In quantum computing, a qubit is a two-level quantum system, described by a two-dimensional complex Hilbert space. From the superposition principles, any state of the qubit may be written as

where,.

Therefore, unlike the classical bit, which can only equal 0 or 1, the qubit resides in a vector space parameterized by the continuous variables and. The normalization condition means that the qubit’s state can be represented by a point on a sphere of unit radius, called the Bloch sphere. The Bloch sphere representation is useful as it provides a geometric picture of the qubit and of the transformations that can be applied to its state. This sphere can be embedded in a three-dimensional space of Cartesian coordinates (, ,). Thus, the state can be written as

The optimization is performed in, so the proposed method can be easily adapted to a variety of optimization problems.

In BQPSO, all particles are encoded by qubits described on the Bloch sphere. Set the swarm size to m, and the space dimension to n. Then the i-th particle is encoded as

where

From the principles of quantum computing, the coordinates x, y, and z of a qubit on the Bloch sphere can be measured by the Pauli operators written as

Let denote the j-th qubit on the i-th particle. The coordinates (x_ij, y_ij, z_ij) of can be obtained by Pauli operators using

In BQPSO, the Bloch coordinates of each qubit are regarded as three paratactic location components, each particle contains three paratactic locations, and each location represents an optimization solution. Therefore, in

the unit space, each particle simultaneously represents three optimization solutions, which can be described as follows

In BQPSO, each particle contains 3n Bloch coordinates of n qubits that can be transformed from the unit space to the solution space of the optimization problem. Each of the Bloch coordinates corresponds to an optimization variable in the solution space. Let the j-th variable of optimization problem be, and

denote the coordinates of the j-th qubit on the i-th particle. Then the corresponding variables in the solution space are computed as follows

where

By substituting the three solutions described by the i-th particle into the fitness function, we may compute its fitness, for Let denote the best fitness so far, and denote the corresponding best particle. Let denote the own best fitness of the i-th particle, and denote the corresponding best particle. Further, let,. If then and. If then and.

In BQPSO, we search on the Bloch sphere. That is, we rotate the qubit around an axis towards the target qubit. This rotation can simultaneously change two parameters and of a qubit, which simulates quantum behavior and enhances the optimization ability.

For the i-th particle, let denote the current location of the j-th qubit on the Bloch sphere, and and denote its own best location and the global best location on the Bloch sphere. According to [27] , for, the two potential well centers in Equation (8) can be obtained using

where m denotes the number of particles, r denotes a random number uniformly distributed in (0, 1), and k denotes the iterative step.

Let O denote the center of the Bloch sphere and denote the angle between and. From the QPSO’s iteration equation, to make move to, the angle needs to be rotated on the Bloch sphere so that

Let the qubit corresponding to the point be. From the above equation we know that the new location of is actually the location of after it is rotated through an angle towards.

To achieve this rotation, it is crucial to determine the rotation axis, as it can directly impact the convergence speed and efficiency of algorithm. According to the definition of the vector product, the rotation axis of rotating towards through an angle can be written as

From the principles of quantum computing, the rotation matrix about the axis that rotates the current qubit towards the target qubit can be written as

and the rotation operation can be written as

where and k denotes the iterative steps.

Many benchmark numerical functions are commonly used to evaluate and compare optimization algorithms. In this section, the performance of the proposed BQPSO algorithm is evaluated on 8 standard, unconstrained, single-objective benchmark functions with different characteristics, taken from [28] - [30] . All of the functions are minimization problems.

1);

2);

3);

4);

5),;

6);

7),;

8).

For all problems, the following parameters are used unless a change is mentioned. Population size: NP = 100 when D = 30 and NP = 80 when D = 20, the precision of a desired solution value to reach (VTR): VTR = 10 − 5

(i.e.) for and; VTR = 100 for and; VTR = 0.1 for; VTR = 10

for and; VTR = 0.001 for. The maximum of the number of function evaluations (MNFE): MNFE = 20000; The control parameter:; Halting criterion: when MNFE is reached, the execution of the algorithm is stopped.

To minimize the effect of the stochastic nature of the algorithms, 50 independent runs on 8 functions are performed and the reported indexes for each function are the average over 50 trials. If an algorithm finds the global minima with predefined precision within the preset MNFE the algorithm is said to have succeeded. Otherwise it fails. All of the algorithms were implemented in standard Matlab 7.0 and the experiments were executed on a P-II 2.0 GHz machine with 1.0 GB RAM, under the WIN-XP platform.

