An improved quantum genetic algorithm and its application duplex

 Although this method is still the traditional genetic algorithm, but the principles of quantum computing has inspired a combination of genetic algorithm studies; Han et al [7] using quantum bits and quantum gates updated coding chromosome, genetic quantum algorithm and the proposed parallel genetic algorithm, and successfully solve the combinatorial optimization problem; Yang Jun-an et al [8], the introduction of genetic algorithms in multi-universe concept, the multi-universe parallel quantum genetic algorithm; Wang Ling et al [9] QGA SGA integration with proposed based on Hybrid quantum computing quantum genetic algorithm; PROCEEDINGS et al [10] to the quantum probability amplitude of the two chromosomes in the chain are the optimal solution as described in the gene chain, double-stranded proposed quantum genetic algorithm (double chains quantum genetic algorithm, DCQGA) , with the cyclical nature of the phase qubits, to some extent, the algorithm can improve the optimization performance. In this paper, [10] study, based on quantum bits by changing the probability amplitude of the cycle, the rotation angle and the mutation strategy, proposed a Improved double-stranded species quantum genetic algorithm (improved DCQGA, IDCQGA). The method can search a number of trigonometric functions in the probability amplitude cycles at the same time, the original algorithm to improve the population diversity, adaptability and stability, and thus further improve the algorithm's search ability.
1 IDCQGA continuous optimization of the basic principles of
1.1 Problem Description
If n-dimensional continuous space optimization problem as n-dimensional space of the point or vector, the continuous optimization problem can be expressed as
min f (x) = f (x1, x2, ..., xn) st ai ≤ xi ≤ bi; i = 1,2, ..., n (1)
If we look at constraints do n-dimensional continuous space bounded closed set Ω, the Ω in each point as the approximate solution of optimization problems, defined as the evaluation function to reflect the extent of the merits of the approximate solution:
fit (x) = Cmax-f (x) (2)
which, Cmax can be a suitable input values, or optimize the process so far f (x) the maximum value.
1.2 double-stranded quantum genetic algorithm
In quantum computing, the smallest unit of information using quantum bits. quantum bit, also known as quantum bits, a qubit state can be expressed as
to satisfy (3) (4) a pair of complex numbers α and β, said as a probability amplitude of quantum bits, qubits can also be used so the probability amplitude is expressed as [α, β] T.
in DCQGA, direct use of probability amplitude as a code. taking into account the equation (4) binding, coding programs are as follows:
pi = cos (ti1) sin (ti1) cos (ti2) sin (ti2) ... cos (tin) sin (tin) (5)
where: tij = 2π * rnd, rnd is ( 0,1) random number between, i = 1,2, ..., m; j = 1,2, ..., nm is the population size, n is the quantum bits. In DCQGA, you will increase the probability of each qubit as is the up and down the genes of two parallel, each chromosome contains the genes of two parallel chains, and each gene represents an optimal solution chain. Therefore, each iteration of the two solutions simultaneously update the same in the case of population size can be extended under the ergodicity of the search space, speed up the optimization process.
DCQGA updated using the revolving door of quantum probability amplitude of quantum chromosome, the angle step formula for
Δθ =- sgn (A) * Δθ0 * ‖ fit max-fit min ‖ ‖ fit (X) - fit min ‖ (6)
including: Δθ0 initial step for the corner; fit (X) function fit for the evaluation of the gradient at point X; fti max and fit min were defined as
fit max = [max {fit (X1) X11, ...,
fit (Xm) X1m}, ..., max {fit (X1) Xn1, ..., fit (Xm) Xnm}] (7 )
fit min = [min {fit (X1) X11, ..., fit (Xm) X1m}, ...,
min {fit (X1) Xn1, ..., fit (Xm) Xnm}] (8)
DCQGA mutation strategy for the use of the quantum NOT gate, the two exchange probability amplitude of quantum bits, so that the strands at the same time variation.
1.3 Improved double-stranded quantum genetic algorithm
this paper, the proposed three-DCQGA point improvement strategies.
1) improvement of the quantum chromosome encoding
in IDCQGA in this probability amplitude in quantum bit to introduce a trigonometric description of the adjustment factor is greater than or equal to 1, the original cycle of improvement for the 2π description of multi-cycle mode. the introduction of the direct purpose of this parameter is to increase the probability of convergence. improved encoding for the
pi = cos (cti1) sin (cti1) cos (cti2) som (cti2) ... cos (ctin ) sin (ctin) (9)
where c ≥ 1.
this encoding phase of each qubit with a constant factor greater than 1, the probability amplitude function to extend the cycle. The algorithm is DCQGA promotion, when c = 1, ie, revert to the original algorithm. Suppose the optimal solution in the optimization process variables corresponding to the probability of a certain amplitude of -0.5 to c = 2, for example, we can see from Figure 1 , the original algorithm (c = 1) has two phase solution: q1 = 7π / 6, q2 = 11π / 6; new algorithm (c = 2) there are four phase solution: p1 = 7π/12, p2 = 11π/12 , p3 = 19π/12, p4 = 23π/12. Therefore, increasing the probability of obtaining the optimal solution.
