## Abstract

The effort that the has been put in the field of complex computation has been brought a lot of changes. The issue of proving or disapproving whether a set of binomially verifiable problems is equal to a set of bounded problems. The paper gives a detailed overview of the myth surrounding the quantum complexity and the implication that has brought to the world of quantum physics. To better understand the concept, it is crucial to introduce a Turing machine which is based on the quantum mechanical concept. In addition, the relationship between quantum oracle and Turing machine is exploded. The paper will also give an overview of quantum analogues in relation to classical computing. The quantum computing circuit or network which is created using quantum Turing machine and quantum gated. This will give the basis of identifying the difference between quantum and classical concept.

The link between the two concepts are formulated from the principle that any gate is specifically recognized by a quantum Turing machine and each quantum Turing machine can be represented in simulation using a quantum network. To realize an in-depth understanding of this concept, a close relationship between resources required by the network simulation and quantum Turing machine using step operation S or T matrices is tested.

## Introduction

The research has brought forward the fundamental idea for classical computing. The circuit model and the Turing machine became classical computing forms, complexity classes and their application in encryption processes. The basis of this is the fact that the long time calculation is the key thing in all the cryptographic security. It has been proven that getting the factor for a big prime number is simple when calculated using the quantum versions. I try to make the whole idea understandable by giving a mathematical description of quantum gates. Though it is not possible to emulate a quantum gate using a classical gate, an arbitrary estimation can be done.

The common classical computer carries out computation using logic circuits. Basically, computation is where the output is generated from the input. Both the input and the output are abstract representation represented using quanta or bits. This is the least possible quantities of non-probabilistic information.

## The quantum circuit model

Quantum circuit is a circular connection of quantum circuit linked using wires. The quantum gate gives an abstraction of the overall quantum operation that involves a fixed number of quits. The wire gives an abstraction of what the gate is acting. The number of inputs is not necessarily equal to the number of output. For instance, you can have five input slots and six output slots in a circuit unitary quantum circuit is a quantum circuit whose gates maps to unitary quantum operation.

## Unitary purification

The link between unitary quantum circuits is better understood by describing unitary purification. This is a more specified realization of stinespring theorem. The theorem implies that a more generalized operation of a quantum is mapped to several unitary operations in a bigger system. The theorem was put in practice by Aharonov. This person gave several meaning of quantum circuit model. Purification is derived from the idea of mixed quantum state purification.

Any process in classical computing follows three steps. The first step is the preparation of the input states in the media. The second state id the interaction in the quantum machine elastic scattering and lastly the measurement of output carriers after a given number of steps. Step two is the most critical though ignored in most application. A logic gate is a computing machine that uses a fixed number of bits in both eh input and the output while a quantum gate is a device whose input and output is categorized in qubits. The lather uses the eigenstate principles where an output relates to an input through an invertible gate function.

A reversible circuit and gate should have equal number of input and output wires. The device functionality can be reversed by inputting the same number of input on the output part. Several gates can be connected into a circuit. The result after computation gives a circuit that resembles this:

Figure 1: A Circuit of several combinations of gates

The output of each computation state can be used as the input of the next step. From the above explanation, it can be said that logic circuit are computing machine which performs computation in fixed times.

## Mathematical description of gates

The initial state of a describing a gate is choosing the computational basis as per the Eigenstates’ principle. It is crucial to ensure that the input operator and the output operator are the same so that non-trivial issues are eliminated. This concept is represented using such a diagram

Figure 2: Eigenstates’ principle computation

The action of a gate can be written down using a truth table where each combination of the input value is mapped to the output as shown below:

Figure 3: Truth table

Classical gate is an XOR gate since the logical XOR function is computed using the first and the second inputs. This will guarantee reversibility. Basically, the action of a quantum gate is realized by inverting the second input when the first input is set and giving the second input unchanged when the first input remains the same. This is called measurement gate or controlled NOT.

The concept can also be represented using permutation. For instance, the computational basis state for a system is set to be {|a, bi}, a, b ∈ {0, 1}. The input and the output states are mapped using the foll0wing method.

Figure 4: Permutation truth table

The other way is represented using an S-Matrix. This is the most appropriate for quantum gates. Basically, a unitary matrix is used to encode the mapping. For example, linear algebra measurement gate represented by a unitary matrix that denotes the eigenstate of the input and output carriers. This can be represented using this diagram.

Figure 5: Scattering matrix formula

## Quantum during machine

The best description of during machine is given by Benoiffit. According to Bennoiff (), a Turing machine is analogues to classical one-tape machine. A one tape machine is made up of an infinite memory, a finite processor, a state control and a firmware program. The computation in a Turing machine follows a particular algorithm where the fine part of the memory and the processor are involved. There is a halt state which occurs when the quantum Turing machine receives two subsequent states which are identical. The halt state is viewed with the binary spectrum. The state should remain measurable regardless of the state of the quantum Turing machine. Since the quantum Turing machine is universal, it can be simulated by any computer. The key difference between a quantum Turing machine and a classical Turing machine is that the quantum Turing machine uses quantum state of its head on the tape toccata the logic state. A superposition is accepted in any lattive site so that the Boolean state is satisfied.

