Based on the concepts and principles of quantum computing, a novel immune clonal algorithm, called a quantum-inspired immune clonal algorithm (QICA), ... Quantum computing - Acceleration - Convergence - Genetic mutations - Radiative recombination - Benchmark testing - Genetic algorithms - Computational efficiency - Multiuser detection - Multiaccess communication - convergence - optimisation - quantum computing - search problems - quantum-inspired immune clonal algorithm - global optimization - quantum computing - multistate gene quantum bit representation - quantum rotation gate strategy - dynamic adjusting angle mechanism - quantum NOT gate - quantum mutation - quantum recombination - search efficiency - global optimum convergence - genetic algorithm - Genetic algorithms (GAs) - global optimization - immune clonal algorithm - multiuser detection (MUD) - quantum computing - Artificial Intelligence - Biomimetics - Computer Simulation - Humans - Models, Biological - Pattern Recognition, Automated - Quantum Theory - Reinforcement (Psychology) - Genetic algorithms (GAs) - global optimization - immune clonal algorithm - multiuser detection (MUD) - quantum computing

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 5, OCTOBER 2008

Quantum-Inspired Immune Clonal Algorithm for Global Optimization Licheng Jiao, Senior Member, IEEE, Yangyang Li, Member, IEEE, Maoguo Gong, Member, IEEE, and Xiangrong Zhang, Member, IEEE

Abstract—Based on the concepts and principles of quantum computing, a novel immune clonal algorithm, called a quantum-inspired immune clonal algorithm (QICA), is proposed to deal with the problem of global optimization. In QICA, the antibody is proliferated and divided into a set of subpopulation groups. The antibodies in a subpopulation group are represented by multistate gene quantum bits. In the antibody’s updating, the general quantum rotation gate strategy and the dynamic adjusting angle mechanism are applied to accelerate convergence. The quantum NOT gate is used to realize quantum mutation to avoid premature convergences. The proposed quantum recombination realizes the information communication between subpopulation groups to improve the search efficiency. Theoretical analysis proves that QICA converges to the global optimum. In the first part of the experiments, 10 unconstrained and 13 constrained benchmark functions are used to test the performance of QICA. The results show that QICA performs much better than the other improved genetic algorithms in terms of the quality of solution and computational cost. In the second part of the experiments, QICA is applied to a practical problem (i.e., multiuser detection in direct-sequence code-division multiple-access systems) with a satisfying result. Index Terms—Genetic algorithms (GAs), global optimization, immune clonal algorithm, multiuser detection (MUD), quantum computing.

I. I NTRODUCTION

L

IFE phenomena and biological-inspired systems have always attracted attention from the scholars of artificial intelligence. This is particularly evident in the last few years, during which there has been an enormous increase of interest in studying biologically inspired systems. Among these, artificial neural networks, evolutionary computation, DNA computation, and now artificial immune systems (AIS) are perhaps the most popular systems. The immune system is a complex of cells, molecules, and organs that has proven to be capable

Manuscript received January 15, 2007; revised July 19, 2007, January 3, 2008, and May 1, 2008. This work was supported in part by the National Natural Science Foundation of China under Grant 60703107, Grant 60703108, and Grant 60736043, by the Provincial Natural Science Foundation of Shaanxi of China under Grant 2007F32, by the National High Technology Research and Development Program (863 Program) of China under Grant 2006AA01Z107 and Grant 2007AA01Z307, by the National Basic Research Program (973 Program) of China under Grant 2006CB705700, by the Program for Cheung Kong Scholars and Innovative Research Team in University (PCSIRT) under Grant IRT0645, by the National Research Foundation for the Doctoral Program of Higher Education of China under Grant 20070701022, and by the Key Project of Ministry of Education of China under Grant 108115. This paper was recommended by Associate Editor C.-T. Lin. The authors are with the Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China, Institute of Intelligent Information Processing, Xidian University, Xi’an 710071, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2008.927271

of performing several tasks, like pattern recognition, learning, memory acquisition, generation of diversity, noise tolerance, generalization, distributed detection, and optimization. Based on immunological principles, new computational techniques are being developed, aiming not only at a better understanding of the system but also at solving engineering problems [1]–[9]. The clonal selection theory [2] is used in the immune system to describe the basic features of an immune response to an antigenic stimulus. It establishes the idea that only those cells that recognize the antigens proliferate, thus being selected against those that do not [2]. Based on the clonal selection theory, some new optimization algorithms have been proposed [5]–[9]. An antibody clonal operator was first introduced in [5]. In [6]–[9], the improved immune clonal algorithms and their applications were proposed. The clonal selection functioning of the above optimization algorithms can be interpreted as a remarkable microcosm of Darwin’s theory of evolution, with three major principles of repertoire diversity, genetic variation, and natural selection [9]. Like other genetic algorithms (GAs), immune clonal algorithm is characterized by the representation of the antibody, the evaluation function, and the population dynamics such as population size, immune genic operators, and clonal selection. To have a good balance between exploration and exploitation, these components should properly be designed. In particular, in this paper, the representation and immune genic operator are investigated to effectively represent the antibodies to explore the search space with a small number of antibodies, and to exploit the global optimal solution in the search space within a short span of time, respectively. For these purposes, some concepts of quantum computing are adopted in the proposed immune clonal algorithm. Quantum mechanics is one of the greatest achievements in the 20th century. Quantum computing is a research area that includes concepts like quantum mechanical computer (QC) and quantum algorithms. Quantum mechanical computers were proposed in the early 1980s [10], and the description of quantum mechanical computers was formalized in the late 1980s [11]. Many efforts on quantum computers have actively progressed since the early 1990s because these computers were shown to be more powerful than classical computers on various specialized problems. The well-known quantum algorithms include the Deutsch–Jozsa algorithm [12], Simon’s algorithm [13], Shor’s quantum factoring algorithm [14], [15], and Grover’s database search algorithm [16], [17]. Research on merging evolutionary computing and quantum computing has been started since the late 1990s. They can be classified into two respects. One concentrates on designing new quantum algorithms using automatic programming techniques such as

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JIAO et al.: QUANTUM-INSPIRED IMMUNE CLONAL ALGORITHM FOR GLOBAL OPTIMIZATION

genetic programming [18]–[20]. The other concentrates on quantum-inspired evolutionary computing for a classical computer, i.e., a branch of study on evolutionary computing that is characterized by certain principles of quantum mechanics such as uncertainty, superposition, interference, etc. [21]–[23]. In [21], a modified crossover operator that includes the concept of interference was introduced. Quantum-inspired genetic algorithms (QGAs) [22] and quantum-inspired evolutionary algorithms (QEAs) [23] were proposed to solve the combinatorial optimization problem, and the experimental results demonstrated that these algorithms were far more superior to conventional GAs (CGAs). In particular, in [23], the improved QEA was proposed to deal with the numerical optimization problem. However, the quantum-rotated gates of these algorithms [21]–[23] are not general. In addition, because there are plenty of multivariable continuous function optimization problems found in various applications, it is very important and necessary to explore new quantum evolutionary algorithms. With no connection to quantum computing, a number of evolutionary algorithms that guide the exploration of the search space by building probabilistic models of promising solutions found have been introduced since the late 1990s [22]. These algorithms have shown to perform well on a variety of problems. In population-based incremental learning (PBIL), which is a method of combining the mechanisms of a generational GA with simple competitive learning, the solutions are represented by binary strings, and the population of solutions is replaced with a probability vector [22]. The compact GA replaces the population with a single probability vector as in PBIL; however, its modification method of the probability vector is different from PBIL [22]. The class of GAs that employ learning and sampling of probability distributions is usually referred to as estimation of distribution algorithms (EDAs) [24]. EDAs are a class of novel stochastic optimization algorithms that have recently become a hot topic in the field of evolutionary computation. EDAs acquire solutions by statistically learning and sampling the probability distribution of the best individuals of the population at each iteration of the algorithm. EDAs have introduced a new paradigm of evolutionary computation without using conventional evolutionary operators such as crossover and mutation. In such a way, the relationships between the variables involved in the problem domain are explicitly and effectively exploited. This paper proposes a novel immune clonal algorithm called a quantum-inspired immune clonal algorithm (QICA), which is based on merging quantum computing and clonal selection theory. Unlike other research areas, there has been relatively little work done in applying quantum computing to AIS. In [25], quantum-inspired computing was proposed. QICA was first introduced in [8] for solving the high-dimensional function optimization problems. It should be noted that although QICA is based on the concepts of quantum computing, QICA is not a quantum algorithm, but a novel optimization algorithm for a classical computer [8]. There are three innovation points listed as follows. 1) The antibody is proliferated and divided into a set of subpopulation groups. Antibodies in a subpopulation group are represented by multistate gene quantum bits. The quantum bit representation has the advantage to repre-

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sent a linear superposition of states (classical solutions) in search space probabilistically. Thus, the quantum bit representation has a better characteristic of population diversity than other representations. 2) In the antibody’s updating, the general quantum rotation gate strategy and the dynamic adjusting angle mechanism are applied to accelerate convergence. The quantum NOT gate is used to realize quantum mutation to avoid premature convergences. Each subpopulation group independently evolves, which enlarges the search space. 3) The proposed quantum recombination realizes the information communication between subpopulation groups to improve the search efficiency. This paper is organized as follows. Section II describes QICA and analyzes its convergence. Sections III and V present the experimental studies on the problems of global numerical optimization and the optimal multiuser detection (MUD) for directsequence code-division multiple-access (DS-CDMA) systems, respectively. Section IV analyzes the parameters of QICA. Finally, concluding remarks are presented in Section VI.

