Quantum evolution method

Abstract

A quantum evolution method includes steps of: according to the quantum evolution method, initializing a generation number t=0, and initializing a population Q(t)={q1t, q2t, . . . , qnt}; observing Q(t) and generating P(t)={x 1t, x2t, . . . , xnt}, wherein represents strings comprising 0 or 1 with a length of m; evaluating each xit with an evaluation function, and inputting evaluating results into a fitness function F(t), F(t)={f1t, f2t. . . , fnt}, wherein fit represents a fitness of each individual; selecting an elite group E(t) from P(t) according to the fitness; evolving Q(t) through U(Δθijt); inputting an optimal solution b of P(t) into B(t), wherein if the optimal solution is better than an original optimal solution in B(t), then replacing the original optimal solution; otherwise remaining the original optimal solution; and judging a shutdown condition, if satisfied, outputting the optimal solution; otherwise returning to the step (2) for further evolution. The method can effectively control a quantum evolution direction and improve method stability.

Description

BACKGROUND OF THE PRESENT INVENTION

Field of Invention

The present invention relates to a field of optimizing methods, and more particularly to a quantum evolution method introducing an elite group and a state preference.

Description of Related Arts

Quantum evolution methods are based on state vector expression of quantum, wherein probability amplitudes of quantum bits are used for representing chromosome encoding, in such a manner that a chromosome is able to express multiple superimposed states; and quantum revolving door and quantum NOT gate are used for chromosome updating, so as to achieve optimized solution of a target. However, conventional quantum convergence direction cannot be effectively controlled, which may cause degradation. Conventionally, there are many improvements for the quantum evolution methods, but none effectively overcomes the problem of convergence direction. Therefore, how to speed up the convergence of the quantum evolution methods, and how to control the convergence direction for preventing degradation, so as to improve method stability, are real key to quantum methods.

SUMMARY OF THE PRESENT INVENTION

An object of the present invention is to overcome the above technical defects, and provide a quantum evolution method which effectively controls a convergence direction, wherein an elite group and a state preference are introduced for controlling the convergence direction, so as to improve method stability.

Accordingly, in order to accomplish the above object, the present invention provides:

a quantum evolution method, comprising steps of:

(1) according to the quantum evolution method, initializing a generation number t=0, and initializing a population Q(t)={q₁^t, q₂^t, . . . , q_n^t}, wherein n is a population size, t is the generation number, q_i^t is a No. i individual in a No. t generation, and i ∈[1,n]; defining

$q_i^t=[\begin{array}{ccc}\begin{array}{c}{\alpha }_{i1}^t\\ {\beta }_{i1}^t\end{array}\left|\begin{array}{c}{\alpha }_{i2}^t\\ {\beta }_{i2}^t\end{array}\right|& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}{c}{\alpha }_{\mathrm{im}}^t\\ {\beta }_{\mathrm{im}}^t\end{array}|],\end{array}$

wherein q_i^t, comprises m quantum bits, α represents a probability of each of the quantum bits that a state thereof is 0, β represents a probability of each of the quantum bits that the state thereof is 1, and |α|²+|β|²=1; wherein the quantum bits are randomly generated, and satisfy an equation:

(α_ij^t, β_ij^t)=(sign(rand[0,1]−0.5)*/√{square root over (2)}, sign(ramd[0,1]−0.5)*/√{square root over (2)}),

wherein α_ij^t represents a probability of a No. j quantum bit of the No. i individual in the No. t generation that a state thereof is 0, and β_ij^t represents a probability of the No. j quantum bit of the No. i individual in the No. t generation that a state thereof is 1; initializing an optimal solution collection B(t), and inputting a string b, which comprises m 0-characters, into B(t) as an initial optimal solution;

(2) observing Q(t), and observing all individuals in the No. t generation, wherein for q_i^t, the m quantum bits are all observed for generating a string x_i^t with a length of m, wherein i is a corresponding individual, t is the generation number, and all individuals in the string x_i^t correspond to the quantum bits of q_i^t; if a quantum bit is 0, then 0 is written to a corresponding location in the string x_i^t, and if the quantum bit is 1, the 1 is written to the corresponding location in the string x_i^t; finally generating P(t)={x₁^t, x₂^t, . . . , x_n^{t };}