There has been a lot of speculation on whether the quantum superposition’s on the tape has some impacts on the principle statements on every computing model. The church Turing hypothesis implies that each functions which can naturally be considered computable can be calculated using a universal Turing machine. When this principle is expressed as a physical principle, any finitely realized physical system is perfectly simulated by a computing machine which is operated by a finite means. The quantum Turing machine satisfies this principle including the initial hypothesis. This is not the case for the second version because of its finite state.

The step operator is the key principle in the operation because the quantum Turing machine follows some specific steps based on the unitary step operator which must be satisfied by the requirements. The requirements should be local so that the interaction between the tape and the head is achieved in a time interval. The head can shift to the left, right pr can interact while it is static. Any displacement that occurs should be described. In addition, the lattice sites’ periodicity should be considered. The whole concept is represented by the following formula

Figure 6: Non-Abelian gauge formula

The first statement implies that an alteration in one step specifically interact with the bits at the head. Any contribution is not accepted because of the S precondition. The lather equation implies that the elementary elements of interaction are equivalent for any input. This is only applicable when the lattice sites are periodic.

Connection between the quantum circuit and the quantum Turing machine is achieved in such a manner that enables two-way transformation. The quantitative statements are derived from the respective simulation originating from the complexity theory.

There are some measurements that exist. The elementary gates in a quantum circuit are given by the size of the circuit while the depth is the highest lent of the projected path from the input to the output register. The par of the interaction circuit provides the number of elementary gates in a given quantum circuit. This will describe the split of the circuit with disjoint pair of input so that all the output is realized from the same site.

The measures are used to give the properties of a quantum circuit. This is more applicable when optimization is needed. When you need a computer with high speed, the communication and the depth cost should be minimal. The speed at which the information is propagated should be very high. This implies that shorter and fewer wires gives a very fast computer. The size should be minimized so that the circuit is optimized.

Figure 7: Circuit optimization

A simulation of a quantum Turing machine by a quantum circuit increases exponentially or polynomials bounded. To differentiate the two, simulation is introduced.

## Space-bounded quantum computation

Unlike the previous classes of quantum computation which are based on the abstraction, this type of quantum computation is more efficient since its performance is based on polynomial time. Quantum complexity classes are also classified according to the space or time. To effectively give a good representation, a variety of computational model is needed. The most effective choice is the combination of classical Turing machine and quantum circuits.

A quantum Turing machine contains read-only classical input tape and a quantum tape that has unlimited qubits which references to zero-state. There are three tape heads which reads the tape facilitating the performance of quantum operation which correspond to qubits. Unit step of quantum Turing machine can include a mere move by the classical sections of quantum and machined operation in the quantum tape.

The operation time of a QTM is represents the mere Turing machine. The number of squares and the classical tape used by that particular machine gives the space of the machine. As I mentioned earlier, the quantum hold the force in close range while the tape releases them. It is easy for a quantum machine to override the force exerted by the head since the point of exertion is very close to the pin joint while the machine pull has longer moment time.

Polynomial 6-power should be estimated by efficiency and power efficient. Assume we want to find out the machine rating in a computer system. The QTM which has a speed 3 is supposed to accelerate from the speed.

There are several factors that considered in implementing the quantum computational technology. These include cost, practicability and speed. For instance, it is cheaper to install QTM in a mini computer but it is impractical when it reaches some particular speed. When constructing a powerful computer system, classical classes are the most appropriate. In addition, classical classes are faster than QTM. On the other hand, it is very simple to customize a classical classes compared to a QTM. Basically, the requirements of the computer to be constructed major determines the economies to be derived from production of mass components.

## Conclusion

The link between the two concepts is formulated from the principle that any gate is specifically recognized by a quantum Turing machine and each quantum Turing machine can be represented in simulation using a quantum network. To realize an in-depth understanding of this concept, a close relationship between resources required by the network simulation and quantum Turing machine using step operation S or T matrices is tested.

The basis of this is the fact that the long time calculation is the key thing in all the cryptographic security. It has been proven that getting the factor for a big prime number is simple when calculated using the quantum versions. I try to make the whole idea understandable by giving a mathematical description of quantum gates. Though it is not possible to emulate a quantum gate using a classical gate, an arbitrary estimation can be done.

The common classical computer carries out computation using logic circuits. Basically, computation is where the output is generated from the input. Both the input and the output are abstract representation represented using quanta or bits. This is the least possible quantities of non-probabilistic information.

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List of Diagrams

Figure 1: A Circuit of several combinations of gates 5

Figure 2: Eigenstates’ principle computation 6

Figure 3: Truth table 6

Figure 4: Permutation truth table 7

Figure 5: Scattering matrix formula 7

Figure 6: Non-Abelian gauge formula 9

Figure 7: Circuit optimization 10