_{1}

^{*}

In this article the concept of phase encoding/decoding is used to analyze and formalize a simple quantum algorithm—the Deutsch’s algorithm. The algorithm is formalized in two different ways through an analysis, based on phase encoding/decoding, carried out by the formalized elementary operators developed by the author of this article. Concrete examples of different possible realizations of the formalized with Raychev’s operators Deutsch’s algorithms are offered.

Quantum computers can be useful in solving many important and complex issues such as solving many fundamental and complex problems such as integer factorization [

Both Identity_{1} and Negation_{1} classes are analogous to the classic operators and therefore the formalization of single qubit operators with these classes, acting as main operators, ensures means for primitive operators operation, set out in the classical concepts for calculations. The main goal is the parts of the state phase to be separated from those of the amplitude in such a way that should be set out as the key models at the interference, generated by the operators. This interference is a key to the quantum calculations and the quantum algorithms development. In addition, the separation permits the strict characterization of the consequences from a change of the phase as the binary information encoding in a phase space and thus permits the characterization in terms of Boolean functions. This report will consider the necessary and sufficient conditions for combining operators from Identity_{1} and Negation_{1} for formation of unitary operators and provision of an abstraction on the relative weights of the base operators. The logical formalization of the main operators will be addressed as regards to their phases and phase changes, which they carry out, on the states, to which they are applied. Whereas this approach to elementary controlled operators is elementary, the formalized system for designing of quantum circuits algorithmic models offers a second approach, which unites the elementary controlled operations with another important aspect of multitude of quantum algorithms, the Oracle operators. In the quantum algorithms design a standard technique is the use of an Oracle operator to enact a phase-kickback operation. Herein an arbitrary, probably irreversible function

・ Superposition (step 2)

・ Quantum parallelism (step 3)

・ Interference (step 4)

Deutsch’s algorithm has been explored many times [

The Deutsch’s algorithm points out the basic principles of the quantum algorithms, while requiring a small and relatively easy to understand circuit. Here it will be studied anew through the prism of the phase encoding and decoding and will be given more common characteristic of the algorithm. First, let’s recall the algorithm and its standard presentation. The problem of Deutsch is to determine if a single bit boolean function is in BAL or CONST at given quantum external source of information, black box, to that function. It requires a single query of the source, where a classical approach would require two.

Particular attention will be paid to three elements of the Deutsch’s algorithm, as it is presented in

1) The Hadamard operator H, which is applied to

2) The composition of

3) The selection of an initial state of the working qubit plays a role in the result of the algorithm. The conditions for the initial state of the working qubits can be expressed and explained in respect of the encoding/de- coding results.

To examine the conclusions from these observations isproposed the three-part decomposition.

The algorithm is divided into three logical steps: the application of

If

If

Similar formulation of V allows the phase changes to be introduced by the Oracle operator in addition to the conditional basis changes. By Theorem 1 VC is α degree

and

In this analysis the two operators of O are effectively indexed via the function f. In a similar way, the amplitude parameter of O can be indexed. If

Then

The final step of the algorithm generates interference between the operator A and B. If

If the general formulas for the probability amplitude of each basis state shown in Equation (2) are given, the Deutsch’s algorithm can be generalized in the context of the formalized Raychev’s operators. The general character of the output data will be maintained in such way that the final state should leave

Until now the decoding was limited to the conditions under which the two operators are combined in order to form a certain basis operator. The requirements for the parameters are shown in

This is done within the limits of the formalized Raychev’s operators. More common structure of the decoding operators occurs when calling the main matrix formulation of the operators.

