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To enhance the optimization ability of particle swarm algorithm, a novel quantum-inspired particle swarm optimization algorithm is proposed. In this method, the particles are encoded by the probability amplitudes of the basic states of the multi-qubits system. The rotation angles of multi-qubits are determined based on the local optimum particle and the global optimal particle, and the multi-qubits rotation gates are employed to update the particles. At each of iteration, updating any qubit can lead to updating all probability amplitudes of the corresponding particle. The experimental results of some benchmark functions optimization show that, although its single step iteration consumes long time, the optimization ability of the proposed method is significantly higher than other similar algorithms.

In 1999, Dr. Eberhart and Dr. Kennedy proposed particle Swarm Optimization (particle swarm optimization, PSO) [

There is M particles in the n-dimensional space. For the

where

For convenience of description, Equation (1) can be rewritten as follows.

where

To make the PSO convergence, all particles must approximation

What is a qubit? Just as a classical bit has a state―either 0 or 1―a qubit also has a state. Two possible states for a qubit are the state

Notation like

where

In the quantum computation, the logic function can be realized by applying a series of unitary transform to the qubit states, which the effect of the unitary transform is equal to that of the logic gate. Therefore, the quantum services with the logic transformations in a certain interval are called the quantum gates, which are the basis of performing the quantum computation. A single qubit rotation gate can be defined as

Let the quantum state

that

Let the matrix

where

In general, for an n-qubits system, there are

where

Let

It is clear from the above equations that, in an n-qubits system, any one of the ground state probability amplitude is a function of n-qubits phase^{n} probability amplitudes.

In our works, the n-qubits rotation gate is employed to update the probability amplitudes. According to the principles of quantum computing, the tensor product of n single-qubit rotation gate

where

Taking

It is clear that

where

In this paper, the particles are encoded by multi-qubits probability amplitudes. Let N denote the number of particles,

For an n-bits quantum system, there are

First, generating randomly N n-dimensional phase vector

where

Let

In this paper, the multi-bit quantum rotation gates are employed to update particles. Let the phase vector of the

global optimal particle be^{th} particle be

the itself optimum the phase vector be

From Equation (11), it is clear that, once

Step 1. Set

Step 2. Set

Step 3. Determine the value of the rotation angle, where the sgn donates the symbolic function.

If

If

If

If

Step 4. Compute the rotation angles, and update all particles according to the following equation,

Step 5. If

Suppose that, N denote the number of particles,

1) Initialize the particles swarm

According to Equation (15) to determine the number of qubits n, according to Equation (16) initialize phase of each particle, according to Equation (11) to calculate the probability amplitude of

Initialization phase update step

2) Calculation of the objective function value

Set the j-dimensional variable range be

where

Calculate the objective function values of all particles. Let the i^{th} particle phase be

objective function value is

tive function value be

3) Update the particle position

For each particle

4) Update the global optimal solution

Let the optimal particle phase be

5) Examine termination conditions

If

In this study, the 20 standard test functions are employed to verify the optimization ability of MQPAPSO, and compare with the general particle swarm optimization (PSO) [

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

The dimension of all test functions is set to

For SFLA, according to Ref. [

where the first number denotes the number of sub-group and the second number denotes the number of frog in sub-group. For each of combination, the SFLA is independent run 30 times, and the average optimization result over 30 runs and the average time of a single iteration are recorded. In these six groups, the best optimization results and the corresponding average time of a single iteration are regarded as a comparison index.

For PSO, according to Ref. [

Experiments conducted using Matlab R2009a. Taking

For the function

For four algorithms, the ratios of the average time of a single iteration are shown in

