# Quantum Genetic Algorithms for Computer Scientists

## Abstract

**:**

## 1. Introduction

## 2. What is Quantum Computing?

#### 2.1. Quantum Information

#### 2.2. Quantum Gates

^{n}binary numbers x with equal probability.

**O**.

#### 2.3. Quantum Algorithms and Quantum Circuits

#### 2.4. Quantum Computing in Practice

#### 2.4.1. Q-Circuit Simulators

#### 2.4.2. Q-Programming Languages

#### 2.4.3. Simulated Q-Computer

#### 2.4.4. Do-It-Yourself: Quantum Computing with Python

## 3. Quantum Computing and Quantum Evolutionary Algorithms

#### 3.1. Quantum Genetic Operators

#### 3.1.1. Qubit (Interference) Rotation Gate

#### 3.1.2. Quantum Mutation (Inversion) Gate

#### 3.1.3. Quantum Mutation (Insertion) Gate

#### 3.1.4. Quantum Crossover (Classical) Gate

#### 3.1.5. Quantum Crossover (Interference) Gate

#### 3.2. A Canonical Classification of Quantum Evolutionary Algorithms

## 4. Towards True Quantum Evolutionary Algorithms

**F**which evaluates the fitness of individuals (Figure 9). In 2008 [63], a similar idea is also applied to other version of a true quantum evolutionary algorithm which was termed as Quantum Genetic Optimization Algorithm (QGOA).

**F**is applied, RQGA searches for the maximum fitness value based on the Grover’s search algorithm [65]. This is one of the most popular quantum algorithms oriented to search in an unstructured database. Without going into details on this algorithm, RQGA performs the following two steps. First, given a register with a set of fitness values an oracle

**O**is designed to mark all the kets:

**G**. This operator aims at finding the marked states, i.e., $f\left({|fitnes{s}_{x}\rangle}_{i}\right)=1$:

## 5. Simulation Experiments

## 6. Results

## 7. Future Directions

^{13}C-iodotrifluroethylene and a spectrometer at 306 K. Furthermore, in the future we will achieve the physical realization [75] of 50–100 qubits and therefore the hardware to build a quantum computer. At present, it is possible to experience with a 5 qubits quantum computer via a cloud computing platform and run experiments, such is the case of IBM’s quantum processor [76]. However, today, even when QGAs are inspired by principles of quantum computing they are eventually executed on a classical computer. In our opinion this scenario will change once we achieve the design and implementation of a RQGA on a quantum computer. Consequently, this will accelerate research on quantum evolutionary algorithms designing higher-order QGA [77], QGAs with entanglement [78] or hybridizing QGAs with the quantum version of optimization algorithms, e.g., artificial bee colony [79], cuckoo search [58], etc. The leap from emulation to the actual implementation of quantum algorithms is a big step that should be accompanied by a good training of future computer scientists on the fundamentals of quantum mechanics [80]. The result of these changes will be a dramatic increase of the applications of quantum evolutionary algorithms, either on specific problems, e.g., the analysis of cancer microarray data [81] or in classical engineering optimization problems [82]. Moreover, it is also expected an increase in the number of applications in the field of Artificial Intelligence, e.g., the N-Queens problem [83], and even in the field of Artificial Life [84]. In the future quantum computing may also have a profound influence on Darwinism [85], transforming our vision of life on Earth. However, until these advancements occur we must exploit the advantages that provides us the current software and hardware technologies, e.g., designing more efficient QGAs with the support of CUDA (from NVIDIA) platform and the Matlab Graphic Processing Unit (GPU) library [86].

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Quantum circuit including (from left to right) a quantum register, input state $|\psi (0)\rangle $, information processing step

**U(t)**, and the output as a result of measuring or observing the state of each qubit.

**Figure 7.**DNA modeled as ${|\psi \rangle}^{i}$ and gene sequence $|\psi \rangle $ represented as qubit in a superposition state.

**Figure 11.**Representative performance graph obtained in the benchmark function optimization experiments conducted with (

**a**) Quantum genetic algorithm (QGA); (

**b**) Hybrid genetic algorithm (HGA) with mutation; (

**c**) Hybrid genetic algorithm (HGA) without mutation; (

**d**) Simple genetic algorithm (SGA).

**Figure 12.**Notched Box-and-Whisker Plots, one boxplot per evolutionary algorithm showing the medians (notches) and means (crosses) of the fitness [70]. Squares indicate outliers (unusual fitness values).

