Effect of Quantum Cooperation in Three Entangled Ants

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 8, August 2015)

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Effect of Quantum Cooperation in Three Entangled Ants

Ichiro Iimura

1

, Kazuhiro Takeda

2

, Shigeru Nakayama

3

1

Prefectural University of Kumamoto, 3-1-100 Tsukide, Higashi-ku, Kumamoto 862-8502, Japan

2_{Kagoshima National College of Technology, 1460-1 Shinkou, Hayato-cho, Kirishima 899-5193, Japan} 3_{Kagoshima University, 1-21-40 Korimoto, Kagoshima 890-0065, Japan}

Abstract—Recently, the physical concept of quantum

com-putation model has inspired the information and computer science domain. Summhammer applied the physical concept of quantum entanglement to cooperative behavior of two ants/agents pushing a pebble which may be too heavy for one ant. According to his results, we have confirmed that the two quantum-inspired ants imitating quantum entanglement state, i.e., two entangled ants, can push the pebble up to twice rela-tive to the two classical ants in independent relation, i.e., two independent ants. In the previous study, the quantum entan-glement state of only two qubits, which is called Bell state, has been applied to cooperative behavior. In this study, we novelly applied the quantum entanglement state of three qubits, which is called GHZ (Greenberger-Horne-Zeilinger) state, to cooperative behavior, and performed its simulation. From the experimental analysis, we have confirmed that the three en-tangled ants can push a pebble up to thrice relative to the three independent ants.

Keywords—agents, cooperative forces, GHZ (Greenberger-Horne-Zeilinger) state, quantum cooperation, quantum en-tanglement

I. INTRODUCTION

In recent years, the physical concept of quantum compu-tation model has inspired the information and computer science domain. For instance, the physical concept of quan-tum interference and Quanquan-tum superposition were innovated to Genetic Algorithm [1] and Evolutionary Algo-rithm [2], respectively. And furthermore, Summhammer applied the physical concept of quantum entanglement to cooperative behavior of two ants/agents pushing a pebble which may be too heavy for one ant [3]. That is, each of ants makes measurements on quantum states to decide whether to execute certain actions. In Summhammer’s model, the ants make odour-guided random choices of pos-sible directions, followed by a quantum decision whether to push or to rest. According to his results, we have confirmed that the two quantum-inspired ants imitating quantum en-tanglement state, i.e., two entangled ants, can push the pebble up to twice relative to the two classical ants in inde-pendent relation, i.e., two indeinde-pendent ants. Then, Nakayama et al. have simulated the cooperation of two ants while changing the condition of important parameters and have clarified its feature [4, 5].

From the experimental analysis, in competitive society where ants with strong force are advantageous, Nakayama et al. have proven that two homogeneous ant brains are good and two heterogeneous ant forces are good. The result of the experimental analysis was similar to the idea in col-lective decision making. In the previous studies described above, the quantum entanglement state of only two qubits, which is called Bell state, has been applied to cooperative behavior. In this study, we novelly applied the quantum entanglement state of three qubits, which is called GHZ (Greenberger-Horne-Zeilinger) state, to cooperative behav-ior, and performed its simulation. This paper describes the experimental results.

II. OVERVIEW OF QUANTUM ENTANGLEMENT IN THE

PHYSICAL DOMAIN USED IN THREE ENTANGLED ANTS

In this paper we deal with the simplest kind of quantum entanglement. It is the correlation of the angular momen-tum between three particles of the same kind. The angular momenta of three such particles can easily be measured along different directions. The possible results are then

, , , , , , ,

and , where and are spin up and spin down,

re-spectively. Quantum theory can only predict the

probabilities, , , , , ,

, , and , for these measurement results.

An important state of quantum entanglement, which will be used in ants, is the so-called GHZ state. Symbolically, this state is written as

| 〉

| 〉 | 〉

√

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III. NOVEL MODEL OF THREE ANTS PUSHING APEBBLE

We assume that three ants must push a pebble towards a certain goal. Each ant ( ) can push with a

force . In order to move the pebble, a minimum force

must be applied. Clearly, if the pebble is too heavy to

be moved by any two of the ants, the three ants must push

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The three ants achieve their task by making a series of simultaneous push attempts. A push attempt is successful if the force applied to the pebble is larger than the required minimum. Then the pebble will move a little path length proportional to the force in the direction of the force.

Before a push attempt, three ants must make two deci-sions as follows. These decideci-sions are made independently by each ant and they are not communicated to the other ants.

1st Decision Choose each direction that each ant

pushes. The direction is chosen in accordance with

a probability distribution ( ) as shown in

( ) ( | |) , (2)

where is an appropriate normalization factor and is a

positive constant of , -.

2nd Decision Decide either to really push at this at-tempt or to have a little rest. These decision processes of three entangled ants and three independent ants are de-scribed in Section IV. B. and IV. C., respectively.

If the ant decides to push, the force applied to a

pebble is given as follows:

[

] , (3)

Where is the strength of the ant, that is, equals to | |.

The direction is , where is the

direc-tion straight to the goal. For example, , , and , which

are the forces of , , and , are depicted in Fig.

[image:2.612.91.244.543.656.2]

1.

Figure 1: Relation between the forces , , and in three ants

, , and , where P and G are the pebble and the goal, respectively.