(3) evaluating each x_i^t with an evaluation function, and inputting evaluating results into a fitness function F(t), F(t)={f₁^t, f₂^t, . . . , f_n^t}, wherein f_i^t represents a fitness of q_i^t which is the No. i individual in the No. t generation, and n is the population size of the No. t generation;

(4) selecting an elite group E(t) from P(t), specifically comprising steps of:

(4.1) comparing all the individuals in the No. t generation with a worst individual of the No. t generation which is evaluated by the fitness function in the step (3), constructing {tilde over (f)}_i^t=abs(f_i^t−min(F(t)));

(4.2) representing a probability that x_i^t enters the elite group by a probability function S_i^t,

$s_i^t={\tilde{f}}_i^t/\sum _{i=1}^n{\tilde{f}}_i^t,$

and constructing S(t)={s₁^t, s₂^t, . . . , s_n^t}; and

(4.3) based on S(t), deciding whether the individuals in P(t) are selected to enter the elite group E(t) by a roulette method, E(t)={e₁^t, e₂^t, . . . , e_p^t}, wherein p is a total individual quantity in the elite group;

(5) evolving the No. t generation population Q(t) through

$U\left(\Delta {\theta }_{\mathrm{ij}}^t\right)=\left[\begin{array}{cc}\cos \left(\Delta {\theta }_{\mathrm{ij}}^t\right)& -\sin \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)\\ \sin \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)& \cos \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)\end{array}\right],$

so as to obtain a No. t+1 generation population Q(t+1),

${\mathrm{\Delta \theta }}_{\mathrm{ij}}^t=\mathrm{sign}\left({\alpha }_{\mathrm{ij}}^t{\beta }_{\mathrm{ij}}^t\right)\frac{1}{p}\sum _{k=1}^p{\mathrm{\Delta \phi }}_{\mathrm{ij}}^k,$

wherein sign(α_ij^tβ_ij^t) represents a quadrant location of a current quantum bit,

$\mathrm{sign}\left({\alpha }_j^1{\beta }_j^1\right)=\{\begin{array}{cc}1& 1\mathrm{st}\mathrm{or}3\mathrm{rd}\mathrm{quadrant}\\ -1& 2\mathrm{nd}or4\mathrm{th}\mathrm{quadrant}\end{array},\mathrm{and}\frac{1}{p}\sum _{k=1}^p{\mathrm{\Delta \phi }}_{\mathrm{ij}}^k$

is a phase angle rotation weight of the elite group E(t), so the elite group actively guides evolution of the whole population; a value of Δφ_ij^k is selected according to: 1) if the individual q_i^t in the No. t generation enters the elite group, then Δφ_ij^k=0; 2) if the individual q_i^t in the No. t generation fails to enter the elite group and x_ij^t=e_kj^t, then Δφ_ij^k=0; 3) if the individual q_i^t in the No. t generation fails to enter the elite group while x_ij^t is in a ‘0’ state and e_kj^t is in a ‘1’ state, then Δφ_ij^k=φ₁, wherein φ₁ is a rotation value evolving towards the ‘1’ state, so as to increase a probability that x_ij^t evolves from the ‘0’ state to the ‘1’ state; and 4) if the individual q_i^t in the No. t generation fails to enter the elite group while x_ij^t is in the ‘1’ state and e_kj^t is in the ‘0’ state, then Δφ_ij^k=φ₀, wherein φ₀ is a rotation value evolving towards the ‘0’ state, so as to increase a probability that x_ij^t evolves from the ‘1’ state to the ‘0’ state; wherein x_i^t is the quantum bits of the individual q_i^t in the No. t generation, which is determined in the step (2); and e_k^t is all individuals of the elite group E(t), which is determined in the step (4), k ∈[1, p], x_ij^t and e_kj^t respectively represent the No. j quantum bit of x_i^t and e_k^t in the No. t generation;