2) Quantum revolving door step function improvements in the corner
equation (6) there are insufficient: if fit (X) = fit min, then ‖ fit (X) - fit min ‖ = 0, Δθ = ∞, which led to shock algorithm. In view of this, we propose the following angle step function:
Δθ =- sgn (A ) Δθ0 exp-f (X) - f min f max-f min (10)
keeping with the style both on the type (5) the consistency of function, but also overcome the existing deficiencies, to improve the algorithm Adaptability and stability.
3) non-gate based on quantum improvement
mutation strategy is the role of the quantum NOT gate to the two probability amplitude of quantum bit conversion, which is essentially a variation of the phase qubit rotation. Let a quantum bit rate angle θ, the rotation angle increases after the π/2-θ, The angle forward rotation of the π/2-2θ. The analysis, in fact, the rotation rate of the π / 2 is not necessary, can be replaced with other angles. In IDCQGA in this paper, based on single-qubit Hadamard's variation of strategy, the single-qubit gate in the effect on the
1211 1-1cos (θij ) sin (θij) =
cos (π4-θij) sin (π4-θij) = cos (θij + π4-2θij) sin (θij + π4-2θij) (11)
which: i ∈ {1 , 2, ..., m}, j ∈ {1,2, ..., n}. from (11) can be seen that this mutation strategy is also a rotation. For the first section on the chromosome i j qubits , corner size Δθij = π/4-2θij, enhance the diversity of population.
2
contrast experiments in the improvement measures in order to verify the effectiveness of IDCQGA, consider the following two typical extreme value function optimization problem compared with the DCQGA.
1) BR-Branin function
f (x, y) = x-5.14π2y2 +5 πy-62 +101-18 πcos y +10 (12)
where: x ∈ [ ,],[, 12.276) and (3.142,2.275), a global minimum point and very close to the local minimum point is (9.425,2.425), the local minimum value is 0.400 4, and their spatial distribution shown in Figure 2. When the error is less than optimal results that the convergence 0.000 4.
2) RA-Rastrigin function
f (x, y) = x2 + y2-cos (18x)-cos (18y) (13)
where: x, y ∈ [-1,1].
optimization objective function is to strike a very small value, type (13) global minimum point is (0,0), the global minimum value of -2.000 in the feasible region of about 50 local minimum points, their spatial distribution shown in Figure 3. When the optimization results when the error is less than 0.005 that the convergence.
3) Generalized Rosenbrock's function
f3 (X ) = Σ29i = 1 [100 (xi +1- x2i) 2 + (1-xi) 2] (14)
where: xi ∈ [-30,30].
optimization objective function for the strike minimum value, type (14) global minimum point is (1,1, ..., 1), the global minimum value is 0. When the optimization results that the convergence is less than 0.005.
the above functions, respectively, optimization with IDCQGA and DCQGA 50 times, then the best statistical results of each algorithm, the average result, convergence times, the average number of steps, the average time as a comparison index. three algorithms are taking the population size of 50, were taking the maximum optimization of algebraic 500, mutation probability were removed and 0.05, initial steps were taken corner 0.01π. performance comparison results are shown in Table 1.
Table 1 IDCQGA, DCQGA algorithm performance is the best result comparing function algorithm
average results converge The average number of steps the average time / s
BR-BraninIDCQGA DCQGA0.397 9 0.397 90.399 1 0.399 649 4655 .040 0 66.440 00.030 3 0.037 7
RA-RastriginIDCQGA DCQGA-2.000 0 -2.000 0-1.994 9 -1.993 748 4559.220 0 72.560 00.029 7 0.031 1
Generalized RosenbrockIDCQGA DCQGA0.000 0 0.000 00.025 3 0.038 643 3913 8.230 0 163.870 00.058 6 0.059 9
Comparative experimental results show that, IDCQGA optimal performance is better than DCQGA, which show that the proposed improvement strategies is feasible and effective.
3 Conclusion
quantum computing quantum genetic algorithm is combined with the product of genetic algorithms, present a new research direction, by the scholars attention. This quantum genetic algorithm for the current double-stranded problems, three improved through the implementation of strategies to establish a new type of double-stranded genetic algorithm, effectively overcome the shortcomings of the original algorithm to further improve the algorithm's search ability. Simulation results demonstrate that the proposed improved algorithm effectiveness.
References:
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Reference
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