II. QICA Before describing QICA, some basics of quantum computing are addressed first. The smallest unit of information stored in a two-state QC is called a quantum bit or qubit [26]. A qubit may be at the “1” state, signed as the state vector |1, at the “0” state, signed as the state vector |0, or at a state of superposition of the above two. This state of a qubit can be represented as |ϕ = α|0 + β|1

(1)

where α and β are complex numbers that specify the probability amplitudes of the corresponding states, satisfying |α|2 + |β|2 = 1.

(2)

|α|2 gives the probability that the qubit will be found at the “0” state, and |β|2 gives the probability that the qubit will be found at the “1” state. In other words, |ϕ is a unit vector in 2-D complex vector space for which a particular basis has been fixed. One of the simplest physical examples of a qubit is the spin 1/2 of an electron. The spin-up and spin-down states of an electron can be taken as the states |0, |1 of a qubit. The state of a qubit can be changed by the operation with a quantum gate. A quantum gate is a reversible gate and can be represented as a unitary operator U acting on the qubit basis states satisfying U + U = U U + , where U + is the Hermitian adjoint of U . There are several quantum gates, such as the NOT gate, controlled NOT gate, rotation gate, Hadamard gate, etc. [26]. If there is a system of n-qubits, the system can represent 2n state at the same time, and therefore, quantum computers are shown to be more powerful than classical computers on various specialized problems. However, the QC has not walked out of the laboratory up to now due to the difficulty of its physical realization [27]. In this paper, QICA is a novel optimization algorithm for a classical computer. The definitions of these elements for global optimization problems are first described.

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A. Problem Definition A global optimization can be formulated as solving the following objective function: minimize f (x),

x = (x1 , . . . , xm ) ∈ S ∩ F

(3)

where f (x) is an objective function, and S ⊆ Rm defines the search space, which is an m-dimensional space bounded by the parametric constraints xi ≤ xi ≤ xi , i = 1, . . . , m, in which xi is lower bound and xi is upper bound, and a feasible region F is defined by F = {x ∈ Rm |gj (x) ≤ 0

∀j ∈ {1, . . . , N }}

(4)

where gj (x) ∀j ∈ {1, . . . , N } are constraints. In this paper, we transform a constrained optimization problem into an unconstrained one by the penalty term, such as the one given by ψ(x) = f (x) + A

N

max {0, gj (x)}

(5)

j=1

where A is the static penalty coefficient. An antibody for numerical optimization problems is defined as follows. Definition 1: An antibody a represents a candidate solution to the optimization problem in hand. The value of its affinity is equal to the negative value of the objective function a∈S

D(a) = −f (a).

Fig. 1.

QICA.

where |αl |2 + |βl |2 = 1 (l = 1, 2, . . . , m). So the phase of the lth qubit is ωl = arctan(βl /αl ).

(11)

C. Immune Clonal Algorithm The immune clonal algorithm used in this paper is an antibody random map induced by the affinity, including three steps, i.e., clone operator, immune genic operator, and clonal selection operator. The state transfer of the antibody population is denoted as clone operator A(t) −−−−−−−−−−−−→ A (t)

(6)

immune genic operator −−−−−−−−−−−−−−−−−−−→ A

(t) clonal selection operator −−−−−−−−−−−−−−−−−−−−→ A(t + 1).

B. Representation In AIS, a number of different representations can be used to encode the solutions as antibodies. The representation can generally be classified as binary, numeric, and symbolic. QICA uses a new representation, i.e., a qubit for the probabilistic representation. It is based on the concept of qubits, and a qubit antibody is a string of qubits, which are defined below. Definition 2: The probability amplitude of one qubit is defined with a pair of numbers (α, β) as α (7) β where α and β satisfy (2). Definition 3: The phase of a qubit is defined with an angle ω as ω = arctan(β/α)

(8)

and the product α ∗ β is represented with the symbol d, i.e., d=α∗β

(9)

where d stands for the quadrant of qubit phase ω. If d is positive, the phase ω lies in the first or third quadrant; otherwise, the phase ω lies in the second or fourth quadrant. Definition 4: A qubit antibody as a string of m qubits is defined as α1 α2 . . . αm (10) β1 β2 . . . βm

According to the affinity function, a point ai (t) that represents the ith antibody in the antibody population A(t) will be copied into Ci same points a i (t) ∈ A (t) in the solution space by using the clone operator. A new antibody population A(t + 1) is attained after performing the immune genic operator and the clonal selection operator. The immune genic operator can include the mutation and recombination operation. It is found that the immune clonal algorithm obtains a good local searching ability at a cost of adding the scale of the population by the clone operator, and the characteristics of population diversity and selective pressure are not easy to be implemented in the immune clonal algorithm [5]. In contrast, a qubit representation with a stochastic characteristic is adopted in QICA, and the corresponding immune genic operation is associated with the probability amplitudes of basic quantum states. Therefore, the selective pressure problem can be avoided, namely, the proposed QICA has automatic balance ability between exploration and exploitation, and the diversity of the population can be maintained in the process of searching optimal. D. QICA Fig. 1 lists the major steps in the proposed algorithm, in which Q(t), P (t), and B(t) denote the antibody population based on qubit at the tth generation, the antibody population based on classical bit at the tth generation, and the best antibody population based on classical bit in the tth generation’s subpopulation, respectively.

JIAO et al.: QUANTUM-INSPIRED IMMUNE CLONAL ALGORITHM FOR GLOBAL OPTIMIZATION

The major elements of QICA are presented as follows. 1) Quantum Population: QICA maintains a quantum population Q(t) = {q1t , q2t , . . . , qnt } at the tth generation, where n is the size of the population, and m is the length of the qubit antibody qit , which is defined as t t α1 α2t . . . αm qit = , i = 1, 2, . . . , n. t β1t β2t . . . βm

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TABLE I LOOKUP TABLE OF F (αL , βL )

At step 1, all αlt and βlt of qit (l = 1, 2, . . . , m; t = 0) are randomly generated between −1 and 1, satisfying |αlt |2 +|βlt |2 = 1. 2) Clonal Operator: The clonal operator Θ is defined as Θ (Q(t)) = [ Θ(q1 ) Θ(q2 ) . . .

Θ(qn ) ]T

(12)

where Θ(qi ) = Ii qi , i = 1, . . . , n, and Ii are the identity matrices of dimensionality Ci . Generally, Ci is given by ⎤ ⎡ ⎢ D(qi ) ⎥ ⎥, Ci = ⎢ n ⎥ ⎢Nc ∗ ⎥ ⎢ D(q ) i ⎥ ⎢

i = 1, 2, . . . , n

(13)

j=1

and can self-adaptively be adjusted by the affinity D(∗ ). Nc is a given value relating to the clone scale. After the clone, the population becomes Q (t) = {Q(t), q1 (t), . . . , qn (t)}

(14)

qi (t) = {qi1 (t), qi2 (t), . . . , qiCi −1 (t)} qij (t) = qi (t), j = 1, 2, . . . , Ci − 1.