Δφijk values
	xijt	ekjt	f (xit) ≦ f (ekt)	Δφijk
	*	*	×	0
	0	0	✓	0
	0	1	✓	φ1
	1	0	✓	φ0
	1	1	✓	0

for controlling an evolution direction so as to uniformly evolve towards the ‘1’ state, introducing a state preference for further weighting, specifically comprising steps of: when the individual q_i^t in the No. t generation fails to enter the elite group while x_ij^t is in the ‘0’ state and e_kj^t is in the ‘1’ state, increasing a value of φ₁ so as to increase a probability that x_ij^t evolves from the ‘0’ state to the ‘1’ state; when the individual q_i^t in the No. t generation fails to enter the elite group while x_ij^t is in the ‘1’ state and e_kj^t is in the ‘0’ state, decreasing a value of φ₀ so as to decrease a probability that x_ij^t evolves from the ‘1’ state to the ‘0’ state; in such a manner that total evolution is towards the ‘1’ state;

(6) using x_i^t with a highest fitness, which is selected from P(t) by the fitness function F(t) in the step (3), as an optimal solution of the No. t generation; comparing the optimal solution of the No. t generation with an optimal solution b obtained before the No. t generation, wherein if the optimal solution of the No. t generation is better than the optimal solution before the No. t generation, then the optimal solution of the No. t generation is inputted into B(t−1) for replacing b, so as to obtain B(t); otherwise, the original optimal solution b in B(t−1) remains, so as to obtain B(t); and (7) judging a shutdown condition, specifically: when the optimal solution b in the B(t) is not a globally optimal solution, b is a string comprising m 1-characters and the generation number t is lower than a certain limit, executing t=t+1, and returning to the step (2) for further evolution; otherwise, outputting the optimal solution b in the B(t).

Preferably, in the step (5), introducing the state preference for controlling a convergence direction of the quantum evolution method specifically comprises steps of: using φ₁ and φ₀ for increasing or decreasing a state value of the current quantum bit, wherein if the current quantum bit is in the ‘1’ state, a tendency that a quantum moves to 0 is decreased by decreasing φ₀ ; if the current quantum bit is in the ‘0’ state, a tendency that a quantum moves to 1 is increased by increasing φ₁.

Preferably, in the step (3), for evaluating each x_i^t the evaluation function, all quantum bits of x_i^t are added together, and a result thereof is inputted into F(t) as a fitness f_i^t of x_i^t.

Preferably, in the step (4.3), for deciding whether the individuals in P(t) are selected to enter the elite group E(t) by the roulette method based on S(t), the fitness f_i^t of all the individuals in the No. t generation is calculated, then a fitness sum

$\sum _{i=1}^nf_i^t$

of all the individuals in the No. t generation is calculated, probabilities that the individuals in P(t) enter the elite group E(t) is

$f_i^t=\sum _{i=1}^nf_i^t,$

p individuals with highest probabilities are selected to enter the elite group E(t).

Preferably, in the step (6), b in the optimal solution collection B(t) is the optimal solution of the No. t generation, and an updating process thereof is: during initializing, the optimal solution b is the string comprising m 0-characters; when the generation number t=0, the optimal solution obtained through the step (2) and the step (3) is surely better than the initial optimal solution; as a result, replacing the initial optimal solution by the optimal solution, and inputting in the optimal solution collection B(t) as the optimal solution b, so as to obtain a current generation optimal solution collection B(0); when the generation number t=1, repeating the step (2) and the step (3), comparing an obtained optimal solution with the optimal solution in B(0), wherein if the optimal solution when t=1 is better than the optimal solution in B(0), then the optimal solution when t=1 is inputted into B(t) as b, so as to obtain a current generation optimal solution collection B(1); if the optimal solution when t=1 is worse than the optimal solution in B(0), then the original optimal solution b in B(1) remains, so as to obtain B(1); when the generation number is t, comparing the optimal solution of the No. t generation with the optimal solution b in B(t−1), so as to obtain B(t).