**Figure 13.**Performance graph obtained in the benchmark function optimization experiment conducted with RQGA, a true quantum genetic algorithm based on Grover’s algorithm.

Step | |
---|---|

1 | Randomly initialize a population P(0) |

2 | Evaluate P(0) |

3 | while (not termination condition) do |

4 | begin |

5 | t ← t + 1 |

6 | Selection of parents from population P(t) |

7 | Crossover |

8 | Mutation |

9 | Evaluate P(t) |

10 | end |

Q1 | Q2 | Q1 | * Q2 |
---|---|---|---|

0 | 0 | 0 | 0 |

0 | 1 | 0 | 1 |

1 | 0 | 1 | 1 |

1 | 1 | 1 | 0 |

*****Observed after performing the measurement.

Q1 | Q2 | Q3 | * Q3 |
---|---|---|---|

0 | 0 | 0 | 0 |

0 | 0 | 1 | 1 |

0 | 1 | 0 | 0 |

0 | 1 | 1 | 1 |

1 | 0 | 0 | 0 |

1 | 0 | 1 | 1 |

1 | 1 | 0 | 1 |

1 | 1 | 1 | 0 |

*****Observed after performing the measurement.

Step | |
---|---|

1 | Prepare an input state |

2 | Apply quantum parallelism |

3 | Performs quantum information processing |

4 | Use interference to exploit the parallelism |

5 | Make a measure |

Quantum Circuit Example | Symbol | Input → Output * |
---|---|---|

$|1\rangle \to 1,1.0$ | ||

$|1\rangle \to 0,1.0$ | ||

$|0\rangle \to 1,1.0$ | ||

$|0\rangle \to 0,1.0$ | ||

$|0\rangle \to \{\begin{array}{c}0\text{\hspace{0.17em}},\text{\hspace{0.17em}}0.5\\ 1\text{\hspace{0.17em}},\text{\hspace{0.17em}}0.5\end{array}$ | ||

$\begin{array}{l}|0\rangle \to |1\rangle \\ |1\rangle \to |0\rangle \end{array}$ | ||

$|01\rangle \to |11\rangle =3,1.0$ | ||

$|110\rangle \to |011\rangle =3,1.0$ |

*****Output value and probability of being observed after performing the measurement.

Step | Quantum Computing | Classical Computing |
---|---|---|

1 | Initialize a quantum population Q(0) | |

2 | Make P(0), measure of every individual Q(0) → P(0) | |

3 | Evaluate P(0) | |

4 | while (not termination condition) do | |

5 | begin | |

6 | t ← t + 1 | |

7 | Update Q(t) applying Q-gates: Q(t + 1) = U(t).Q(t) | |

8 | Make P(t), measure of every individual Q(t) → P(t) | |

9 | Evaluate P(t) | |

10 | end |

${\mathit{x}}_{\mathit{j}}\text{\hspace{0.17em}\hspace{1em}}{\mathit{b}}_{\mathit{j}}$ | $\mathit{f}({\mathit{x}}_{\mathit{j}})\ge \text{\hspace{0.17em}}\mathit{f}({\mathit{b}}_{\mathit{j}})$ | $\mathbf{\Delta}{\mathit{\theta}}_{\mathit{j}}$ | $\mathit{s}\mathit{g}\left({\mathit{\alpha}}_{\mathit{j}}\text{\hspace{0.17em}}{\mathit{\beta}}_{\mathit{j}}\right)$ | |||
---|---|---|---|---|---|---|

${\alpha}_{j}{\beta}_{j}>0$ | ${\alpha}_{j}{\beta}_{j}<0$ | ${\alpha}_{j}=0$ | ${\beta}_{j}=0$ | |||

