One-way quantum finite state automaton seem to model very well the way very simple quantum processors work (Ambainis and freivalds, 1998), and also the way simple classical/quantum processors are expected to work: the classical part reads an input, picks up the corresponding quantum operator (a transition mapping) and performs it on a quantum memory of fixed size, independent of the size of input. 1QFA are very simple but less powerful than classical 1-way finite automaton.

Measure many quantum finite state automata (1QFA): We consider 1-way quantum finite automata (QFA) as defined in [2] .

Definition 1. Namely, a 1-way QFA is a tuple where

1) Q is a finite set of states,

2) is an input alphabet,

3) is a transition function,

4) is a starting state,

5) and are sets of accepting and rejecting states.

The states in and are called halting states and the states in are called non- halting states. and are symbols that do not belong to. We use and as the left and the right end marker, respectively. The working alphabet of M is.

A superposition of M is any element of (the space of mappings from Q to with norm). For, denotes the unit vector with value 1 at q and 0 elsewhere. All elements of can be expressed as linear combinations of vectors. We will use to denote elements of. The transition function maps to C. The value is the amplitude of in the superposition of states to which M goes from after reading a. For, is a linear transformation on defined by

The computation of a QFA starts in the superposition. Then transformations corresponding to the left endmarker, the letters of the input word x and the right endmarker are applied. The transformation corresponding to a consists of two steps. 1) First, is applied. The new superposition is where is the superposition before this step. 2) Then is observed with respect to the observable where, ,. This observation gives, with the probability equal to the amplitude of the projection of. After that, the superposition collapses to this peojection. If we get, the input is accepted. If we get, the input is rejected. If we get, the next transformation is applied.

Theorem 1. Let L be any language recognized by 1QFA with bounded error. Then L is regular.

Proof. The proof is in [2] .

Proposition 1. Given a language, it is not possible in general case to build a 1QFA that recognizes this language.

Proof. The proof is in [20] .

Theorem 2. The language a cannot be recognized by 1QFA with bounded error.

Proof. This is shown in [2] .

The model of 2-way quantum finite state automaton (2QFA) is first introduced by Watrous [2] . 2QFA is more powerful than their classical counterpart. A 2QFA consists of a finite state control and a 2-way tape head― which scans a read only input tape.

Definition 2. Formally, a 2-way QFA is specified by 6-tuplet where

1) Q is a finite set of states.

2) is an input alphabet.

3) is a transition function which has a mapping of the form. In addition to input symbols, and are symbols that do not belong to. We use and as the left and the right end marker, respectively. The working alphabet of M is.

4) is a starting state.

5) are sets of accepting states.

6) are sets of rejecting states.

The states in and are called halting states and the states in are called non- halting states.

The 2QFA satisfies the following conditions (of well-formedness) for any, ,:

1) Local probability and orthogonality condition

.

2) Separability condition I

.

3) Separability condition II

.

4) Separability condition III

.

In order to process an input word by M, we assume that the input is written on the tape with the endmarkers in the form w_x = #x$ and such a tape of length |x| + 2 is circular, i.e., the symbol to the right of $ is #.

For an integer n let C_x be the set (of size (n + 2)|Q|) of all possible configuration of M, for inputs of length x. represents the amplitude with which a machine currently in state q and scanning symbol will change state to and move its tape head in direction d. For any tape induces an operator (called the time-evolution operator U on tape x) on as follows:

, k + d mod |x| for each and is extended to all by

linearity. Consider the Hilbert space l₂(Q), where Q is the set of internal states of a 2QFA M. Suppose that we havea linear operator for each and a function D:. Define tran- sition function as

and

M is well-formed when is unitary.

Theorem 3. Every regular language is accepted by a 2QFA.

Proof. The proof has been shown in [2] .

Multi-letter quantum finite state automata has been introduced in [15] . In [16] , Qie etal. further study the decidability of the equivalence and minimization problems of multiletter QFAs. In [17] , hierarchy and equiva- lence of multiletter quantum finite state automaton are studied. k-letter QFA can be thought of as an analogue of quantum automata with classical memory containing the previously read letters. k-letter QFA is not limited to see only one, just incoming input letter, but can see several up to k of the earlier letter as well.

Definition 3. Formally, a k-letter QFA M is specified by a 5-tuple where

1) Q is a finite set of states,

2) are set of accepting states.

3) is the initial quantum state from H_Q.

4) is an input alphabet.