The present invention has a simple structure, and introduces the elite group and the state preference for weighting quantum evolution, which finally achieves optimized solution. By weighting control of the convergence direction, quantum degradation is inhibited and method stability is improved.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a quantum evolution method according to a preferred embodiment of the present invention.

FIG. 2 illustrates controlling an evolution direction by adjusting φ₀ and φ₁.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to drawings and a preferred embodiment, the present invention is further illustrated.

Referring to FIG. 1, a quantum evolution method of the present invention comprises steps of:

(1) executing a step 101, specifically: initializing a generation number t=0, wherein a population size n=20, a parallel population number N=30, and a max generation number is 10000 (i.e. t ∈[0,10000]);

$q_i^t=\left[\begin{array}{c}{\alpha }_{i1}^t\\ {\beta }_{i1}^t\end{array}\left|\begin{array}{c}{\alpha }_{i2}^t\\ {\beta }_{i2}^t\end{array}\right|\begin{array}{cc}\dots & {\alpha }_{\mathrm{im}}^t\\ \dots & {\beta }_{\mathrm{im}}^t\end{array}|\right]$

is a No. i individual in a No. t generation (i ∈[1, n]) , and q_i^t comprises m quantum bits; α and β respectively represent probabilities of a state of 0 or 1, and |α|²+|β|²=; wherein wherein the quantum bits are randomly generated, and satisfy an equation:

(α_ij^t, β_ij^t)=(sign(rand[0,1]−0.5)*1/√{square root over (2)}, sign(rand[0,1]−0.5)*1/√{square root over (2)}),

wherein α_ij^t represents a probability of a No. j quantum bit of the No. i individual in the No. t generation that a state thereof is 0, and β_ij^t represents a probability of the No. j quantum bit of the No. i individual in the No. t generation that a state thereof is 1; inputting a string b, which comprises m 0-characters, into B(t) as an optimal solution;

(2) executing a step 102, specifically: observing all individuals in the No. t generation, and generating P(t)={x₁^t, x₂^t, . . . , x_n^t}, wherein x_i^t represents strings comprising 0 or 1 with a length of m, 0 means the individual is valueless and 1 means valuable;

(3) executing a step 103, specifically: evaluating each x_i^t of P(t) in the No. t generation, constructing a fitness function F(t)={f₁^t, f₂^t, . . . , f_n^t}, wherein f_i^t represents a fitness of the No. i individual in the No. t generation;

(4) executing a step 104, specifically: constructing an elite group E(t), and comparing all values in the fitness function F(t) with a min value in the F(t) for obtaining {tilde over (f)}_i^t with an equation:

{tilde over (f)} _i ^t =abs(f_i^t−min(F(t)));

constructing a probability function S_i^t, which represents a probability that x_i^t of P(t) enters the elite group, with an equation:

$s_i^t={\tilde{f}}_i^t/\sum _{i=1}^n{\tilde{f}}_i^t;$

constructing S(t)={s₁^t, s₂^t, . . . , s_n^t};

deciding whether the individuals in P(t) are selected to enter the elite group E(t) by a roulette method, E(t)={e₁^t, e₂^t, . . . , e_p^t}, wherein p is a total individual quantity in the elite group; here, p=20;

(5) executing a step 105, specifically: evolving Q(t), and evolving the quantum bits based on the elite group with

$U\left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)=\left[\begin{array}{cc}\cos \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)& -\sin \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)\\ \sin \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)& \cos \left({\mathrm{\Delta \theta }}_{\mathrm{ij}}^t\right)\end{array}\right]$

for rotating Q(t), wherein

${\mathrm{\Delta \theta }}_{\mathrm{ij}}^t=\mathrm{sign}\left({\alpha }_{\mathrm{ij}}^t{\beta }_{\mathrm{ij}}^t\right)\frac{1}{p}\sum _{k=1}^p{\mathrm{\Delta \phi }}_{\mathrm{ij}}^k,$

sign(α_ij^tβ_ij^t) represents a quadrant location of a current quantum bit,

$\mathrm{sign}\left({\alpha }_{\mathrm{ij}}^t{\beta }_{\mathrm{ij}}^t\right)=\{\begin{array}{cc}1& 1\mathrm{st}\mathrm{or}3rd\mathrm{quadrant}\\ -1& 2\mathrm{nd}\mathrm{or}4\mathrm{th}\mathrm{quadrant}\end{array},\mathrm{and}\frac{1}{p}\sum _{k=1}^p{\mathrm{\Delta \phi }}_{\mathrm{ij}}^k$

is a phase angle rotation weight of the elite group E(t), so the elite group actively guides evolution of the whole population; values of Δφ_ij^k are listed in Table 1;