Five performance criteria were selected from [31] to evaluate the performance of the algorithms. These criteria are also used in [32] and are described as follows.

Error: The error of a solution is defined as, where is the global optimum of the function. The error was recorded when the MNFE was reached and the average (mean) and standard deviation (std dev) of the 50 error values were calculated.

NFE: When the VTR was reached, the number of function evaluations (NFE) was recorded. If the VTR was not reached within the preset MNFE then the NFE is equal to MNFE. The average (mean) and standard deviation (std dev) of the 50 NFE values were calculated.

Number of successful runs (SR): The number of successful runs was recorded when the VTR was reached before the MNFE was satisfied.

Running time (time (s)): Running time indicates the average time over one function evaluation.

In this section, we compare our approach with the classical QPSO of [26] , to demonstrate the superiority of BQPSO. The parameters used for the two algorithms are described in Section 4.2. The results were calculated using 50 independent runs. Table 1 shows the mean and standard deviation of the errors of BQPSO and QPSO on 8 benchmark functions. The mean and standard deviation of NFE are shown in Table 2.

From Table 1 and Table 2, we can see that BQPSO performs significantly better than QPSO for 8 functions. For, , , BQPSO succeed in finding the minimum in all runs. For the other functions, BQPSO succeed much more often than QPSO. Furthermore, BQPSO obtains smaller mean and standard deviations than QPSO

for 8 functions. Especially, for, , , , , and, BQPSO succeeds many times while all runs of QPSO fail. In Table 1, we can see that there are significant differences in quality between the BQPSO and QPSO solutions of the high-dimensional functions.

In Table 2, the MNFE is fixed at 20000 for 8 functions. From this table it can be observed that, for all functions, BQPSO requires less NFE than QPSO. For some high-dimensional functions (such as, , , , and, QPSO fails to reach the VTR after 20,000 NFE while BQPSO is successful. It is worth noting that, from Table 1, the running time of BQPSO is about 10 to 20 times longer than that of QPSO. According to the no free lunch theorem, the superior performance of BQPSO is at the expense of a long running time.

It can be concluded that the overall performance of BQPSO is better than that of QPSO for all 8 functions. The improvement based on quantum computing can accelerate the classical QPSO algorithm and significantly reduce the NFE to reach the VTR for all of the test functions.

In this subsection, we compare BQPSO with other state-of-art algorithms to demonstrate its accuracy and performance. These algorithms include a genetic algorithm with elitist strategy (called GA), a differential evolution algorithm (called DE), and a bee colony algorithm (called BC). The BQPSO’s control parameter was. For the genetic algorithm, the crossover probability was and the mutation probability was. For the differential evolution algorithm, the scaling factor was, and the crossover probability was. For the bee colony algorithm, let denote the population size of the whole bee colony, and and denote the population size of the employed bee and onlooker bee, respectively.

We have taken. The threshold of a tracking bee searching around a mining bee was. The other parameters used for the four algorithms are the same as described in Section 4.2. The eight high-dimensional functions were used for these experiments, which had 50 independent runs. Table 3 shows the mean of these 50 errors and the number of successful runs. The mean and standard deviation of the NFE are shown in Table 4.

From Table 3 and Table 4 it can be argued that the BQPSO performed best among the four algorithms. It obtained the best results for all eight benchmark functions. The best algorithm is not as obvious for the remaining

three algorithms. The DE algorithm performed well on average. It obtained the best results among the three algorithms for some benchmark functions, but it did not successfully optimize the functions f₂, f₄, and f₇, because it got trapped in a local optimum. The BC achieved the best results among the three algorithms for the 30-dimensional functions f₂, f₄, and f₇. The GA achieved the best results among three algorithms for the 20-dimensional functions f₂ and f₇. The DE achieved the best results among three algorithms for the 20-dimensional function f₄. According to the experimental results, the algorithms can be ordered by optimizing performance from high to low as BQPSO , DE , BC, GA. This demonstrates the superiority of BQPSO.

These results can be easily explained as follows. First, In BQPSO, two parameters and of a qubit can be simultaneously adjusted by means of rotating the current qubit through an angle about the rotation axis. This rotation can automatically achieve the best matching of two adjustments. In other words, when the current qubit moves towards the target qubit, the path is the minor arc of the great circle on the Bloch sphere, which is clearly the shortest. Obviously, this rotation with the best matching of two adjustments has a higher optimization ability. Secondly, the three chains structure of the encoding particle also enhances the ergodicity of the solution space. These advantages are absent in the other three algorithms.