5) is a transition function that assign a unitary trasition matrix on to each string

and where Cⁿ denotes Euclidean space consisting of all n-dimensional complex vectors.

A k-letter QFA M works in the same way as an measure-once 1-way quantum finite state automaton [1] except that it applies unitary transformation corresponding not only to the last letter but to the last k-letters received. When k = 1, it is exactly same as measure once 1-way quantum finite state automaton. According to [16] , all languages accepted by k-letter QFAs with bounded error are regular language for any k.

To calculate the probability P_M(x) that a k-letter QFA accepts an input string, it fol-

lows that for any, is a unitary matrix. By they define a map from to the set of

of all unitary matrices. is induced by in the following way. For

and

which specifies the computing process of M for an input string x. They identify the states in Q with an orthonormal basis of the complex Euclidean space and let P_a denote the projection operator on the subspace spanned by Q_a where

where denotes the conjugate transpose of vector.

In these section we write transition matrices of different automaton and discuss different properties of these automaton in terms of their transition matrices.

1) Deterministic finite state automaton

A deterministic finite automaton [20] consists of five tuple tuple where

1) Q is a finite set of states,

2) is an input alphabet,

3) is a transition function that takes as arguments a state and an input symbol and return a state,

4) is a starting state,

5) is a set of final or accepting states.

We design a deterministic finite state automaton which accepts all of string having at least one alphabet “b” [see Figure 1] where

The transition matrix of the deterministic finite state automaton is shown in Figure 2.

Here each row of each transition matrix contain exactly one non-zero entry i.e. 1 for deterministic finite state automaton.

2) Non-deterministic finite state automaton

An non-deterministic finite automaton [20] is represented essentially like a deterministic finite state auto- maton. It consists of five tuple where

1) Q is a finite set of states,

2) is an input alphabet,

3) is a transition function that takes a state in Q and an input symbol in as arguments and returns a subset of Q,

4) is a starting state,

5) is a set of final or accepting states.

The only difference between an non-deterministic finite state automaton and deterministic finite state auto- maton is the value of that returns a set of states in the case of an non-deterministic finite state auto- matonand single state in the case of deterministic finite state automaton.We design a non-deterministic finite state automaton (shown in Figure 3) which accepts all of string having at least one alphabet “b” where

The transition matrix of the deterministic finite state automaton is shown in Figure 4.

There is atleast one row in a transition matrix for non-deterministic automaton which contain more than one non-zero entry.

3) Reversible finite state automaton

An automaton [21] is reversible if, for every state p in Q for every letter a in M there existsat most one transition in that comes from p (respectively goes to p) with label a. Here

1) Q is a finite set of states,

2) is an input alphabet,

3) is a transitions,is a subset of,

4) is a set of final states,

5) is a set of final states.

A reversible automaton is a finite automaton in which each letter induces a partial one-to-one map from the set of states into itself. A reversible automaton may have several initial or final states. As a consequence, the minimal automaton of a reversible language may not reversible.

We define reversible automaton shown in Figure 5 which accept string of a’s of length 3 where

The transition matrix of the above automaton is shown in Figure 6.

In case of reversible automaton dot product of any two row is zero and there are no cycles within the transi- tion/output matrix that can’t accessed from one of the input states.

4) Probabilistic finite state automaton

A probabilistic finite state automaton [22] over the alphabet is a system consists of where

1) Q is a finite set of states,

2) is a transition function from into such that for, , ,

3) is a starting state,

4) is a set of final or accepting states.

In case of probabilistic finite state automaton we allow the fractional values in transition matrix with the provision that sum of each row give 1 [see Figure 7].

5) Quantum finite state automaton

We consider 1-way quantum finite state automata (QFA) as defined in [23] is a tuple where

1) Q is a finite set of states,

2) is an input alphabet,

3) is a transition function,

4) is a starting state

5) and are sets of accepting and rejecting states.

The states in and are called halting states and the states in are called non- halting states. and are symbols that do not belong to. We use and as the left and the right end marker, respectively. The working alphabet of M is. Quantum finite state automaton is obtainby letting the transition matrix with complex entries.We also require each of the matrices to be unitary.

The transition matrix of the quantum finite state automaton looks like [Figure 8]:

The transition matrix is unitary since the sum of the squares of the norms in each row adds up to 1 and the dot product of any two row is 0.If all matrices only have 0 or 1 entries and the matrices are unitary,then the automaton is deterministic and reversible.