0 0 | False | 0 | - | - | - | - |

0 0 | True | 0 | - | - | - | - |

0 1 | False | $\delta $ | +1 | −1 | 0 | $\pm $1 |

0 1 | True | $\delta $ | −1 | +1 | $\pm $1 | 0 |

1 0 | False | $\delta $ | −1 | +1 | $\pm $1 | 0 |

1 0 | True | $\delta $ | +1 | −1 | 0 | $\pm $1 |

1 1 | False | 0 | - | - | - | - |

1 1 | True | 0 | - | - | - | - |

Step | Quantum Computing | Classical Computing |
---|---|---|

1 | Initialize a quantum population Q(0) | |

2 | Make P(0), measure of every individual Q(0) $\to $ P(0) | |

3 | Evaluate P(0) | |

4 | while (not termination condition) do | |

5 | begin | |

6 | t $\leftarrow $ t + 1 | |

7 | Rotation Q-gate | |

8 | Mutation Q-gate | |

9 | Make a measure Q(t) $\to $ P(t) | |

10 | Evaluate P(t) | |

11 | end |

Step | Quantum Computing | Classical Computing |
---|---|---|

1 | Initialize a quantum population Q(0) | |

2 | Make P(0), measure of every individual Q(0) $\to $ P(0) | |

3 | Evaluate P(0) | |

4 | while (not termination condition) do | |

5 | begin | |

6 | t $\leftarrow $ t + 1 | |

7 | Rotation Q-gate | |

8 | Crossover operator | |

9 | Mutation Q-gate | |

10 | Make a measure Q(t) $\to $ P(t) | |

11 | Evaluate P(t) | |

12 | end |

Step | Quantum Computing |
---|---|

1 | Initialize a superposition of all possible chromosomes |

2 | Evaluates fitness with operator F |

3 | Apply Grover’s algorithm |

4 | Ask to the oracle O |

5 | Apply Grover’s diffusion operator G |

6 | Make a measure |

Algorithm | Sample Size | Mean Rank |
---|---|---|

SGA | 50 | 325.5 |

HGA1 | 50 | 119.12 |

HGA2 | 50 | 111.52 |

HGA3 | 50 | 202.98 |

QGA1 | 50 | 151.3 |

QGA2 | 50 | 184.38 |

QGA3 | 50 | 133.7 |

State | ${|\mathit{x}\rangle}_{\mathit{i}}$ | ${|\mathit{f}\mathit{i}\mathit{t}\mathit{n}\mathit{e}\mathit{s}{\mathit{s}}_{\mathit{x}}\rangle}_{\mathit{i}}$ |
---|---|---|

$|0\rangle =\dots $ | ${|0000\rangle}_{0}$ | ${|250\rangle}_{0}$ |

$|1\rangle =\dots $ | ${|0001\rangle}_{1}$ | ${|140\rangle}_{1}$ |

$|2\rangle =\dots $ | ${|0010\rangle}_{2}$ | ${|103\rangle}_{2}$ |

$|3\rangle =\dots $ | ${|0011\rangle}_{3}$ | ${|93\rangle}_{3}$ |

$|4\rangle =\dots $ | ${|0100\rangle}_{4}$ | ${|80\rangle}_{4}$ |

$|5\rangle =\dots $ | ${|0101\rangle}_{5}$ | ${|0\rangle}_{5}$ |

$|6\rangle =\dots $ | ${|0110\rangle}_{6}$ | ${|58\rangle}_{6}$ |

$|7\rangle =\dots $ | ${|0111\rangle}_{7}$ | ${|75\rangle}_{7}$ |

$|8\rangle =\dots $ | ${|1000\rangle}_{8}$ | ${|100\rangle}_{8}$ |

$|9\rangle =\dots $ | ${|1001\rangle}_{9}$ | ${|165\rangle}_{9}$ |

$|10\rangle =\dots $ | ${|1010\rangle}_{10}$ | ${|343\rangle}_{10}$ |

$|11\rangle =\dots $ | ${|1011\rangle}_{11}$ | ${|599\rangle}_{11}$ |

$|12\rangle =\dots $ | ${|1100\rangle}_{12}$ | ${|478\rangle}_{12}$ |

$|13\rangle =\dots $ | ${|1101\rangle}_{13}$ | ${|330\rangle}_{13}$ |

$|14\rangle =\dots $ | ${|1110\rangle}_{14}$ | ${|300\rangle}_{14}$ |

$|15\rangle =\dots $ | ${|1111\rangle}_{15}$ | ${|377\rangle}_{15}$ |

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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Lahoz-Beltra, R.
Quantum Genetic Algorithms for Computer Scientists. *Computers* **2016**, *5*, 24.
https://doi.org/10.3390/computers5040024

**AMA Style**

Lahoz-Beltra R.
Quantum Genetic Algorithms for Computer Scientists. *Computers*. 2016; 5(4):24.
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Lahoz-Beltra, Rafael.
2016. "Quantum Genetic Algorithms for Computer Scientists" *Computers* 5, no. 4: 24.
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