(15)

where

3) Immune Genic Operator: We adopt the quantum mutation and recombination operation as the immune genic operator. Quantum mutation is first used at the inner subpopulation, and information between the subpopulation is exchanged by adopting the quantum recombination. Quantum Rotation Gate Strategy: In quantum theory, the transform of states is fulfilled by the quantum transformation matrix. For example, the updating operator can be denoted by the quantum rotation gates. In addition, the best antibody in the current generation is important, i.e., the communication among antibodies is needed to speed up the convergence. Quantum mutation focuses on a wise guidance by the current best antibody in the subpopulation. A quantum rotation gate U is

cos(θ) − sin(θ) U (θ) = (16) sin(θ) cos(θ) where θ is the rotate angle that controls the convergence speed, and θ is defined as θ = k ∗ f (αl , βl )

(17)

where k is a coefficient determining the speed of convergence. If k is too big, the search grid of the algorithm is large, and the solutions may diverge or have a premature convergence to a local optimum. On the other hand, if it is too small, the search grid of the algorithm is also small, and the algorithm may stay at a stagnant state. Here, k is defined as a variable that is relative to the clone scale. The better the antibody quality, the smaller the mutating size, and thus the local search is more advantageous.

For example, k = 10 ∗ exp(−Ci /Nc ), where Ci is the clone scale, and Nc , a constant, is the clonal size. The function f (αl , βl ) determines the search direction. The lookup table (Table I) can be used as a strategy for convergence. It should be indicated that it is a general way of mutation for different problems. Why can it converge to a better solution? For example, when d1 > 0 and d2 > 0, it means that the best solution and the current solution are in the first or third quadrant, and when |ω1 | > |ω2 |, θ should rotate counterclockwise, so f (αl , βl ) = 1, and else f (αl , βl ) = −1. The update procedure can be described as

t t αl t αl ∗ = U θ . (18) l βl

t βl t Quantum NOT Gate: Quantum NOT gate is used to realize quantum mutation and to avoid premature convergences. According to the mutation probability pm , each qubit of the updated quantum subpopulation by quantum rotation gate is mutated as 1 − |Q

i (t)|2 , i = 1, 2, . . . , m. (19) Q

i (t) = Equation (19) states that the NOT gate changes the probability of the 1 (or 0) state to that of the 0 (or 1) state with probability pm . Quantum Recombination: In order to construct more robust crossover operators, we introduce quantum recombination, i.e., All Interference Crossover [21], in light of the quantum interference, which allows more antibodies involve in the recombination. For easy comprehension, an example is given. Let the population size be 5 and the antibody length be 8, as described in Table II. After quantum recombination with probability pr , the new population is shown in Table III. All Interference Crossover occurs as follows: take the first element of antibody 1, take the second element of antibody 2, take the third element of antibody 3, take the fourth element of antibody 4, etc. This kind of recombination represents the antibodies’ participation in the operation, which aims to overcome the locality of its classical version. It comes from quantum theory and can be used to prevent prematurity. If two antibodies are the same, the conventional recombination will not work, whereas this new recombination is able to generate new antibodies to

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TABLE II BEFORE

TABLE III AFTER

impel the evolution. In the worst case, when all the antibodies in the population are identical, an All Interference Crossover is used. It can be described as follows. Randomly generate an integer v in the range [1, m] (m is the length of the antibody) as the crossover point. Then, the vth bit of the first antibody is selected as the first bit of the new antibody, continuing this operation on the antibodies in turn until a string of length m, i.e., a new antibody, is obtained. It proves to be more powerful against prematurity in practical experiments. To summarize, the quantum mutation can guide the quantum antibody in the subpopulation to evolve to a better antibody with a larger probability. The quantum recombination is used on the classical population B(t) to improve the search efficiency. 4) Observing Operator: At Step4, in the act of observing a quantum state, it collapses to a single state (namely, classical representation). We introduce the following two kinds of observing method. 1) For the binary coding problem, we observe the updated Q (t), which is signed as Q

(t), and produce the binary string population P (t) = {xt1 , xt2 , . . . , xtn }, where xti (i = 1, 2, . . . , n) is the numeric string of length m derived from the amplitude αl

t or βl

t (l = 1, . . . , m). The process is described as follows. a) Generate a random number p ∈ [0, 1]. b) If it is larger than |αl

t |2 , the corresponding bit in P (t) takes “1”; otherwise, it takes “0.” 2) For the nonbinary coding problem, we first convert the amplitude of the qubit antibody to the region of the given problem. In other words, we construct a new observation. We observe the updated Q (t) [namely, Q

(t)] and produce the numerical string population P (t) = {xt1 , xt2 , . . . , xtn } and xti (i = 1, 2, . . . , n), which are numeric strings of length m derived from αl

t or βl

t (l = 1, . . . , m). The process is described as follows. a) Generate a random number p ∈ [0, 1]. b) If it is larger than the given selected probability, the corresponding bit in P (t) takes (βl

t )2 ; otherwise, it takes (αl

t )2 . In fact, each bit that belongs to [0, 1] in antibody xti denotes an encoded variable, and only the corresponding bit in xti is mapped to the feasible solution space for converting from numeric

strings to a real value. In other words, a quantum antibody of length m is equal to the number of variables. Then, similar to binary coding, we compare each qubit phase of antibody in the population with that of the current best one, and then make a modification of its appearing probability, which aims at the evolution toward the fitter antibody with a larger probability. 5) Clonal Selection Operator: Assuming we are searching the maximal value of the object function, the operation is given as follows. a) For ∀i = 1, 2, . . . , n, if there is a mutated and observed classical antibody bi and D(bi ) = max{D(xij )|j = 2, 3, . . . , Ci }, namely, D(bi ) > D(x i ), then bi replaces the antibody x i in the original population. b) Record the corresponding qubit of bi as the next generation population Q(t + 1) at the same time. Thus, the antibody population is updated, and the information exchange between the antibody population is realized. 6) Termination Condition: It is defined as ∗ |f − f best | < ε|f ∗ |, f ∗ = 0 (20) f∗ = 0 |f best | < ε, where f best and f ∗ represent the best solution found in the current generation and the global optimum, respectively. ε is the acceptable error. The termination condition is dependent on both (20) and the given number of generation. Fig. 2 shows the overall structure of QICA. 7) Convergence of QICA: Definition 5: Assume that the size of a population is n, and X is a searching space to which all the antibodies belong. Let Xt = (x1 , x2 , . . . , xn ) be the population at the tth generation, D be the affinity function on X, and f ∗ be the global optimum. Let M = X|D(X) = max {D(xi ), i ≤ n} (21) xi ∈Xt

where M is called the satisfied set of population Xt . Definition 6: Let Dt = maxxi ∈Xt {D(xi ) : i = 1, 2, . . . , n}. For any initial distribution, if the following equation holds: lim P {Dt = f ∗ } = 1

t→∞

(22)

where P denotes the probability, then we say that the algorithm is convergent with probability 1. This definition of convergence means that if the algorithm implements for enough iteration, then the probability with which the population contains the optimal antibody will verge on 1. Lemma 1: The population series of QICA {Xt , t ≥ 0} is a finite homogeneous Markov chain. Proof: Like the evolutionary algorithms, the state transfer of QICA is processed on finite space. Therefore, the population is finite since Xt+1 = T (Xt ) = Ts ◦ Tg ◦ Θ(Xt ).

(23)

Ts , Tg , and Θ denote the clonal selection operator, the immune genic operator, and the clone operator, respectively. Note that Ts , Tg , and Θ have no relation with t, and Xt+1 is only related

JIAO et al.: QUANTUM-INSPIRED IMMUNE CLONAL ALGORITHM FOR GLOBAL OPTIMIZATION

Fig. 2.

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Overall structure of QICA.

to Xt . Therefore, {Xt , t ≥ 0} is a finite homogeneous Markov chain. Lemma 2: The M Markov chain of QICA is monotonically increasing, namely, ∀t ≥ 0, D(Xt+1 ) ≥ D(Xt ). Proof: It is apparent that the antibody of QICA does not degenerate since holding the best strategy is used in the algorithm. Theorem 1: The QICA is convergent. Proof: We consider the population with size n as a point in the state space S = X n , in which each coordinate is an antibody in X. Let |S| denote the number of states in S. si ∈ S (i = 1, 2, . . . , |S|) is a certain state si = {x1 , x2 , . . . , xn }. Xti is a random variable (i.e., the population) at state si of the tth generation. Let pij (t) be the probability of transition from Xti j to Xt+1 , i.e., j pij (t) = P Xt+1 /Xti .