TABLE 1
Δφijk values
	xijt	ekjt	f (xit) ≦ f (ekt)	Δφijk
	*	*	×	0
	0	0	✓	0
	0	1	✓	φ1
	1	0	✓	φ0
	1	1	✓	0

for preventing degeneration of conventional quantum evolution methods, introducing a state preference for further weighting, specifically comprising steps of: using φ₁ and φ₀ for increasing or decreasing a state value of the current quantum bit, wherein if the current quantum bit is in the ‘1’ state, a tendency that a quantum moves to 0 is decreased by decreasing φ₀ ; if the current quantum bit is in the ‘0’ state, a tendency that a quantum moves to 1 is increased by increasing φ₁; wherein values of φ₁ and φ₀ are determined by the population size and the individual quantity in the elite group, which is generally sufficient when 0≦φ₀≦φ₁, as shown in FIG. 2;

(6) executing a step 106, specifically: using x_i^t with a highest fitness of P(t) as an optimal solution of the No. t generation; comparing the optimal solution of the No. t generation with an optimal solution b obtained before the No. t generation, wherein if the optimal solution of the No. t generation is better than the optimal solution before the No. t generation, then the optimal solution of the No. t generation is inputted into B(t−1) for replacing b, so as to obtain B(t); otherwise, the original optimal solution b in B(t−1) remains, so as to obtain B(t); and

(7) executing a step 107, specifically: judging a shutdown condition, specifically: when the optimal solution b in the B(t) is not a globally optimal solution, b is a string comprising m 1-characters and the generation number t is lower than a certain limit, executing t=t+1, and returning to the step (2) for further evolution; otherwise, outputting the optimal solution b in the B(t).

EXAMPLE

A famous NP problem—knapsack problem is adapted as an example. The problem is: under a certain knapsack volume, how to reach a max total price of items with different prices and sizes. There are five knapsacks with different volumes of 600, 1200, 1800, 2400 and 3000 in the example. Comparison is provided between the quantum evolution method of the present invention with the elite group and the state preference (PEQIEA), a quantum evolution method (QIEA), a quantum evolution method with H quantum (HQIEA), improved quantum evolution method (IQIEA), quantum evolution method with fitness (FQIEA), hybrid quantum evolution method (QEP), and a comprehensively learning quantum evolutionary approach (CLQIEA) for comparison.