6) 1-way multihead deterministic finite state automaton

A 1-way k-head deterministic finite state automaton is a deterministic finite state automaton with k- independent read heads on a single input tape with the end markers. On each move the machine can si- multaneously read the k input cells scanned by k-heads,move each head one square to the right or keep stationary.

A 1-way multihead deterministic finite state automaton (1DFA(k)) [13] is a tuple where

1) Q is a finite set of states,

2) is an input alphabet,

3) is the number of heads.

4) is the partial transition function;where 1 means to move the head one square to the right and 0 means to keep the head on the current square,

5) is the left and is the right endmarkers.

6) is a starting state,

7) is a set of final or accepting states.

We define a 1DFA(2) shown in Figure 9 which accept where

The transition matrix of the above automaton is [see Figure 10].

Each row of transition matrix contain only one 1 which has the same property as deterministic finite state automaton.

7) 1-way Reversible multihead finite state automaton

A 1-way reversible multihead finite state automaton (1REV-DFA(k)) [13] is a tuple which has same structure as 1DFA(k) where

1) Q is a finite set of states,

2) is an input alphabet,

3) is the number of heads.

4) is the partial transition function;where 1 means to move the head one

square to the right and 0 means to keep the head on the current square,

5) is the left and is the right endmarkers.

6) is a starting state,

7) is a set of final or accepting states.

Let M be a 1DFA(k) and D be the set of all reachable configuration that occur in any computation of M beginning with an initial configuration and with, and. D be the set of all reachable configurations that occur in any computation. M is said to be reversible if the following two conditions are fulfilled:

1) For any two transitions:

and

it holds if.

2) There is at most one transition of the form

.

The non-context free language is accepted by REV-1DFA(2)

shown in Figure 11 where the transition function is asfollows:

The transition matrix of the above automaton is shown in Figure 12.

Dot product of any two row is zero for multihead reversible finite state automaton.

8) 1-way multihead quantum finite state automaton

1-way multihead quantum finite state automaton is a 1-way k-head quantum finite state automaton where k-heads of the automaton can move to the right or stay on the current tape square but not beyond the end markers.The language is accepted by the 1QFA(k)

[see Figure 13] where

The transition matrix of the above automaton is shown in Figure 14.

The sum of the square of the norms in each row adds up to 1 and dot product of any two row is zero formultihead quantum finite state automaton.

In this section we show that 1QFA(k) has more language recognizing power than 1QFA. 1QFA(2) can recognize regular language and context-sensitive language respectively.

Theorem 4. 1QFA(2) can accept all unary regular languages.

Proof. In [12] it has been shown that any unary regular language is accepted by some 1-way reversible 2-headdeterministic finite automaton. We find from the previous section that in a 1-way multihead quantum

finitestate automaton where the transition matrices are only 0 and 1 entry, it is essentially a 1-way reversible multihead finite state automaton. So 1-way 2-head quantum finite state automaton accept all unary language.

Example 1. A 1-way 2-head quantum finite state automaton is a automaton can accept in the folowing manner:

Let, , ,

Define:

The automaton acts as follows: Initially both heads of the automaton M are at #. After reading the input symbols, the automaton M remain at state. The first head remain stationary where the second head move one square to the right of the input tape whenever the automaton reaches at state. The movement of heads are similar as the previous case. Due to the movement of heads the second head may be at “a” or “b”.

For both cases the automaton M move to state from state and both heads move one square to theright of the input tape. When the first head at “a” and second head at “$”, the automaton M goes to final state from and the string will be accepted by the automaton M with probability 1.

Consider a string w not in L. As w is not in L the heads of the automaton M will arrived in such a way that for that particular position of heads and state, no transition rules are defined. So, for a string w, which M does not accept, there is no sequence of transitions that makes M to its final state after consumption of w. So, M rejects with probability 1. Each pairs of is unitary.

Example 2. A 1-way 2-head quantum finite state automaton is a automaton can accept in the following manner:

Let, , ,

Define:

The automaton acts as follows: Initially both heads of the automaton M are at #. After reading the input symbols, the automaton M remains at and the first head remain stationary and the second head moveone square to the right of the input tape.Whenever the automaton M reach at state the movement of heads are similar as the above case.The automaton M performs similar task as previous one for all inputletter “a” until the next input letter read by second head is “b”. For reading input letter “b” by the second head of the automaton M, it moves to state and remains at state until the next input letter read by thesecond head is “c”. This steps counts the number of letter “b” with number of letter “a”. Both heads move one square to the right of the input string for state. After reading the letter “c” by second head where the first is reading letter “a” of the input string, the automaton M goes to state from state. Both heads moveone square to the right of the input string whenever state will be reached. The automaton M now checks the number of letter “b” and “c” of the input string with two heads of the automaton until the second head readthe right endmarker $, for which the automaton goes to state. When the automaton M is in state and both heads read the right endmarker $, then the automaton accept the input string with probability 1.

