ISSN: 2229371X
Tirtharaj Dash^{*1} and Tanistha Nayak^{2}

Corresponding Author: Tirtharaj Dash, Email: [email protected] 
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Nowadays every computing device is based on the Turing Machine. It is known that classical physics is sufficient to explain macroscopic phenomena, but is not to explain microscopic phenomena like the interference of electrons. In these days, speedup and downsizing of computing devices have been carried out using quantum physical effects; however, principles of computation on these devices are also based on classical physics. This paper tries to analyze mathematically a possibility that the Universal Quantum Turing Machine (UQTM) is able to compute faster than any other classical models of computation. Basically we focused on comparative study on computation power of Universal Turing Machine (UTM) and UQTM. Namely, in the equal, we tried to show that the UQTM can solve any NPcomplete problem in polynomial time. The result analysis showed that UQTM is faster for any computation.
Keywords 
Quantum; Quantum Computer; Quantum Turing Machine; Universal Turing machine; Universal Quantum Turing Machine. 
INTRODUCTION 
Quantum is a discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents [1]. An analogous discrete amount of any other physical quantity, such as momentum or electric charge is known as Quantum [1,3]. Bit (Binary Digit) is the smallest unit of information within computer, the only thing that computer can understand Bit is basic unit of Classical Computer. Classical Computer only works binary values called Bits, each having a value usually denoted by 0 or 1. One of the most intuitive representation of bit is an open or closed switch or by extension, and „onÃ¢ÂÂ or „offÃ¢ÂÂ portion of the circuit .In todayÃ¢ÂÂs modern computer ,this representation remains in transistors, with a high voltage possibly denoting a 1 and low voltage possibly denoting a zero. A two state system (0 →1) is the building block of classical computational device. Quantum bit (Qubit) is a unit of quantum information. Qubit represents both the state memory and the state of entanglement in a system. Quantum entanglement is experimentally verified property of nature. 
Quantum Entanglement occurs when the particles such as electron, photon, molecules interacts physically and then become separated. This interaction is called entanglement. Quantum bit (Qubit) can exist in a superposition of states which can be represented as α0>+β1> where α, β represents complex number satisfying α2 +β2=1.Any state measurement results in 0> with probability α2 and 1> with probability β2.A nQubit system can exist in any superposition of the 2 basis states. Quantum entanglement is a form of quantum superposition. A Quantum Computer also works with two Eigen values, a 0 and 1, but these units are termed as Qubit or Quantum Bits due to their non classical behavior. A Qubit can take any value i.e. linear combination of Qubit 0 or Qubit 1. 
Therefore while classical bits can take any value of north and south and any orientation would be illegal. A Quantum bits (Qubit) can take the value of northeast [2]. However upon measurement, only one of these values, north or south is read. To determine the coefficient of the linear combination of the relative portions of the amount read being north and south must be found out through repeated measurements of the original Qubit state. Only something that exhibit quantum phenomena that can be used to hold a Qubit. It is impossible for a simple on/off switch to hold a linear combination of on or off. On the other hand electron or nuclear spin and polarization state of photon make natural candidates. These quantum states either spin or polarization must also somehow be able to manipulate so that that value could be assigned to each bit [3]. Figure1 shows a binary transition of states incorporating physical phenomenon. 
TURING MACHINE 
We are interested in designing an automaton (machine) that can solve our two objectives. (i)Recognizing (ii) Computation. A Turing Machine (TM) is a generalization of Pushdown Automata (PDA) a tape instead of a tape and stack. The potential length of the tape is assumed to be infinite and divided into cells, and one cell hold one input symbol. The head of TM is capable to read as well as write on the tape and can move left or right or can remain static. Turing Machine(TM) is an abstract version of a computer; this has been used to define formally what is computable. The ChurchÃ¢ÂÂs Thesis [4] is supported the fact that a Turing Machine(TM) can stimulate the behavior of general purpose computer system. Alan Turing, an English mathematician in 1936, suggests the concept of Turing Machine. Turing Machine accepts the language defined by the grammars that are called Type 1 languages or Recursively Enumerable languages. Turing Machine is idealized in that they have an infinite memory, which comes in the form of an “infinite tape” (infinite on both the sides). The idea is that this tape consists of cells each of which can take one symbol from some underlying alphabet Σ, and that there is a „read/write headÃ¢ÂÂ which moves over the tape one position at a time and can either read from or write to the tape. It is practical to assume that the input string is written on the tape (from left to right). 
When the machine starts the read/write head is positioned on the cell holding the first symbol of the input string. We further assume that the tape contains no further symbols that are all remaining cells are empty. In order to describe what the machine does (that is, under which circumstances it moves the head one position to the left or the right, or it reads from the tape, or writes to the tape) we need to have a notion of state, just as we had with the other automata. As before there is a specific start state, and this time we need an action stop to tell us when the machine has finished its computation. 
Abstract Model of a UTM: 
A Turing Machine (TM) consists of a finite control, which can be in any of a finite set of states. There is a tape divided into square or cells; each cell can hold any one of a finite number of symbols. Figure 2 shows a model of TM and its components. 