(24)

Let pi (t) = P {Xt = si }, where P stands for the probability, and pt = i∈I pi (t). According to Lemma 1, we have pt+1 =

pi (t)pij (t)

si ∈S j∈I

=

pi (t)pij (t) +

pi (t)pij (t).

(27)

pi (t)pij (t)+ pi (t)pij (t) = pi (t) = pt

(28)

i∈I j∈I

i∈I j∈I

Because i∈I j∈I

i∈I j∈I

we have

i∈I

pi (t)pij (t) = pt −

(29)

pi (t)pij (t) + pt = pt .

(30)

i∈I j∈I

According to (25)–(29), we can obtain

1) If i ∈ I, j ∈ I, according to Lemma 1, then pij (t) = 0.

pi (t)pij (t).

i∈I j∈I

Suppose that I = {i|si ∩ s∗ = ∅}. Two special cases will first be discussed.

0 ≤ pt+1 <

i∈I j∈I

(25) Therefore

j )> 2) If i ∈ I, j ∈ I, according to Lemma 2, then D(Xt+1 i D(Xt ). Therefore

pij (t) > 0.

(26)

Next, we discuss the general case except for the above two cases.

lim pt = 0.

(31)

t→∞

Because lim P {Dt = f ∗ } = 1− lim

t→∞

t→∞

i∈I

pi (t) = 1− lim pt t→∞

(32)

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TABLE IV BASIC CHARACTERISTICS OF TEST FUNCTIONS

the following control experiment. We execute a standard immune clonal algorithm (SICA) to solve the above test functions with the following parameter values or scheme. 1) For QICA, the size of the initial antibody population is 10, the clonal size Nc is 30, the recombination probability pr = 1, and the mutation probability pm = 0.5. The number of qubits for the ten test functions is m. The observing operator is set to the observing method 2). 2) SICA does not apply qubit antibody design and general quantum rotation gate strategy. Gaussian mutation and quantum recombination are used on the classical numeric representation. The used parameters and the termination criterion are the same as those of QICA.

Since QICA is compared with quantum-inspired evolutionary algorithm (QEA) [23], orthogonal genetic algorithm with quantization (OGA/Q) [34], fast evolutionary programming (FEP) [35], covariance matrix adaptation evolution strategy (CMA-ES) [36], breeder genetic algorithm (BGA) [37], and adaptive evolutionary algorithm (AEA) [38] in the following experiments, we first give a brief description of the six algorithms.

We execute the QICA to solve the unconstrained test functions f1 −f10 (see Appendix A). Table IV lists the basic characteristics of these test functions. When m > 30, they appear to be one of the most difficult problems for many optimization algorithms including evolutionary programming, differential evolution, Tabu Search, and Ant Colony Optimization [28]–[33], where m is the dimension. They are all high-dimensional problems, in which f1 −f4 are unimodal functions, f5 is a noisy quartic function [where random[0, 1) is a uniformly distributed random variable in [0, 1)], and f6 −f10 are multimodal functions, where the number of local minima increases with the problem dimension. For example, the number of local minima of f6 is about 10 m in the given search space. The above test functions were examined in [23] and [34]–[38]. In this paper, we execute QICA to solve these functions with the following dimensions. 1) The problem dimension for f10 is 100, and the problem dimension for the rest functions is 30. In this case, these test functions have so many local minima that they are challenging enough for performance evaluation, and the existing results reported in [23], [34], and [35] can be used for direct comparison. 2) Because the size of the search space and the number of local minima increase with the problem dimension, the higher the dimension, the more difficult the problem. Therefore, this experiment studies the performance of QICA for functions f1 −f10 with 20–1000 dimensions. In this case, the existing results reported in [37] and [38] can be used for direct comparison. 1) Compared Algorithms and Parameter Settings: To identify the improvement due to the qubit antibody design and the general quantum rotation gate strategy, we design and carry out

1) QEA [23]: This is an improved GA based on merging quantum theory with evolutionary algorithm. It is different from the CGA in encoding. QEA adopts the quantum Hε gate as the mutation operator. 2) OGA/Q [34]: This is a modified version of CGA. It is the same as CGA except that it uses the orthogonal design to generate the initial population and the offspring of the crossover operator. 3) FEP [35]: This is a modified version of the classical evolutionary programming (CEP). It is different from CEP in generating new individuals. FEP uses a Cauchy instead of Gaussian mutation as the primary search operator. 4) CMA-ES [36]: The basic idea of CMA-ES is to deterministically use the most recent descent direction, i.e., the direction between the best parents at two consecutive generations. The covariance matrix C is gradually updated with rank-one matrices whose unique nonzero eigenvalue has the last descent direction as the eigenvector. 5) BGA [37]: It is based on artificial selection similar to that used by human breeders and is a recombination of evolution strategies (ES) and GAs. BGA uses truncation selection as performed by breeders. This selection scheme is similar to the (μ, λ) strategy in ES. The search process of BGA is mainly driven by recombination, making BGA a GA. Thus, BGA can be described by (Pg0 , N, T, Γ, Δ, HC, F, term), where Pg0 is the initial population, N is the size of the population, T is the truncation threshold, Γ is the recombination operator, Δ is the mutation operator, HC is the hill-climbing method, F is the fitness function, and term is the termination criterion. 6) AEA [38]: This is a modified version of BGA. In addition to the new recombination operator and the mutation operator, each individual of AEA is coded as a vector with all components in the unit interval, and inversion is applied with some probability to the parents before recombination is performed.

we have lim P {Dt = f ∗ } = 1.

t→∞

This implies that QICA converges to the global optimum.

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III. E XPERIMENTAL S TUDIES ON G LOBAL N UMERICAL O PTIMIZATION A. Unconstrained Optimization

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TABLE V PERFORMANCE OF QICA

TABLE VI COMPARISON BETWEEN QICA AND SICA ON THE FUNCTIONS

2) Results and Evaluations: Comparison of QICA, SICA, QEA, OGA/Q, FEP, and CMA-ES on Functions With 30 and 100 Dimensions: QEA [23], OGA/Q [34], FEP [35], and CMA-ES [36] are recently proposed methods with good performance on numerical optimization problems. In [34], OGA/Q is compared with CGAs and five existing algorithms, and the results showed that OGA/Q outperformed all the other methods. In [35], the termination criterion of FEP was set to be 1500 generations for f1 and f9 , 2000 generations for f2 and f8 , 3000 generations for f5 , 5000 generations for f3 and f6 , and 9000 generations for f7 . In [23], QEA was tested on the above five test functions involving f1 and f6 −f7 , and the termination criterion was the same as FEP. In [34], the termination criterion of OGA/Q was that the quality of the solution cannot further be improved in the successive 50 generations after 1000 generations. Since the termination criteria of FEP and OGA/Q are different, to make a fair comparison, we compare both the qualities of their final solutions and the computational cost at

the same time. Accordingly, the termination criterion of QICA is that the quality of the solution cannot be further improved in successive 50 generations for each function. We use the termination criterion of CMA-ES in http://www.icos.ethz.ch/ software/evolutionary_computation/cma. The results averaged over 50 trials are shown in Tables V–X, where the dimensions of the test functions are 30 except for those of f10 being 100. We performed QICA on each test function and recorded 1) the mean number of function evaluations, 2) the best solution in the 50 trials, 3) the mean function value (i.e., the mean of the function values found in the 50 trials), 4) the standard deviation of function values, 5) the worst solution in the 50 trials, and 6) the mean running time in the 50 trials. All experiments are executed on a 2.4-GHz Pentium-IV PC with 1-GB RAM. Table V shows the performance of QICA. We find that the best solution and the mean function values even in the worst case are equal or close to the optimal ones, and the standard deviations of the function values are relatively small. In particular, the mean running time is between 0.1 and 0.2 s

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TABLE VII COMPARISON BETWEEN QICA AND QEA, WHERE THE RESULTS FOR QEA ARE OBTAINED FROM [23]

TABLE VIII COMPARISON BETWEEN QICA AND OGA/Q, WHERE THE RESULTS FOR OGA/Q ARE OBTAINED FROM [34]

TABLE IX COMPARISON BETWEEN QICA AND FEP, WHERE THE RESULTS FOR FEP ARE OBTAINED FROM [35]

except for f10 . The mean running time of f10 is 2.03 s because of its high cost of computation. Consider f7 as an example. The mean function value is −12569.49, which is close to the

global minimum −12569.5, whereas the standard deviation of function value is 1.035 × 10−10 only. These results indicate that QICA can find optimal or close-to-optimal solutions at

