During the quantum computing, a qubit is a two-level quantum system which could be described in two-dimen- sion complex Hilbert space. Based on principle of superposition, a qubit can be defined as:

where and.

Owing to the continuity of and, a qubit can be in infinitely many different states and described by a point on the Bloch sphere. As shown in Figure 1.

In this paper, we create a search mechanism on the Bloch sphere which is used to rotate the qubit about the rotation axis to the target qubit on the Bloch sphere. The rotation can simultaneously change two parameters of a qubit, and automatically achieve the best match between two adjustment quantities, thus we can enhance the performance of optimization. The key to the above rotation is the design of rotation axis. The design method given by this paper can be expressed as the follows [13] [14] .

Assuming and denote two points on the Bloch sphere respectively, the rotation axis about which P rotates to Q along the shortest path is the vector product of P and Q, namely,. As shown in Figure 2.

Based on the principle of quantum computing, the rotation matrix makes the qubit rotate about a rotation axis

along the unit vector with radian of rotation, and it’s defined by

where I is the unit matrix, and, are Pauli matrices [15] .

Therefore, on the Bloch sphere, the rotation matrix that rotates the about to can be described as follows:

The rotation operation is given by

where t is iteration step.

In QIBC, the individual coding adopts qubit based on the Bloch sphere. Let denote population size, D de-

note the number of variables, and denote the t-th generation colony. During in

itialization, the i-th individual can be coded as follows:

where.

On the basis of the principle of quantum computing, by applying Pauli matrices to, we can get the Bloch coordinates of. The qubits projection measurement is denoted by the following equations:

where and.

In QIBC, each qubit are regarded as three paratactic genes, each one contains three paratactic gene chains, and each of gene chains represents an optimization solution. Therefore, each entity represents three optimization solutions at the same time.

In QIBC, three optimization solutions given by each entity can be expressed by Bloch coordinates. Because the value of coordinate is in (−1, 1), we must map it to solutions to the problems. Assuming the j-th variable is

, the solution transformation can be expressed as the following three equations:

where and.

Taking minimum optimization as example, according to the value of target function, we rank the optimized population in descending order, and select the first entities to compose employed bee colony. The rest of them compose onlooker colony. For the i-th employed bee, first of all, randomly select employed bees

and , and also randomly select a dimension d. Then calculate the rotation axis and the rotation angle between and. Taking as target, rotate about through. Let denote the employed bee after rotation. By greedy selection operator, we select the better ones between and for the next generation from the following equations:

where,

It is similar with employed bee search. First, we calculate selective probability of each employed bee with the equation as follows:

To each onlooker bee, first, we select employed bee according to the roulette method, and in its neighborhood we search for new position in the same method as the employed bee search. If is better than, set, and set the number of searches. If is less than threshold, set, otherwise, we initialize and set.

To this algorithm, the termination criterion is the number of iterations. Whether it meets the pre-set accuracy or not, the algorithm will stop when the limited number of iterations reaches.

To verify the superiority of QIBC, the following ten Benchmark functions are used in this experiment. All of ten functions are for the minimum optimization, and is the minimum extreme value point.

(1), , ,;

(2), , ,;

(3), , ,;

(4), , ,;

(5), , ,;

(6), , , ,;

(7), ,

, , ;

(8), , ,;

(9), , ,;

(10), , , ,.

For convenient comparison, based on the complexity of benchmark functions, we set up precision thresholds for each function. Only when optimization effect is less than the thresholds, the algorithm is call converges. In this experiment, the precision thresholds are set as follows. for,; for ; for,; for,; for,; for,; for,. If the algorithm converges, the current optimization steps are called the number of iteration. If not, the number of iteration is equal to the pre-set limited number of iterations.

The dimensions of ten functions are set to, and population sizes of five algorithms are equal to 40. For QIBC and BC, we have, ,. For BQGA, on the basis of Ref. [12] , the initial value of rotation angle is, and the mutation probability is 10^-3. For QDPSO, based on Ref. [16] , the control parameter is. For EGA, the crossover probability is 0.8, the mutation probability is 0.01. The limited number of iterations of five algorithms is.

To enhance the objectivity of comparisons, each algorithm independently runs 50 times, and we have comparisons of statistical results. The average time for each of iteration are shown in Table 1. To make comparisons be sufficient, besides showing the average results, the best and the worst ones are also are given. Comparison of the number of iterations, average iterations, and optimization results are shown in Table 2.

From Table 1, for the running time, we rank five algorithms in descending order as QIBC, BQGA, BC, EGA, QDPSO. The reason is that, in QIBC and BQGA, we use coding mechanism of qubit with three chains. During each iteration, each entity respectively do updating, solution to space transformation, calculating value of benchmark functions etc. and these operations take a long time. QIBC is longer than BQGA because QIBC needs the projection measurement, design of rotation axis and rotation matrices, and its calculating quantity is

larger.

In Table 2, for the ability of optimization, obviously, QIBC is better than QIBC, BQGA is better than EGA. QDPSO is worse than QIBC, but it’s better than BQGA, and it’s similar to BC. For the optimization performance of five algorithms, we rank them in descending order as QIBC, BC, QDPSO, BQGA, EGA.

For the above results, we present the following analysis. Firstly, we introduce the coding mechanism of qubit with three chains, and it effectively enhances the ergodicity of algorithm to solution space. Based on the geometric properties of the Bloch sphere, this mechanism can enlarge the number of global optimal solutions and the probability of attaining global optimal solutions. That is the reason why these two algorithms are better than their classical ones respectively. Secondly, QIBC has a better performance of optimization than BQGA, because they adjust qubit in the different ways.

In BQGA, qubits are updated by directly adjusting and with the same adjustment amount (namely). Obviously, in this method, it is hard to close to target qubits along the shortest path, because there must be some matching relation between and when it’s along the shortest path. But this matching relation is hard to be expressed clearly by analytic equations. In QIBC, we adjust qubit in indirect method. Namely, we rotate qubit to target qubit about the rotation axis on the Bloch sphere, and obviously, this path is

the shortest. Although it does’t adjust two parameters of qubit directly, it achieves the best match between and automatically and accurately. Hence, this method has better optimizing efficiency than classical ones.

It’s also worth pointing out that, although the QIBC has better optimizing performance, the complexity of this algorithm is obviously higher than classical ones. Comparing with classical ones, the QIBC needs some extra operations, such as calculating rotation axis, rotation angel, rotation matrices and the solution to space transformation. Considering run time and optimizing performance, QIBC earns better optimizing efficiency by sacrificing running time, which is the same as No Free Lunch. For many off-line optimizing tasks which may ignore running time, the QIBC has wide application prospects.

We need investigate and study the influence after dimensions and parameters change. Taking and as example, while, population size is, and,. Precision threshold of and is. The QIBC and the BC run 10 times respectively. Optimization results is shown in Table 3.

As the results are shown, for the high-dimensional optimization problems, the proposed algorithm also shows its better performance than classical ones.