Mathematical Problems in Engineering

Volume 2018, Article ID 8614073, 8 pages

https://doi.org/10.1155/2018/8614073

## Quantum Genetic Algorithm Based on Qutrits and Its Application

Vasyl Stefanyk Precarpathian National University, 57 Shevchenko Str., Ivano-Frankivsk 76018, Ukraine

Correspondence should be addressed to Valerii Tkachuk; moc.liamg@0vkuhcakt

Received 29 November 2017; Accepted 11 March 2018; Published 29 April 2018

Academic Editor: Fazal M. Mahomed

Copyright © 2018 Valerii Tkachuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Typical approaches to designing quantum genetic algorithms are based on a concept of a qubit, a two-level quantum system. But many-valued quantum logic is more perspective from the point of view of the computational power. This paper proposes a quantum genetic algorithm based on a three-level quantum system in order to accelerate evolutionary process. Simulation using a set of standard test functions proves that the given algorithm is more effective and precise than the conventional quantum genetic algorithm.

#### 1. Introduction

Evolutionary algorithms are one of the most widely used methods for solving the global optimization or search problem [1]. In its classical implementation, the smallest unit of information is a bit, a structure which may be in one of the following states: 0 and 1. It is important that a classical bit does not allow any intermediate states.

Quantum genetic algorithm (QGA) is a new evolutionary algorithm based on a combination of quantum computation and conventional genetic algorithms technology [2, 3]. This algorithm can be applied to the same set of problems the conventional genetic algorithm is used for, but it allows significantly accelerating the evolutionary process through quantum parallelization and entanglement of the quantum state. Probabilistic mechanism of the quantum computations combined with the evolutionary algorithm provides a global search for a solution with a rapid convergence and a small value of population. These algorithms have demonstrated their effectiveness for solving combinatorial and functional optimization problems, mechanical engineering optimization problems, image processing, and many others [4–8].

QGA main concepts were first proposed by Narayanan and Moore [9]. The basic unit of information used in a quantum computations is a qubit, a quantum system, which may be in the basis state or the basis state. Quantum nature of the qubit lies in the superposition principle, under which the qubit may be in any state which is a linear combination of basis states:with a normalization constraint:

As a consequence of the superposition principle, qubit state space is incomparably greater than state space of a classical bit. Information, stored in and amplitudes, is the quantum part of the information. Practically, та are numbers that specify the probability amplitudes of a qubit being in the states and , respectively.

The outcome of quantum measurement is a qubit in one of the basis states. However, it is important that the result of the measurement is not deterministic, like it is in the classical calculation, but probabilistic.

QGA uses a matrix representation for practical implementation:

Qubit (1) state in this representation can be defined as

A structured set containing qubits represents a quantum chromosome. State vector of such chromosome is a superposition state of basis states of the register , :

All the information about a qubits system is represented by state vector . The only thing that can be done to such a system is transforming the initial state vector into a new state by applying a quantum gate. So quantum genetic algorithm is a state transition process from some starting state to an ending state using a quantum gate algorithm. The information about the problem solution is only contained by the vector of the system ending state .

The main disadvantage appearing during quantum state rotation is the need to use a table to look up the rotating angle, which limits the universality of the search. Fixed rotating angle has a negative impact on the search speed, which is the reason for QGA sometimes being implemented as an adaptive process for determining the rotating angle. To enhance the local searching ability and to get out of the local optimal solutions, GDA can be extended with conventional operators traditionally used in the classical genetic algorithm, such as quantum mutation operation and quantum disaster operation [10].

#### 2. Quantum Genetic Algorithm Based on Qutrits (QGA_3)

Qudit is a structure with several, more than two, states, which can also be used to encode a chromosome. A qudit is basically a unit of quantum information, which may be in any of states or in any superposition of those.

Ternary quantum logic is the simplest type of the many-valued logic. The basic unit of memory is called a qutrit. It has three basis states, , , and . The state of the qutrit can be represented as a superposition in the form of a linear combination:with a normalization constraint:

Matrix representation:

For the practical implementation, qutrit (6) state can be represented as

A system containing qutrits has basis states (as opposed to states for binary logic). When switching from binary to many-valued logic, we get an exponential increase in the number of basis states, which leads to an increase in the algorithm performance for the same search accuracy.

Quantum chromosome length is determined by the desired search accuracy , search area , and the number of quantum system basis states:where is a number of quantum system basis states.

With , search accuracy , and search area , the length of the quantum chromosome has to be at least 21 qubits, while with only 14 qutrits are required.

##### 2.1. Qutrit Encoding

Matrix representation of a quantum chromosome is a structure which consists of qutrits.

⋯ | ⋯ | ⋯ | ||||||

⋯ | ⋯ | ⋯ | ||||||

⋯ | ⋯ | ⋯ |

Here determine the state of th qutrit, аnd qutrits determine one individual in the population. The initial state of the qutrits holds no information about the system state, so the easiest way to initialize the base population is by setting all probability state amplitudes to be equal to one another [2]. This means that at the end of the initialization each qutrit is in the state

##### 2.2. Observation of Genes

Classical information about the problem’s solution is located in the ending state vector and can be retrieved as a result of quantum observation. Qutrit in one of the basis states, which is obtained as a result of quantum observation, is a classical representation of a quantum chromosome.

Based on the approach described in [11], the following pseudocode for quantum -qutrit chromosome state observation can be suggested (Algorithm 1). The product of its work provides the ground state (0, 1 or 2) with a probability of , and , respectively.