Initially, the input, which is a finitelength string of symbols chosen from the input alphabet, is placed on the tape. All other tape cells, extending infinitely to the left and right, initially hold a special symbol called the blank. The blank is a tape symbol but not an input symbol, and there may be other tape symbols besides the input symbols and the blank, as well. There is a tape head symbol scanned. In one move, the Turing Machine will work as following steps. 
a. The next state optionally may be the same as the current state. 
b. Write a tape symbol in the cell scanned. This tape symbol replaces whatever symbol was in that cell. Optionally, the symbol written may be the same as the symbol currently there. 
c. Move the tape head left or right. In our formalism we require a move, and do not allow the head to head to remain stationary. This restriction does not constrain what a Turing Machine(TM) can compute, since any sequence of moves with a stationary head could be condensed, along with the next tape head could be condensed, along with the next tape Head move, into a single state change, a new tape symbol, and a move left or right. 
Mathematical Description of Turing Machine: 
TM can perform computation like addition, subtractions, multiplication, and division. Number on the tape is represented in form of Zero. Suppose i is a positive integer, then oi represents this integer. It is assumed that a 1 separates two integer numbers. Suppose f is a computable function and has arguments a1,a2,a3 such that f (a1,a2,a3) =m, where a1,a2,a3 and m are some natural numbers then f is said to be computed by TM M .Such a function f is said to be Turing Computable or simply computable. TM computes integer function as defined below.f1 (a1, a2, a3......,an)→m. If f1 is defined for all arguments a1, a2, a3......an then total function is analogous to recursive language. A function f1(a1, a2, a3......,an) computed by a TM is known as partial recursive function, if f is defined for some, not for all values of a1, a2, a3......,an. The TM that compute this function halts for some and may not halts for other input like n!, log2n etc... total recursive function. Figure6 shows a sample difference between Total Recursive Function and Partial Recursive Function. 
Modification of Turing Machine: 
There are many modification of Turing Machine. 
a. Twoway infinite TM. L is recognized by a Turing Machine with a twoway infinite tape if and only if it is recognized by oneway infinite tape. Figure 7 shows this machine model. 
b. Multiple TM: If a language L is accepted by multiple tape Turing Machine, it is accepted by single tape Turing Machine. Figure 8 shows a theoritical structure of this type of TM. 
the region between c and s, it should be obvious that the offline TM is just special case of multiple TM. An offline TM can simulate any TM by using one more tape than M. The first thing of offline TM does is copy its own input onto the extra tape and it then stimulates M as if extra tape were MÃ¢ÂÂs input. 
QUANTUM TURING MACHINE 
The Quantum Turing Machine (QTM) was defined by David Deutsch in 1985.QTM is a precise model of a quantum physical computer. There are two ways of thinking about QTM, (i)the quantum physical analogue of a Probabilistic Turing Machine; (ii) computation as transformation in a space of complex superposition of configuration. The QTM is the most general model for a computing device based on Quantum physic. Like an ordinary Turing machine. A Quantum Turing Machine M consists of a finite control, an infinite tape, and a tape head. a quantum Turing machine is defined to be a quantum system consisting of a processor, a moving head, and a tape, obeying a unitary time evolution determined by local interactions between its components, and allowing to be in a superposition of computational configuration. Deutsch pointed out that the global transition function between computational configurations should be determined by a local transition function which depends only on local configurations.. Bernstein and Vazirani found a simple characterization of the local transition functions for the restricted class of quantum Turing machines in which the head must move either to the right or to the left at each step. Since the above characterization constitutes an alternative definition of quantum Turing machines more tractable in the field of theoretical computer science, it is an interesting problem to find a general characterization valid even when the head is not required to move or more generally when the machines has more than one tape [7,8]. 
superposition of MÃ¢ÂÂs configuration. This linear mapping is specified by the following matrix MÃ¢ÂÂs configuration. This linear mapping is specified by the following matrix M6. Each row and column of M6 is corresponds to configuration of M. Let c1 and c2 be configuration of M, then the entry corresponds to c2 row and c1 column of M6 is δ evaluated at the tuples which transforms c1 and c2 in a single step. If no such tuples exists, then the corresponding entry will be Zero. We call this matrix M6 a time evolution matrix of M. With a Quantum Turing machine, δ associates with each state and symbol, and each possible next move, a complex probability amplitude (which we require to be a feasible complex number). We also require that the linear transformation defined by the machine is unitary. BQP is the collection of languages L recognized by a Quantum Turing machine, running in polynomial time, under the bounded probability rule. The class BQP is not changed if we restrict the set of possible amplitudes to {0, ±2/3, ±4/5, 1} BPP ⊆ BQP. Shor has shown that the factorization problem is in BQP. It is not known to be in BPP. 
QTM can be considered as a physical model: 
A quantum Turing machine is a quantum system consisting of a processor, a bilateral infinite tape, and a head to read and write a symbol on the tape. Its configuration is determined by the processor configuration q from a finite set Q of symbols, the tape configuration T is represented by an infinite string from a finite set Q of symbols, the tape configuration T is represented by an infinite string from a finite set ÃÂ© symbols, and the discretized head position Ã¡Â¶Â. Figure 10 is a sample example of QTM considered as a physical system [6,7,8]. 
However, various researches are going on throughout the globe to make this hypothesis of Quantum computing to implement it practically like development of a quantum computer. It is still in its infancy stage. 
CONCLUSION 
This paper presents a comparative analysis on the two computational models, „Turing MachineÃ¢ÂÂ and „Quantum Turing MachineÃ¢ÂÂ. The study is based on computational power of both the models. It shows that a Turing machine can be modified according to application. Based on this, some models have been proposed viz. singletape, multitape, kheaded, multiple etc. However, Quantum Turing machine is better than any other computing models based on its computational speed and number of languages accepted. As future works, it would be interesting to work in depth of the Quantum Turing machine and find out its more advantages over traditional computational models. 
References 