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TABLE X COMPARISON BETWEEN QICA AND CMA-ES1

TABLE XI MEAN NUMBER OF FUNCTION EVALUATIONS (THE STANDARD DEVIATIONS) OF QICA WITH 20–1000 DIMENSIONS

a low computational cost, and its solution quality is quite stable. Table VI shows the results of the control experiment. Recall that SICA is the same as QICA, except that it uses classical representation (numerical strings) and Gaussian mutation instead of qubit antibody and quantum mutation. It is found that SICA requires more function evaluations than QICA, and hence, SICA has a larger time cost. However, SICA gives larger mean function values than QICA under the same termination criterion, and hence, its mean solution quality is poorer. In addition, SICA gives larger standard deviations of function values than QICA, and hence, its solution quality is less stable. These results indicate that the qubit antibody design and the general quantum rotation gate strategy can effectively improve the performance of SICA. Table VII compares QICA with QEA. For f7 , the solutions of QICA are as good as those of QEA, and for the other functions, the solutions of QICA are better than those of QEA. Moreover, the mean number of function evaluations of QICA is about 10 000 for all functions, whereas that of QEA ranges from 100 000 to 900 000. It indicates that the improved quantum rotation gate strategy is applied to the complex numerical optimization problems. Tables VIII–X compare QICA with OGA/Q, FEP, and CMAES. As can be seen, QICA finds the exact global optimum (0) in all trials for six out of ten functions. For all the test functions,

both the mean function value and the mean number of function evaluations of QICA are much better than those of FEP. For f9 , the solution of QICA is as good as those of OGA/Q, and for the other functions, the solutions of QICA are better than those of OGA/Q. Moreover, the mean number of function evaluations of QICA is about 20 000 for all functions, whereas that of OGA/Q ranges from 100 000 to 300 000. Therefore, the computational cost of QICA is much lower than that of OGA/Q. For f8 , the solutions of QICA are as good as those of CMAES, but the mean number of function evaluations of CMA-ES is lower than those of QICA, and for the other functions, the solutions of QICA are better than those of CMA-ES. To summarize, the results show that QICA outperforms FEP, OGA/Q, and CMA-ES, and is suitable for the numerical optimization problems. Performance of QICA on Functions With 20–1000 Dimensions: This experiment studies the performance of QICA on the ten test functions with 20–1000 dimensions. The termination criterion is that one of the objectives of (20) is achieved, where ε = 10−4 is used for all functions except for f10 , where ε is 10−2 . Because the global minimum of f10 is 99.6, ε = 10−2 is used. Table XI gives the mean number and the standard deviations of function evaluations of QICA over 50 trials. As can be seen, for f7 and f8 , when the dimension increases from 20 to 1000, the number of evaluations increases to about 20 000. For f10 , when the dimension increases to 1000, the

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TABLE XII MEAN NUMBER OF FUNCTION EVALUATIONS OF QICA BGA AND AEA ON F6 −F9 WITH 20–1000 DIMENSIONS

number of evaluations only increases to 6232, and the standard deviation is 3.9. For the other seven functions, QICA obtains high-quality solutions (ε = 10−4 ) only with thousands of evaluations at all selected dimensions. In addition, because ε is assigned to 10−4 in the termination criterion, small standard deviations are obtained for the test functions at all selected dimensions. BGA [37] and AEA [38] are also tested on f6 –f9 with dimension 20, 100, 200, 400, and 1000, respectively. We use some existing results gleaned from [37] and [38] for direct comparisons in Table XII. As can be seen, the number of evaluations of QICA is much smaller than that of BGA for all the four functions. For f6 , when m ≤ 200, the number of evaluations of QICA is greater than that of AEA, whereas when 400 ≤ m ≤ 1000, it is smaller than that of AEA. For f7 , when m ≤ 100, the number of evaluations of QICA is slightly greater than that of AEA, whereas when 200 ≤ m ≤ 1000, it is slightly smaller than that of AEA. For both f8 and f9 , the number of evaluations of QICA is much smaller than that of AEA at all dimensions. In general, QICA obtains better solutions ε = 10−4 at a lower computational cost than BGA and AEA, and displays good performance in solving high-dimensional problems. B. Constrained Optimization In order to further test the performance of QICA, 13 constrained benchmark functions have been used (see Appendix B). f22 is the harder version studied in [39], where the feasible region of the search space consists of 93 disjointed spheres with radius of 0.25. f13 , f15 , f21 , and f23 include the equality constraints, and all equality constraints have been converted into inequality constraints |h(x) − δ| ≤ 0 using the degree of violation δ = 104 . We transform a constrained optimization problem into an unconstrained one by (5), where A is

⎛

Af11 = 0.5 ⎝ Af16 = 5000 Af21 = 10

Af12 = 100 Af17 = 1000 Af22 = 100

given by (34), shown at the bottom of the page. For QICA, the size of the initial antibody population is 20, the clonal size Nc is 60, the recombination probability pr = 1, and the mutation probability pm = 0.5. The number of qubits for each of the 13 test functions is m, and the observing operator is set to the observing method 2). Runarsson–Yao (RY) [40] is a recently proposed method and obtains good performance on constrained optimization problems. In [40], the termination criterion of RY was set to be 175 generations for f22 and 1750 generations for the other 12 functions. Thus, to make a fair comparison, we compare both the qualities of their final solutions and the computational cost at the same. Accordingly, the termination criterion of QICA is that the quality of the solution cannot further be improved in the successive 50 generations for each function. The results averaged over 100 trials for QICA are shown in Table XIII, where all the solutions for QICA are the feasible solutions. Note that the results listed in Table XIII are the results averaged over 30 trials for RY. In Table XIII, we performed QICA on each test function and recorded the following: 1) the best solution in 100 trials (Best); 2) the mean function value of the 100 trials (Mean); 3) the standard deviation of function values (St. dev.); 4) the worst solution in the 100 trials (Worst); 5) the running time (Time); and 6) the mean number of function evaluations (Evaluations). All of the experiments are executed on a 2.4-GHz Pentium-IV PC with 1-GB RAM. Table XIII also compares QICA with RY. As can be seen, QICA and RY can find the exact global optimum in all the trials for f11 , f13 , f14 , f18 , f21 , and f22 . QICA gives a smaller standard deviation of function values than RY, and hence, QICA has a more stable solution quality. For f12 , f16 , f17 , f19 , f20 , and f23 , the solutions of QICA are much better than those of RY. Moreover, the mean number of function evaluations of QICA is about 20 000 or 60 000 for all the functions, whereas that of

Af13 = 105 Af18 = 1000 Af23 = 0.05

Af14 = 104 Af19 = 500

⎞ Af15 = 10 Af20 = 5 × 106 ⎠

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TABLE XIII COMPARISON BETWEEN QICA AND RY, WHERE THE RESULTS FOR RY ARE OBTAINED FROM [40]

RY is 350 000 except for f22 , where it is 35 000 (it is 25 637 for QICA). Therefore, the computational cost of QICA is much lower than that of RY. In particular, for all the test functions, the mean running time is between 1 and 2 s. For f15 , the best solution of QICA is as good as that of RY, and both are smaller than the global optimum because equality constraints are converted into inequality constraints, but the mean and worst solutions of QICA are better than those of RY. To summarize, the results listed in Table XIII show that QICA outperforms RY and is suitable for the numerical optimization problems. Microgenetic algorithm as a generalized hill-climbing operator for GA (GA-MGA) [41] is a recently proposed novel GA. In [41], GA-MGA is compared with 12 existing algorithms on f11 , f17 , f19 , f20 , and f23 . The results showed that GA-MGA outperforms all the other methods. We will use these existing results for direct comparison in Table XIV. As can be seen, QICA and GA-MGA can find the exact global optimum in all trials for f11 . For the other functions, including f17 , f19 , f20 , and f23 , the solutions of QICA are much better than those of GA-MGA. In particular, for f20 , the

percentage of feasible solutions supplied by GA-MGA is 90% and that of QICA is 100%. IV. A NALYSES OF P ARAMETERS IN QICA In this section, we want to test the sensitivity of the algorithm to the parameters. For QICA, the main parameters are listed as follows: the size of population n, the clonal size Nc , the mutation probability pm , and the recombination probability pr . The effects of Nc , pm , and pr are complicated. In this section, we carried out experiments for the unimodal function f1 , the multimodal function f6 , and the two constrained optimization problems, which are functions f14 and f23 . Here, we define the mean success ratio R as R = MS /M , where M is the number of independent runs for QICA, and MS is the number of runs of finding a satisfied solution in M runs. The satisfied solution is defined as the solution that satisfies (20). Obviously, the mean success ratio describes the performance of the algorithm; the higher the R value, the better the stability and the performance of the algorithm. In this

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TABLE XIV COMPARISON BETWEEN QICA AND GA-MGA, WHERE THE RESULTS FOR GA-MGA ARE OBTAINED FROM [41]

experiment, for f23 , let ε = 10−3 , and for the other functions, let ε = 10−5 .