TABLE 2
solution comparison of Knapsack problem
		mean square
volume	method	deviation	best	middle	worst	GS/UL
600	QIEA	10000	3676.1290	3675.6275	3671.1261	GS = 3679.8790
	HQIEA	10000	3676.1280	3670.6278	3666.1266	UL = 3681.1291
	IQIEA	10000	3666.1287	3663.1251	3656.1290
	FQIEA	7814	3681.1258	3679.2501	3670.9387
	QEP	10000	3631.1214	3617.6242	3596.1278
	CLQIEA	10000	3676.1289	3676.1277	3676.1256
	PEQIEA	173	3681.1286	3681.1284	3681.1283
1200	QIEA	10000	7365.4952	7357.9944	7355.4917	GS = 7371.8498
	HQIEA	10000	7340.4905	7334.4700	7320.4842	UL = 7375.4961
	IQIEA	10000	7320.4867	7310.9122	7295.1624
	FQIEA	8894	7375.4902	7370.6721	7363.1028
	QEP	10000	7230.4937	7208.9874	7185.4958
	CLQIEA	10000	7365.4959	7363.4941	7360.4930
	PEQIEA	216	7375.4961	7375.4960	7375.4957
1800	QIEA	10000	11023.6642	11007.6652	10978.6658	GS = 11036.5618
	HQIEA	10000	10963.6448	10942.1029	10913.6130	UL = 11043.6659
	IQIEA	10000	10893.3635	10864.4902	10848.6453
	FQIEA	8271	11043.6588	11039.4819	11025.4341
	QEP	10000	10743.6641	10711.1526	10653.6633
	CLQIEA	10000	11028.6620	11021.6642	11008.6656
	PEQIEA	271	11043.6658	11043.6656	11043.6650
2400	QIEA	10000	14699.7198	14679.7188	14649.7187	GS = 14746.7553
	HQIEA	10000	14579.6499	14546.8915	14494.5388	UL = 14749.7202
	IQIEA	10000	14403.3049	14376.8923	14344.2411
	FQIEA	9699	14749.6244	14739.5228	14723.7171
	QEP	10000	14274.7198	14206.6810	14164.7197
	CLQIEA	10000	14709.7199	14703.7185	14684.7152
	PEQIEA	353	14749.7201	14749.7197	14749.7184
3000	QIEA	10000	18301.2696	18280.2692	18246.2699	GS = 18374.7172
	HQIEA	10000	18061.2126	18031.8713	17984.6050	UL = 18381.2708
	IQIEA	10000	17816.0103	17777.0292	17736.2377
	FQIEA	10000	18379.8046	18370.6711	18357.2014
	QEP	10000	17721.2648	17628.2355	17570.9420
	CLQIEA	10000	18321.2701	18310.7693	18301.2676
	PEQIEA	324	18381.2708	18381.2707	18381.2705

wherein GS is a max value obtained by a greedy method, and UL is an ideal limit.

Referring to Table 2, values obtained by PEQIEA of the present invention are better than solutions in any other cases, while stability is also greater.

Claims

1-4. (canceled)

5. A quantum evolution method, comprising steps of: (1) according to the quantum evolution method, initializing a generation number t=0, and initializing a population Q(t)={q1t, q2t, . . . , qnt}, wherein n is a population size, t is the generation number, qit is a No. i individual in a No. t generation, and i ∈[1, n]; defining

6. The quantum evolution method, as recited in claim 5, wherein in the step (3), for evaluating each xit with the evaluation function, all quantum bits of xit are added together, and a result thereof is inputted into F(t) as a fitness fit of xit.

7. The quantum evolution method, as recited in claim 5, wherein in the step (4.3), for deciding whether the individuals in P(t) are selected to enter the elite group E(t) by the roulette method based on S(t), the fitness fit of all the individuals in the No. t generation is calculated, then a fitness sum

8. The quantum evolution method, as recited in claim 6, wherein in the step (4.3), for deciding whether the individuals in P(t) are selected to enter the elite group E(t) by the roulette method based on S(t), the fitness fit of all the individuals in the No. t generation is calculated, then a fitness sum

9. The quantum evolution method, as recited in claim 5, wherein in the step (6), b in the optimal solution collection B(t) is the optimal solution of the No. t generation, and an updating process thereof is: during initializing, the optimal solution b is the string comprising m 0-characters; when the generation number t=0, the optimal solution obtained through the step (2) and the step (3) is surely better than the initial optimal solution; as a result, replacing the initial optimal solution by the optimal solution, and inputting in the optimal solution collection B(t) as the optimal solution b, so as to obtain a current generation optimal solution collection B(0); when the generation number t=1, repeating the step (2) and the step (3), comparing an obtained optimal solution with the optimal solution in B(0), wherein if the optimal solution when t=1 is better than the optimal solution in B(0), then the optimal solution when t=1 is inputted into B(t) as b, so as to obtain a current generation optimal solution collection B(1); if the optimal solution when t=1 is worse than the optimal solution in B(0), then the original optimal solution b in B(1) remains, so as to obtain B(1); when the generation number is t, comparing the optimal solution of the No. t generation with the optimal solution b in B(t−1), so as to obtain B(t).