σ and w Σ +. Computation. The computation of OT-QFA A = (Σ, Q, δ, q 0, F) on an input w = w 1 w n proceeds as follows. A begins with an initial state q 0 q 0 and the input word transforms the initial state into δ(w) q 0 q 0 = δ(w n ) δ(w 1 ) q 0 q 0 6. l + Tr (l(h Q )) denotes the set of completely positive, trace-preserving mappings on the state space 24

39 3. Quantum finite automata Observable. The state of an automaton A is determined by the observable { q q q Q}. If the automaton is in a quantum state φ, the probability that state q q, q Q is seen after the observation is P(q) = Tr( q q φ). Acceptance of languages. Let P = q q be the projection onto q F the subspace spanned by the basis vectors corresponding to the final states, then the acceptance probability of an OT-QFA A on an input word w is Tr(Pδ(w) q 0 q 0 ) We say that A accepts a language L, if there exists ɛ [0, 1 2 ) such that for each w L automaton A accepts w with probability at least 1 ɛ and accepts each w / L with probability at most 1 2. Languages accepted by OT-QFA. of regular languages. OT-QFA accept exactly the class Power of OT-QFA. OT-QFA are the most powerful quantum finite automata. All classical and quantum finite automata can be simulated by OT-QFA, some of them (PFA, MO-1QFA, MM-1QFA) without increasing the number of states. 3.6 Other models of quantum finite automata In this section we briefly introduce other known models of 1QFA, namely Latvian QFA, QFA with ancilla qubits and multiletter QFA. Latvian quantum finite automata. Ambainis et al. [28] presented model of one-way quantum finite automata called Latvian quantum finite automata (LQFA). LQFA can be seen as a generalization of MO- 1QFA. The definition of LQFA is almost the same as a definition of MO- 1QFA (Definition 10). The only one difference is again in the transition function. The transition function in LQFA is for each σ Σ a combination of a unitary matrix and projective measurement. 25

40 3. Quantum finite automata The class of languages accepted by LQFA is a proper subset of the class of languages recognized by MM-1QFA. Unlike MM-1QFA, LQFA cannot recognize regular language aσ *. Quantum finite automata using ancilla qubits. Paschen [29] introduced a model of one-way quantum finite automata using ancilla qubits (A-QFA). A-QFA are defined in a similar way like MO-1QFA, but in addition, A-QFA also use ancilla qubits and have defined an output tape. A-QFA recognize all regular languages with certainty. Some nonregular languages can be recognized with one-sided unbounded error probability by A-QFA. Any classical finite automata can be simulated by A-QFA. Multiletter quantum finite automata. Belvos et al. [30] proposed model of one-way multi-letter quantum finite automata (ML-QFA). ML-QFA can be seen as a quantum analog of one-way multi-head finite automata [31]. Unlike MO-1QFA, in ML-QFA a unitary transformation matrix is assigned to each string w of length k, where k is a number of symbols, that the automaton can read in one step. The class of languages accepted by ML-QFA is again a proper subset of regular languages. However, ML-QFA can accept some languages that MM-1QFA and MO-1QFA cannot. 26

41 4 Conlusion We have presented in a uniform way several models of classical finite automata and quantum finite automata. The most attention was given to quantum finite automata. We have studied five main models of quantum finite automata in detail. Three other models of quantum finite automata were also briefly presented. We have also presented basic results concerning the power of each model, closure properties, mutual relations, and results concerning decision problems. This field still remains open for further research. This thesis could be important for deriving a new model of quantum finite automata which would be satisfactory. One way how to derive a new model of quantum finite automata is for example by modifying one of the presented models by restricting or extending the set of allowed used quantum tools. 27

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