A. Analysis of the Effect of the Clonal Size Nc For unconstrained optimization problems (f1 and f6 ), Nc is changed from 5 to 100, where the fixed parameters are as follows: n = 10, pm = 0.5, and pr = 1. For constrained optimization problems (f14 and f23 ), Nc is changed from 20 to 200, where the fixed parameters are as follows: n = 20, pm = 0.5, and pr = 1. For each size, we performed M = 100 runs of the algorithm. The results are shown in Fig. 3. Fig. 3 shows that for the multimodal function f6 and the constrained optimization problems (f14 and f23 ), the addition of Nc produces an improvement in the performance of the algorithm, but for the unimodal function f1 , the mean success ratio R is holding 1. Furthermore, we find from Fig. 3(a) that when Nc is bigger than 20, R is bigger than 0.8. Moreover, it is found from Fig. 3(b) that when Nc is bigger than 40, R is also bigger than 0.8 for the constrained optimization problems. In conclusion, the clonal size Nc influences the ability of local searching. When Nc is bigger than 2n, the mean success ratio can approach 1, which means that the algorithm is less sensitive to the clonal size. In addition, QICA performs well with a small population n in Section III. Therefore, when we choose Nc = 3n, this increment can gain the excellent performance and could not pay for the increased computational cost of QICA.

B. Analysis of the Effect of the Mutation Probability pm For unconstrained optimization problems, the fixed parameters are as follows: n = 10, Nc = 30, and pr = 1. For constrained optimization problems, the fixed parameters are as follows: n = 20, Nc = 60. and pr = 1. The results of QICA with pm changing from 0 to 1 for 100 runs are shown in Fig. 4(a). Fig. 4(a) shows that the addition of pm between 0 and 0.3 produces an improvement in the performance of the algorithm, and R is close to 1 between 0.3 and 0.8. As a result, we often choose pm ∈ [0.3, 0.8].

Fig. 3. Relation curve between the clonal size Nc and the mean success ratio R of QICA.

C. Analysis of the Effect of the Recombination Probability pr For unconstrained optimization problems, the fixed parameters are as follows: n = 10, Nc = 30, and pm = 0.5. For constrained optimization problems, the fixed parameters are as follows: n = 20, Nc = 60 and pm = 0.5. The results of

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capacity of CDMA systems, and, therefore, has gained significant research interests since the Optimal MUD (OMD) was proposed by Verdu [42]. Reference [43]–[45], respectively, proposed their multiuser detectors based on the BP Neural Network, Hopfield Neural Network, and GA. All of them can significantly reduce the computational complexity and get good performance. They provided new ideas and techniques for solving MUD. In this section, we apply QICA to solve MUD and named by quantum-inspired immune clonal MUD (QICAD). We present some simulation results and comparisons to demonstrate the advantage of our algorithm. The performance of QICAD is evaluated via computer simulations and compared with that of SICA [immune clonal MUD (ICAD)] and Optimal Multiuser Detector (OMUD) as well as with that of conventional matched filter detector (MFD) in asynchronous DS-CDMA systems. A. Problem Statements Consider a baseband digital DS-CDMA network with K active users operating with a coherent BPSK modulation format. The signal received at the output of the sensor is r(t) =

M −1 K

Ak bk (i)sk (t − iTb ) + n(t) = S(t, b) + n(t)

i=0 k=1

(35)

Fig. 4. Relation curve between probability and the mean success ratio R of QICA.

QICA with pr changing from 0 to 1 for 100 runs are shown in Fig. 4(b). Fig. 4(b) shows that the influence of pr on R is not significant in unimodal functions f1 , but the increasing of pr produces an improvement of R in the other three kinds of optimization problems. In conclusion, the more complicated the optimization problems, the more significant the effect of the recombination operator on the performance of the algorithm. Often, we set pr = 1. Although the choice of the parameter is different for different optimization functions, it is sure that we can find a high quality of solution by choosing appropriate parameters in a certain scope. Parameter analysis demonstrates that QICA has stable performance and high success ratio, and is insensitive to parameters. V. E XPERIMENTAL S TUDIES ON THE O PTIMAL MUD IN DS-CDMA In recent years, DS-CDMA systems have emerged as one of the prime multiple-access solutions for 3G. In the DS-CDMA framework, multiple-access interference (MAI) existing at the received signal creates “near–far” effects. MUD techniques can efficiently suppress MAI and substantially increase the

where n(t) is the additive white noise vector whose standard deviation is σ, Tb is the symbol interval, M is the packet length, Ak is the signal’s amplitude of the kth user, bk (m) is the mthcoded modulated symbol of the kth user and bk (m) ∈ {±1}, and sk (t) is the kth user’s signature sequence. The matched filter output corresponding to the mth bit of the kth user is given by ∞ r(t)sk (t − mTb − τk )dt.

yk (m) =

(36)

−∞

If we set y(m) = [y1 (m), y2 (m), . . . , yK (m)]T , b(m) = [b1 (m), b2 (m), . . . , bK (m)]T , A(m) = diag(A1, A2 , . . . , AK ), and R(q) = n(m) = [n1 (m), n2 (m), . . . , nK (m)]T , T +τk (ρkl (q))K×K , where ρkl (q) = τk sk (t − τk )sl (t + qT − τl )dt, then y = RAb + n

(37)

where y = [y(m), y(m + 1), . . . , y(m + M − 1)]T b = [b(m), b(m + 1), . . . , b(m + M − 1)]T A = diag (A(m), A(m + 1), . . . , A(m + M − 1)) n = [n(m), n(m + 1), . . . , n(m + M − 1)]T . The OMD produces an estimate for the information vector transmitted at the discrete-time instant m. In asynchronous

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Fig. 5. Simulation results. (a) Performance in “near–far” resistance. (b) Performance in eliminating the noise disturbance. (c) Performance in accommodating users. (d) Performance in accommodating packet length.

systems, it holds that ˆboptimal = arg

max

bk (m)∈{±1} 1≤k≤K,1≤m≤M

{2bT Ay − bT ARAb}.

(38)

Note that, if solved by the Viterbi algorithm, its computational complexity will exponentially increase with the number of users and the packet length. B. QICA for MUD Assuming that K active users share the same channel and the packet length is M , then the question (38) can be described as a combinatorial optimization problem as (39) max f (b) = 2bT Ay − bT ARAb, b ∈ I (1)

(1)

(1)

(M )

(M )

(M )

where b = {[b1 , b2 , . . . , bK ], . . . , [b1 , b2 , . . . , bK ]}, (m) bk ∈ {±1} are the variants to be optimized, and I denotes the antibody space. The value of affinity D(b) is equal to the value of the objective function f (b), and the set I n denotes the antibody population space as I n = {B : B = (b1 , b2 , . . . , bn ), bk ∈ I, 1 ≤ k ≤ n}

(40)

in which B = (b1 , b2 , . . . , bn ) is the antibody population, n is the size of the antibody population, and antibody bi = (1) (1) (1) (M ) (M ) (M ) {[b1i , b2i , . . . , bKi ], . . . , [b1i , b2i , . . . , bKi ]}. In this ex-

periment, the main operators of QICA are the same as those given in Section II, but observing that the operator is set to the observing method 1). The process is as follows. a) Generate a random number p ∈ [0, 1]. b) If it is larger than |αlt |2 , the corresponding bit in P (t) takes “1”; otherwise, it takes “−1.” C. Results and Comparison It is assumed that the number of users is K, and the packet length is M . Gold sequences of length 31 are used as code sequences. The signal-to-noise ratio of the kth user is SNRk = A2k /2 ∗ σ 2 , where σ = 1. For QICAD and ICAD, we will terminate the search at the Y th generation where Y = 2 × K × M . In the two algorithms above, the size of the initial population is 5, the clonal size is 10, pm = 0.3, and we only adopt quantum rotation gate and quantum NOT gate as the strategy of updating population. We take all the experiments on 10 000-bit signals. Our performance metric is the average bit error ratio (BER). All experiments are executed on a 2.4-GHz Pentium-IV PC with 1-GB RAM. A) In order to gain the results of OMUD, we assumed that K = 3, M = 3, and SNR = 10 dB. The first user is the desired user, whereas the other users are the disturbing users, and all users have the same power. The ratio of power between disturbing users and desired user denotes the ratio of “near–far.” The performance in “near–far” resistance of mentioned receivers is shown in Fig. 5(a).