From

f_{i} | MQPAPSO | QDPSO | PSO | SFLA | ||||
---|---|---|---|---|---|---|---|---|

D = 50 | D = 100 | D = 50 | D = 100 | D = 50 | D = 100 | D = 50 | D = 100 | |

f_{1} | 0.0186 | 0.0290 | 0.0011 | 0.0019 | 0.0009 | 0.0016 | 0.0014 | 0.0020 |

f_{2} | 0.0187 | 0.0292 | 0.0012 | 0.0020 | 0.0012 | 0.0016 | 0.0017 | 0.0025 |

f_{3} | 0.0248 | 0.0428 | 0.0064 | 0.0127 | 0.0099 | 0.0227 | 0.0068 | 0.0168 |

f_{4} | 0.0188 | 0.0296 | 0.0011 | 0.0019 | 0.0009 | 0.0016 | 0.0014 | 0.0024 |

f_{5} | 0.0230 | 0.0397 | 0.0049 | 0.0095 | 0.0016 | 0.0025 | 0.0022 | 0.0043 |

f_{6} | 0.0235 | 0.0387 | 0.0018 | 0.0031 | 0.0028 | 0.0048 | 0.0019 | 0.0032 |

f_{7} | 0.0187 | 0.0291 | 0.0013 | 0.0021 | 0.0016 | 0.0022 | 0.0014 | 0.0024 |

f_{8} | 0.0191 | 0.0295 | 0.0015 | 0.0023 | 0.0019 | 0.0028 | 0.0024 | 0.0036 |

f_{9} | 0.0234 | 0.0382 | 0.0016 | 0.0024 | 0.0019 | 0.0028 | 0.0017 | 0.0027 |

f_{10} | 0.0193 | 0.0301 | 0.0019 | 0.0031 | 0.0028 | 0.0048 | 0.0017 | 0.0028 |

f_{11} | 0.0234 | 0.0383 | 0.0016 | 0.0024 | 0.0016 | 0.0025 | 0.0017 | 0.0027 |

f_{12} | 0.0262 | 0.0441 | 0.0096 | 0.0173 | 0.0054 | 0.0089 | 0.0033 | 0.0070 |

f_{13} | 0.0193 | 0.0298 | 0.0020 | 0.0030 | 0.0025 | 0.0041 | 0.0017 | 0.0030 |

f_{14} | 0.0233 | 0.0372 | 0.0048 | 0.0089 | 0.0028 | 0.0044 | 0.0024 | 0.0033 |

f_{15} | 0.0256 | 0.0449 | 0.0031 | 0.0057 | 0.0051 | 0.0096 | 0.0038 | 0.0060 |

f_{16} | 0.0248 | 0.0418 | 0.0065 | 0.0124 | 0.0028 | 0.0048 | 0.0027 | 0.0043 |

f_{17} | 0.0378 | 0.0706 | 0.0246 | 0.0486 | 0.1116 | 0.2192 | 0.0292 | 0.0651 |

f_{18} | 0.0234 | 0.0382 | 0.0017 | 0.0024 | 0.0019 | 0.0025 | 0.0022 | 0.0028 |

f_{19} | 0.0206 | 0.0314 | 0.0028 | 0.0037 | 0.0038 | 0.0054 | 0.0033 | 0.0041 |

f_{20} | 0.0212 | 0.0320 | 0.0028 | 0.0039 | 0.0041 | 0.0057 | 0.0043 | 0.0052 |

f_{i} | MQPAPSO | QDPSO | PSO | SFLA | |||
---|---|---|---|---|---|---|---|

G = 100 | G = 100 | G = 1000 | G = 100 | G = 1000 | G = 100 | G = 1000 | |

f_{1} | 1.9E−08 | 1.5E+03 | 3.4E−05 | 3.4E+03 | 6.0E−05 | 8.5E+02 | 0.00108 |

f_{2} | 1.3E−04 | 9.4E+10 | 33.1953 | 3.8E+15 | 1.3E+02 | 2.8E+02 | 2.6E+02 |

f_{3} | 3.7E−09 | 3.9E+04 | 1.1E+04 | 6.7E+04 | 1.6E+04 | 6.3E+03 | 2.5E+03 |

f_{4} | 0.00101 | 36.9364 | 10.2029 | 61.9675 | 54.6625 | 12.1258 | 9.71406 |

f_{5} | 73.2154 | 1.2E+08 | 1.3E+02 | 2.7E+08 | 2.0E+02 | 1.1E+07 | 4.8E+02 |

f_{6} | 4.1E−11 | 7.2E+07 | 2.1E+02 | 3.7E+08 | 1.5E+05 | 1.2E+04 | 2.3E−09 |

f_{7} | 7.9E−06 | 1.9E+03 | 2.9E+02 | 3.3E+03 | 3.7E+02 | 1.4E+03 | 1.0E+03 |

f_{8} | 3.3E−05 | 21.1629 | 20.5964 | 21.2778 | 21.1744 | 17.0524 | 15.7169 |

f_{9} | 4.9E−10 | 11.1857 | 0.00352 | 23.6757 | 0.03275 | 2.00564 | 0.01209 |

f_{10} | 18.5824 | 2.7E+04 | 10.5743 | 6.5E+04 | 23.4260 | 1.8E+03 | 12.7798 |

f_{11} | 2.8E+04 | 1.6E+06 | 8.4E+04 | 5.1E+06 | 4.2E+05 | 9.7E+05 | 2.4E+04 |

f_{12} | 0.18150 | 2.9E+07 | 0.28276 | 6.5E+07 | 1.65872 | 1.1E+04 | 17.3770 |

f_{13} | 7.6E−07 | 4.2E+03 | 0.00231 | 1.0E+04 | 4.00830 | 3.2E+03 | 13.9008 |

f_{14} | 3.9E−11 | 2.9E+06 | 7.3E+04 | 3.9E+07 | 4.5E+07 | 8.1E+05 | 3.4E+04 |

f_{15} | 0.25413 | 2.3E+02 | 26.5175 | 3.0E+02 | 1.7E+02 | 1.7E+02 | 1.5E+02 |

f_{16} | 2.2E−06 | 1.9E+03 | 3.5E+02 | 3.5E+03 | 3.9E+02 | 1.3E+03 | 1.0E+03 |