Consider a string w not in L. As w is not in L the heads of the automaton M will arrived in such a way that for that particular position of heads and state, no transition rules are defined. So, for a string w, which M does not accept,there is no sequence of transitions that makes M to its final state after consumption of w. So, M rejects with probability 1. Condition of unitarity is satisfied for all pairs of.

Theorem 5. 1QFA(2) is more powerful than 1QFA with respect to recognition of language.

Proof. In Theorem 2 it was proved that the language cannot be recognized by 1QFA with bounded error. In Example 1 we proved that it can be done with 1QFA(2).

Theorem 6. Given a language it is possible in general case to build a 1QFA(k) that re- cognizes this language.

Proof. In [24] it has been shown that this language consists of subset of words from language whichis recognize by 1QFA(2) as shown in Example 1.

Example 3. A 1-way 2-head quantum finite state automaton is a automaton can accept in the following manner:

Let, , ,

Define:

The automaton acts as follows: at each reading of the symbol the automaton goes into a super-

position of two states and with amplitude a respectively.This is done with the objective that at

each reading of the symbol the automaton guesses x to be the first character which does not match. Thus if the guess is rightthen the path corresponding to goes to an accepting states and the guess is wrong the path correspondingto goes to an rejecting state. Even if the guess is wrong the path corresponding to allows us to makeanother guess. So if the input word belongs to L and k^th letter does not match with theend.

Then at the k^th depth will reach an accepting states with probability. But if the input word is an

palindrome and does notbelong to L then no matter what depth we traversed will never go to an accepting state. As the aboveautomaton is 1-way and the length of the input word is finite, the above automaton will always halt in finitetime and the input string belonging to the language is accepted by the automaton if the probability of state is greater than 0. We arrive at this conclusion because if the word is palindrome the probability of will be 0. Each is unitary by inspection. So M is well-formed. Figure 15 and Figure 16 shown below describe the working steps of the automaton in pictorial form.

Theorem 7. where is the set of all languages accepted by 1-way 2-head quantum finite state automaton and is the set of all languages accepted by 1-way 2-headdeterministic finite state automaton.

Proof. The language cannot be recognized by 1DFA(2) ( [18] [19] ). In Example 3 we proved that it can be done with 1QFA(2).

Theorem 8. For every 1-way reversible 2-head finite state automaton M which accepts a language L, thereexists a 1-way 2-head quantum finite state automaton M’ which accepts the same language L.

Proof. We know that the transition matrix of 1-way reversible multihead finite state automaton has the following properties:

1) Dot product of any two row is zero for 1-way reversible multihead finite state automaton.

2) All matrices only have 0 or 1 entries.

Therefore the above two properties of the transition matrix ensures that the transition matrix is also unitary. As a result given a 1-way reversible 2-head finite state automaton M we get a 1-way 2-head quantum finite state automaton M’ which has the same transition matrix,same set of states, same set of accepting states and start state as M. as the transition matrix, start state and accepting states of M and M’ are same,they accept the same language.

Theorem 9. The set of all languages accepted by 1-way reversible 2-head finite state automata (1RMFA(2)) is a proper subset of set of all language accepted by 1-way 2-head quantum finite state automata. (1QFA(2))

Proof. Theorem 8 tells us that for every 1RMFA(2) which accept a language L there exist 1QFA(2) which accept the same language. So, the set of all languages accepted by 1RMFA(2) is a subset of set of all languages accepted by 1QFA(2). From ( [18] [19] ) we know that the language is not accepted by any 1DFA(2). Also from ( [13] ) we know that the set of all languages accepted by 1RMFA(2) is a proper subset of 1DFA(2). Therefore, there is no 1RMFA(2) which accepts the language. In Example 3 ithas been shown 1QFA(2) accept the language L. Thus, the subset relation is proper.

Corollary 1. 1QFA(2) is computationally more powerful than 1RMFA(2).