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B) It is assumed that K = 10 and M = 10. All users have the same power, changing the value of SNR from −2 to 10 dB. The performance in eliminating the noise disturbance of the mentioned receivers is shown in Fig. 5(b). C) It is assumed that M = 10 and SNR = 10 dB, the number of users K is changed from 5 to 30, and all users have the same power. The performance in accommodating the users of the mentioned receivers is shown in Fig. 5(c). D) It is assumed that SNR = 10 dB, K = 10, the packet length is changed from 5 to 30, and all users have the same power. The performance in accommodating the packet length of the mentioned receivers is shown in Fig. 5(d). As shown in Fig. 5(a), the conventional detector produces the receivable estimate only when the powers of the users are close to each other. QICAD and ICAD are better than the conventional detector. However, ICAD’s performance is unacceptable when the powers of the disturbing users are much larger than that of the desired user. As expected, QICAD exhibits the best performance and seldom fails to produce the correct estimate for the transmitted symbols, so its performance is almost as good as that of OMD. When the cumulative BER is evaluated versus the value of the SNR of all the users, we can see, from Fig. 5(b), that the QICAD receiver achieves acceptable performance, whereas the performance of conventional detector and ICAD is very poor. When the number of users or the transmitted packet length is relatively large, the advantaged of QICAD are shown in Fig. 5(c) and (d). The simulations suggest that QICAD detector still performs quite well when K and M are relatively large. VI. C ONCLUSION This paper has proposed a novel immune clonal algorithm (QICA) inspired by the concept of quantum computing. Our objective was to apply quantum theory to enhance the immune clonal algorithm so that it could be more robust and statistically sound. In particular, a qubit antibody was defined as a string of quantum bits for probabilistic representation. Due to the novel representation, we put forward the quantum mutation operator that was used in the inner subpopulation to accelerate convergence of the immune clonal algorithm. Information between the subpopulation was exchanged by adopting the quantum recombination for improving the diversity of the population and avoiding premature convergence. As a result, the proposed QICA has the automatic balance ability between exploration and exploitation. We executed QICA to solve benchmark problems. The dimensions of these problems are 20–1000, and some of them have numerous local minima. The results show that QICA could find optimal or close-to-optimal solutions. In addition, parameter analysis demonstrated that QICA was insensitive to parameters and had stable performance and high success ratio. Furthermore, we applied QICA to solving the MUD problem. Monte Carlo simulations show that the QICA could significantly reduce the computational complexity and achieve better performance in eliminating MAI and “near–far” resistance over other algorithms such as SICA and the conventional detection. It greatly improved the system capacity in acceptable computational cost for practical implementation in

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CDMA systems. The application of QICA to other problems such as the multiobjective optimization problem deserves our further research. A PPENDIX A U NCONSTRAINED O PTIMIZATION P ROBLEMS The following unconstrained optimization functions were considered in this paper. A) Sphere Model m x2i . f1 = Minimize i=1

B) Schwefel’s Problem 2.22 m m |xi | + |xi | . f2 = Minimize i=1

i=1

C) Schwefel’s Problem 1.2 ⎛ ⎞2 ⎞ ⎛ m i ⎝ xj ⎠ ⎠ . f3 = Minimize ⎝ i=1

j=1

D) Rosenbrock’s Function m−1 f4 (x) = Minimize 100(x2i − xi+1 )2 + (1 − xi )2 . i=1

E) Quartic Function, i.e., Noise m 4 ixi + random[0, 1) . f5 (x) = Minimize i=1

F) Generalized Rastrigin’s Function m

f6 (x) = Minimize x2i − 10 cos(2πxi ) + 10 . i=1

G) Generalized Schwefel’s Problem 2.26 m f7 (x) = Minimize − xi sin |xi | . i=1

H) Generalized Griewank Function m m x2 xi i − cos √ + 1 . f8 (x) = Minimize 4000 i=1 i i=1 I) Ackley’s Function ⎛

! ⎞ " m "1 f9 (x) = Minimize ⎝−20 exp ⎝−0.2# x2 ⎠ m i=1 i − exp

⎛

m 1 cos(2πxi ) + 20 + exp(1) . m i=1

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J) Michalewicz’s Function m i × x2i 20 f10 = Minimize − sin(xi ) sin . π i=1

subject to g1 (x) = 85.334407 + 0.0056858x2 x5 + 0.0006262x1 x4 − 0.0022053x3 x5 − 92 ≤ 0 g2 (x) = 85.334407 − 0.0056858x2 x5 − 0.0006262x1 x4

A PPENDIX B C ONSTRAINED O PTIMIZATION P ROBLEMS The following constrained optimization functions were considered in this paper: 4 4 13 2 xi − 5 xi − xi f11 = Minimize 5 i=1

i=1

i=5

+ 0.0022053x3 x5 ≤ 0 g3 (x) = 80.51249 + 0.0071317x2 x5 + 0.0029955x1 x2 + 0.0021813x23 − 110 ≤ 0 g4 (x) = − 80.51249 − 0.0071317x2 x5 − 0.0029955x1 x2 − 0.0021813x23 + 90 ≤ 0 g5 (x) = 9.300961 + 0.0047026x3 x5 + 0.0012547x1 x3

subject to

+ 0.0019085x3 x4 − 25 ≤ 0 g1 (x) = 2x1 + 2x2 + x10 + x11 − 10 ≤ 0 g2 (x) = 2x1 + 2x3 + x10 + x12 − 10 ≤ 0 g3 (x) = 2x1 + 2x3 + x11 + x12 − 10 ≤ 0 g4 (x) = −8x1 + x10 ≤ 0 g5 (x) = −8x2 + x11 ≤ 0 g6 (x) = −8x3 + x12 ≤ 0 g7 (x) = −2x4 − x5 + x10 ≤ 0 g8 (x) = −2x6 − x7 + x11 ≤ 0 g9 (x) = −2x8 − x9 + x12 ≤ 0

where the bounds are 0 ≤ xi ≤ 1 (i = 1, . . . , 9), 0 ≤ xi ≤ 100 (i = 10, 11, 12), and 0 ≤ x13 ≤ 1. The global minimum value is −15 at x = (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1). In addition, we have %⎞ ⎛ % % m % m & % % ⎜ % cos4 (xi ) − 2 cos2 (xi ) %⎟ %⎟ ⎜ % i=1 i=1 % %⎟ ' f12 = Minimize ⎜ %⎟ ⎜− % m %⎠ ⎝ % 2 % % ix i % % i=1

g6 (x) = − 9.300961 − 0.0047026x3 x5 − 0.0012547x1 x3 − 0.0019085x3 x4 + 20 ≤ 0 where 78 ≤ x1 ≤ 102, 33 ≤ x2 ≤ 45, and 27 ≤ xi ≤ 45 (i = 3, 4, 5). The global minimum value is −30665.539 at x = (78, 33, 29.995256025682, 45, 36.775812905788). Then we have

f15 = Minimize 3x1 + 0.000001x31 + 2x2 + (0.000002/3)x32 subject to g1 (x) = −x4 + x3 − 0.55 ≤ 0 g2 (x) = −x3 + x4 − 0.55 ≤ 0 h3 (x) = 100 sin(−x3 − 0.25) + 1000 sin(−x4 − 0.25) + 894.8 − x1 = 0 h4 (x) = 100 sin(−x3 − 0.25) + 1000 sin(x3 − x4 − 0.25)

subject to g1 (x) = 0.75 − g2 (x) =

m

m

xi ≤ 0

i=1

+ 894.8 − x2 = 0 h5 (x) = 1000 sin(x4 − 0.25) + 1000 sin(x4 − x3 − 0.25)

xi − 7.5 m ≤ 0

+ 1294.8 = 0

i=1

where m = 20, and 0 ≤ xi ≤ 10 (i = 1, . . . , m). The global minimum is unknown. Moreover, we have m √ m xi f13 = Minimize − m i=1

subject to h(x) =

m

x2i − 1 = 0

where 0 ≤ x1 ≤ 1200, 0 ≤ x2 ≤ 1200, −0.55 ≤ x3 ≤ 0.55, and −0.55 ≤ x4 ≤ 0.55. The best known solution [39] is 5126.4981 at x = (679.9453, 1026.067, 0.1188764, −0.3962336). Next we have

f16 = Minimize (x1 − 10)3 + (x2 − 20)3 subject to

i=1

where m = 10 and 0 ≤ xi ≤ 1(i √ = 1, . . . , m). The global minimum value is −1 at x = 1/ m (i = 1, 2, . . . , m). Next we have f14 = Minimize 5.3578547x23 + 0.8356891x1 x5 + 37.293239x1 − 40792.141)

g1 (x) = −(x1 − 5)2 − (x2 − 5)2 + 100 ≤ 0 g2 (x) = (x1 − 6)2 + (x2 − 5)2 − 82.81 ≤ 0 where 13 ≤ x1 ≤ 100. and 0 ≤ x2 ≤ 100. The global minimum value is −6961. 81388 at x = (14.095, 0.84296). Then