f_{17} | 0.51201 | 67.3310 | 47.5805 | 78.8131 | 75.8892 | 48.0274 | 33.5393 |

f_{18} | 1.1E−05 | 8.0E+04 | 4.1E+04 | 1.2E+05 | 9.8E+04 | 8.0E+03 | 5.4E+03 |

f_{19} | 1.5E−04 | 0.49997 | 0.49959 | 0.49999 | 0.49998 | 0.49469 | 0.49168 |

f_{20} | 1.1E−06 | 46.2759 | 34.4948 | 47.2014 | 45.7578 | 44.0433 | 43.4175 |

f_{i} | MQPAPSO | QDPSO | PSO | SFLA | |||
---|---|---|---|---|---|---|---|

G = 100 | G = 100 | G = 1000 | G = 100 | G = 1000 | G = 100 | G = 1000 | |

f_{1} | 5.7E−08 | 2.3E+04 | 1.2E+02 | 3.9E+04 | 2.7E+02 | 3.3E+03 | 5.48043 |

f_{2} | 6.3E−04 | 1.0E+20 | 6.0E+02 | 5.4E+25 | 1.2E+15 | 5.9E+02 | 5.7E+02 |

f_{3} | 2.2E−08 | 1.9E+05 | 1.1E+05 | 3.0E+05 | 2.4E+05 | 2.4E+04 | 1.4E+04 |

f_{4} | 0.00133 | 67.7123 | 44.9334 | 85.5049 | 85.4186 | 15.2185 | 13.0471 |

f_{5} | 1.3E+02 | 4.1E+09 | 5.3E+05 | 9.4E+09 | 6.5E+07 | 3.0E+07 | 1.0E+05 |

f_{6} | 1.6E−09 | 4.0E+09 | 2.2E+08 | 1.5E+10 | 5.9E+08 | 2.4E+06 | 28.1635 |

f_{7} | 2.5E−05 | 2.1E+04 | 1.4E+03 | 4.3E+04 | 2.5E+03 | 4.4E+03 | 3.7E+03 |

f_{8} | 5.4E−05 | 21.2627 | 21.0887 | 21.4234 | 21.3745 | 18.4078 | 17.1200 |

f_{9} | 1.5E−08 | 1.4E+02 | 1.42371 | 2.4E+02 | 2.74191 | 18.7604 | 0.22863 |

f_{10} | 21.6660 | 4.7E+05 | 1.0E+02 | 9.4E+05 | 6.7E+03 | 2.6E+03 | 16.4156 |

f_{11} | 2.4E+05 | 2.0E+08 | 2.1E+07 | 4.9E+08 | 1.1E+08 | 3.0E+08 | 1.8E+06 |

f_{12} | 0.25614 | 1.8E+09 | 2.7E+03 | 4.2E+09 | 1.3E+07 | 7.5E+04 | 26.5760 |

f_{13} | 1.6E-06 | 6.9E+04 | 7.8E+02 | 1.1E+05 | 1.0E+03 | 1.0E+04 | 58.7252 |

f_{14} | 9.6E−11 | 9.3E+07 | 4.4E+06 | 1.0E+09 | 1.5E+09 | 3.8E+06 | 3.3E+05 |

f_{15} | 0.67362 | 6.7E+02 | 3.5E+02 | 8.1E+02 | 5.5E+02 | 3.8E+02 | 3.4E+02 |

f_{16} | 1.0E−05 | 2.2E+04 | 1.4E+03 | 4.3E+04 | 3.2E+03 | 3.9E+03 | 3.7E+03 |

f_{17} | 1.06924 | 1.4E+02 | 1.1E+02 | 1.7E+02 | 1.6E+02 | 1.2E+02 | 1.0E+02 |

f_{18} | 1.3E−05 | 1.9E+05 | 1.4E+05 | 3.0E+05 | 2.4E+05 | 2.1E+04 | 1.9E+04 |

f_{19} | 0.00127 | 0.49999 | 0.49998 | 0.49999 | 0.49999 | 0.49774 | 0.49830 |

f_{20} | 0.00222 | 96.3208 | 83.6293 | 97.0198 | 95.6162 | 92.8530 | 90.5664 |

D | |||
---|---|---|---|

50 | 9.863512 | 9.800094 | 9.620418 |

100 | 9.785713 | 10.03969 | 9.752004 |

AVG | 9.824613 | 9.919894 | 9.686211 |

D | ||||||
---|---|---|---|---|---|---|

50 | 0.001361 | 0.165868 | 0.000671 | 0.067214 | 0.002587 | 0.140945 |

100 | 0.000623 | 0.012133 | 0.000510 | 0.000795 | 0.001124 | 0.073973 |

AVG | 9.92E-04 | 0.089001 | 5.91E-04 | 0.034004 | 0.001856 | 0.107459 |

amplitude coding and evolutionary mechanisms can indeed improve the optimization capability. From

In this paper, a quantum-inspired particle swarm optimization algorithm is presented encoded by probability amplitudes of multi-qubits. Function extreme optimization results show that under the same running time, the optimization ability of proposed algorithm has greatly superior to the traditional methods, revealing that the multi-qubits probability amplitude encoding method indeed greatly enhances the ability of traditional particle swarm optimization performance.

This work was supported by the Youth Foundation of Northeast Petroleum University (Grant No. 2013NQ119) and the National Natural Science Foundation of China (Grant No. 61170132).