When B is a decoder of A, can be found the interference pattern, generated by B, from

N | −N | X | −X | |
---|---|---|---|---|

000 | 000 | 011 | 110 | 101 |

001 | 010 | 001 | 100 | 111 |

010 | 001 | 010 | 111 | 100 |

011 | 011 | 000 | 101 | 110 |

100 | 110 | 101 | 000 | 011 |

101 | 100 | 111 | 010 | 001 |

110 | 111 | 100 | 001 | 010 |

111 | 101 | 110 | 011 | 000 |

I | Z | N | −X | |
---|---|---|---|---|

000 | 001 | 100 | 000 | 110 |

001 | 000 | 101 | 010 | 100 |

010 | 011 | 110 | 001 | 111 |

011 | 010 | 111 | 011 | 101 |

100 | 100 | 001 | 110 | 000 |

101 | 101 | 000 | 100 | 010 |

110 | 110 | 011 | 111 | 001 |

111 | 111 | 010 | 101 | 011 |

If an attention is paid to the Oracle operator and how f controls its impact on the development of the system. If f is a constant function, then

In Equation (4) is seen that the encoding, performed by the Oracle operator, in each term of the sum of the probability amplitudes is the same. Thus, no interference is generated by the Oracle operator and the final state of the circuit is reduces down to the decoding interaction between A and B. The intervention carried out relative

to the states

the state in a superposition of

If the goal is the probability amplitudes of

clear that

terms of the probability amplitude sums in

will occur; the probability amplitudes of

ference can be guaranteed regardless of the value of

to be defined not as the

Problem: Upon given Oracle like operator

Preconditions:

1) To be defined В and А such that

2)

Result: If the function f is balanced

For Boolean function

Theorem 1 If operator

Proof. If it is assumed that A Î The set of the identity formalized Raychev’s operators and B Î The set of the negation formalized Raychev’s operators, therefore for each n qubit basis

Theorem 2 If operator

Proof. If it is assumed that

Theorem 1 leads to general means for constructing of

Theorem 3 If V is an n qubit

an amplitude parameter

Proof. The proof follows from theorems 1 and 2, by noting that when

Theorem 3 is important because, it captures a common occurrence in the quantum algorithms: setting the target bit of an Oracle operator to a superposition and then applying the oracle operator. In the formalized system for designing of algorithmic models for quantum circuits this can be addressed in the context of an α degree

The composition of random formalized operators with main operators can be expressed in terms of the transformations of parameters.

Formal prerequisite 1 If the operator

Proof. If A is a base operator. Then

If the consideration is limited only to the original formulation of the problem, i.e.

target big of

ators of the composite operator

and

Remark The generalization of the Deutsch’s algorithm given in

Problem: Upon given Oracle operator

Preconditions:

1) To be defined В and А such that

2)

3) For operators

Result: If the function f is balanced

If

Two generalized algorithms for solving the Deutsch’s problem are developed, by re-examining anew the standard formulation of the Deutsch’s algorithm. In both cases are used the specific phase encoding and decoding ideas, realized by the developed by the author of the report formalized operators. Both algorithms set the initial state to be equal to a superposition of all possible two qubit states and in this way set the working bit

an own state of the Oracle operator. Furthermore, the use of Hadamard-like gates with amplitude parameters

Finally, the formalizations of the Deutsch’s algorithm, given in

The first example deals with a slight variation of the traditional Deutsch’s algorithm.

Let operator

preparation of target qubit are equivalent to traditional Deutsch’s algorithm. From Equation (2) can determine the final state of the system, if the input data are given for the traditional algorithm

When f is constant, then the phase encoding executed by Oracle operator is the same in both terms of any amount. Without loss of generality, let us assume that

When f is balanced then

that for

If we consider the example of the algorithm represented in

input

Without loss of generality, let us assume that for a balanced

In the case where f is a constant, there is the same overall result as the standard formulation of the Deutch’s algorithm; the Oracle operator makes no interference between the subspace and the calculation is reduced to the decoding effect B has on A.

The first version of formalized algorithm uses phase change operations where the second version demonstrates how different phases might be used in the context of the conventional Deutsch’s algorithm.

The formalized Raychev’s operators separate the parts of phase and amplitude and allow for expression of amplitude in terms of probabilities as opposed to more general probability amplitudes. This formalization is further improved by the characterization of classes with defined relations and incarnation of formalized logic parameter γ on the global phase. Under the needed and sufficient conditions to construct a unitary, in this case orthogonal operators are incorporated into the formalization.

NikolayRaychev, (2015) Formalized Operators with Phase Encoding. Journal of Quantum Information Science,05,114-126. doi: 10.4236/jqis.2015.53014