JIAO et al.: QUANTUM-INSPIRED IMMUNE CLONAL ALGORITHM FOR GLOBAL OPTIMIZATION

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where −10 ≤ xi ≤ 10 (i = 1, 2, . . . , 7). The global minimum value is 680.6300573 at

we have

f17 = Minimize x21 + x22 + x1 x2 − 14x1 − 16x2

x = (2.330499, 1.951372, −0.4775414, 4.365726,

+ (x3 − 10)2 + 4(x4 − 5)2 + (x5 − 3)2 + 2(x6 − 1) + 2

5x27

+ 7(x8 − 11)

2

+ 2(x9 − 10)2 + (x10 − 7)2 + 45

− 0.6244870, 1.038131, 1.594227) f20 = Minimize(x1 + x2 + x3 ) subject to g1 (x) = −1 + 0.0025(x4 + x6 ) ≤ 0

subject to g1 (x) = −105 + 4x1 + 5x2 − 3x7 + 9x8 ≤ 0

g2 (x) = −1 + 0.0025(x5 + x7 − x4 ) ≤ 0

g2 (x) = 10x1 − 8x2 − 17x7 + 2x8 ≤ 0

g3 (x) = −1 + 0.01(x8 − x5 ) ≤ 0

g3 (x) = −8x1 + 2x2 + 5x9 − 2x10 − 12 ≤ 0

g4 (x) = −x1 x6 + 833.3252x4 + 100x1 − 83333.33 ≤ 0

g4 (x) = 3(x1 − 2)2 + 4(x2 − 3)2 + 2x23 − 7x4 − 120 ≤ 0

g5 (x) = −x2 x7 + 1250x5 + x2 x4 − 1250x4 ≤ 0

g5 (x) = 5x21 + 8x2 + (x3 − 6)2 − 2x4 − 40 ≤ 0

g6 (x) = −x3 x8 + 1250000 + x3 x5 − 2500x5 ≤ 0

g6 (x) = x21 + 2(x2 − 2)2 − 2x1 x2 + 14x5 − 6x6 ≤ 0 g7 (x) = 0.5(x1 − 8)2 + 2(x2 − 4)2 + 3x25 − x6 − 30 ≤ 0 g8 (x) = −3x1 + 6x2 + 12(x9 − 8)2 − 7x10 ≤ 0 where −10 ≤ xi ≤ 10(i = 1, 2, . . . , 10). The global minimum value is 24.3062091 at x = (2.171996, 2.363683, 8.773926, 5.095984,

where 100 ≤ x1 ≤ 10000, 1000 ≤ xi ≤ 10000 (i = 2, 3), and 10 ≤ xi ≤ 1000 (i = 4, 5, . . . , 8). The global minimum value is 7049.3307 at x = (579.3167, 1359.943, 5110.071, 182.0174, 295.5985, 217.9799, 286.4162, 395.5979)

= Minimize x21 + (x2 − 1)2

f21 subject to

0.9906548, 1.430547, 1.321644,

f18

9.828726, 8.280092, 8.375927) 3 sin (2πx1 ) sin(2πx2 ) = Minimize x31 (x1 + x2 )

subject to g1 (x) = x21 − x2 + 1 ≤ 0 g2 (x) = 1 − x1 + (x2 − 4)2 ≤ 0 where 0 ≤ x1 ≤ 10, and 0 ≤ x2 ≤ 10. The global minimum value is −0.095825 at x = (1.2279713, 4.2453733). Next we have f19 = Minimize (x1 − 10)2 + 5(x2 − 12)2 + x43

h(x) = x2 − x21 = 0 −1 ≤ x2 ≤ 1. The global minimum where −1 ≤ x1 ≤ 1, and √ value is 0.75 at x = (±1/ 2, 1/2). Next we have f22 = Minimize − 100 − (x1 − 5)2

− (x2 − 5)2 − (x3 − 5)2 /100 subject to g(x) = (x1 − p)2 + (x2 − q)2 + (x3 − r)2 − 0.0625 ≤ 0 where 0 ≤ xi ≤ 10 (i = 1, 2, 3), and p, q, r = 1, 2, . . . , 9. The feasible region of the search space consists of 93 disjointed spheres. The global minimum value is −1 at x = (5, 5, 5). Then we have

+ 3(x4 − 11)2 + 10x65 + 7x26 + x47 − 4x6 x7 − 10x6 − 8x7 subject to g1 (x) = −127 + 2x21 + 3x42 + x3 + 4x24 + 5x5 ≤ 0 g2 (x) = −282 + 7x1 + 3x2 + 10x23 + x4 − x5 ≤ 0 g3 (x) = −196 + 23x1 + x22 + 6x26 − 8x7 ≤ 0 g4 (x) = 4x21 + x22 − 3x1 x2 + 2x23 + 5x6 − 11x7 ≤ 0

f23 = Minimize (exp(x1 x2 x3 x4 x5 )) subject to h1 (x) = x21 + x22 + x23 + x24 + x25 − 10 = 0 h2 (x) = x2 x3 − 5x4 x5 = 0 h3 (x) = x31 + x32 + 1 = 0 where −2.3 ≤ xi ≤ 2.3 (i = 1, 2), and −3.2 ≤ xi ≤ 3.2 (i = 3, 4, 5). The global minimum value is 0.0539498 at x = (−1.717143, 1.595709, 1.827247, −0.7636413, −0.763645).

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 38, NO. 5, OCTOBER 2008

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Licheng Jiao (SM’89) received the B.S. degree from Shanghai Jiao Tong University, Shanghai, China, in 1982, and the M.S. and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1984 and 1990, respectively. From 1990 to 1991, he was a Postdoctoral Fellow with the National Key Laboratory for Radar Signal Processing, Xidian University, Xi’an. He is currently the Dean of the Electronic Engineering School and the Director of the Institute of Intelligent Information Processing, Xidian University. His current research interests include signal and image processing, machine learning, natural computation, and intelligent information processing.

JIAO et al.: QUANTUM-INSPIRED IMMUNE CLONAL ALGORITHM FOR GLOBAL OPTIMIZATION

Yangyang Li (M’07) received the B.S. and M.S. degrees in computer science and technology and the Ph.D. degree in pattern recognition and intelligent system from Xidian University, Xi’an, China, in 2001, 2004, and 2007, respectively. She is currently an Associate Professor with the School of Electronic Engineering, Xidian University. Her current research interests include quantuminspired evolutionary computation, artificial immune systems, and data mining.

Maoguo Gong (M’07) received the B.S. degree in 2003 from Xidian University, Xi’an, China, where he is currently working toward the Ph.D. degree. He is currently an Associate Professor with the Innovative Research Team, Ministry of Education of China, Xidan University. He has published more than 30 papers in journals and conference proceedings. His research interests are broadly in the area of computational intelligence and hybrid intelligent systems. The areas of special interests include artificial immune systems, evolutionary computation, data mining, optimization, and some other related areas.

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Xiangrong Zhang (M’07) received the B.S. and M.S. degrees in computer science and technology and the Ph.D. degree in pattern recognition and intelligent system from Xidian University, Xi’an, China, in 1999, 2003, and 2006, respectively. She is currently an Associate Professor with the School of Electronic Engineering, Xidian University. Her current research interests include SAR image analysis and understanding, pattern recognition, and machine learning.