IV. DECISION PROCESS OF WHETHER EACH ANT REALLY

PUSHES APEBBLE AT THIS ATTEMPT OR HAS ALITTLE

REST

A. Real Space of a Force and Brain Space of a Qubit in a Quantum-Inspired Ant

In general, a qubit is described by two-dimensional col-umn vector in the complex vector space where the inner product is defined. It uses the following computational ba-sis states | 〉 and | 〉 as orthonormal base vectors.

| 〉 [ ]| 〉

| 〉 | 〉 [ ] | 〉

| 〉 . (4)

The qubit can have a stochastic superposition state (vec-tor sum) of the two vec(vec-tors | 〉 and | 〉 with each complex probability amplitude. The superposition state | 〉 of the qubit can be shown as follows:

| 〉 | 〉 | 〉 [ ]| 〉

| 〉 , (5)

Where and are the complex probability amplitudes

observa-tion result corresponds to resting without pushing a

pebble.

Next let us think the relation between “the force direc-tion to push a pebble in real space of a force” and “the behavior (pushing a pebble or resting) of a

quantum-inspired ant in brain space of a qubit” by using a

sin-gle qubit. As shown in Fig. 2(a), let ( ) be the angle between the goal direction and the force direction

to push a pebble. The is the parameter relating to

behav-ioral decision of a quantum-inspired ant which either pushes a pebble or rests. The ratio of the probability ampli-tudes in the superposition state | 〉 changes depending on

the , and its change is performed by unitary

transfor-mation. To perform the unitary transformation, the

following rotation matrix can be used. That is, the ratio

of the probability amplitudes is changed by rotating a qubit depending on the force direction.

| 〉 | 〉 [ ( ⁄ ) ( ⁄ )

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For instance, the rotation of the initial state | 〉 is given as follows:

| 〉 [ ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ )] [ ]

| 〉 | 〉

( ⁄ ) | 〉 ( ⁄ ) | 〉 , (7)

and Fig. 2(b) shows the rotation result. That is to say, the probability to push a pebble becomes higher if | | be-comes smaller, and the probability to push a pebble

becomes lower if | | becomes larger. Thus, the real space

of a force is associated with the brain space of a qubit.

(a) Real space of a force.

[image:3.612.67.261.296.632.2]

(b) Brain space of a qubit.

Figure 2: Relation between force direction to push a pebble in real space of a force and behavior of a quantum-inspired ant in

brain space of a qubit.

B. Cooperative Behavior of Three Entangled Ants

The probability whether to push a pebble or to rest at each attempt, this is the place where we permitted the phys-ical concept of quantum entanglement to come in, just like Summhammer. Cooperative behavior occurs to three quan-tum-inspired ants, which have the above-mentioned behavioral decision process, by imitating quantum entan-glement state. Here, the decision process of the cooperative behavior is explained. First, we think three qubits state

| 〉 *| 〉 | 〉+ √ ⁄ called GHZ state in

which three ants , , and are entangled.

The force directions to push a pebble of three quantum-inspired ants , , and are , , and ,

respectively. The rotation matrices corresponding to , ,

and are , , and , respectively, where

. Then, the state | 〉 is translated by the

linear operator as follows:

| 〉

→ ( )| 〉 | 〉 √

√ * | 〉 | 〉 | 〉

| 〉 | 〉 | 〉+

√

[

]

| 〉

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| 〉 | 〉 | 〉

√

[

( ⁄ ) ( ⁄ ) ( ⁄ )

( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ]

| 〉

₍₉₎

and

| 〉 | 〉 | 〉

√

[

( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ ) ( ⁄ )

( ⁄ ) ( ⁄ ) ( ⁄ )

( ⁄ ) ( ⁄ ) ( ⁄ )]

| 〉

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Therefore, the probability to observe | 〉 in

which , , and cooperatively push a pebble is given as follows:

| |

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Next, the probability to observe | 〉 in

which only and cooperatively push a pebble is

given as follows:

| |

(12)

Similarly, other six probabilities, , , ,

, , and , are also given.

C. Behavior of Three Independent Ants

The decision process of the behavior in three independ-ent ants is very simple. In three classical ants in independent relation, , , and , each ant

independently decides either to push a pebble or to rest in accordance with the probability ⁄ . As a result, the eight

probabilities in three independent ants are given as

follows:

( ) ₍₁₃₎

where the mark is or .

Incidentally, the expected displacement of a pebble after one push attempt in three independent ants is same as that of three entangled ants.

V. BASIC EXPERIMENT IN QUANTUM COOPERATION OF

THREE ENTANGLED ANTS

In order to confirm the effects of cooperative behavior in three entangled ants, we have performed the numerical

simulation while changing the important parameter .

We changed from to in every steps,

where is the strength of . The other parameter values

used are shown in Table I. The number of push attempts in

one trial, , was and we analyzed the

perfor-mance based on the average values of trials. The

experimental result is shown in Fig. 3. The figure shows the mean distance pushed by ants in a straight line between a starting point and a last point of a pebble as a function of

the minimum force necessary to push a pebble, , in the

case of . Incidentally, the cases of and

have also shown the same tendency as .

TABLEI

PARAMETER VALUES USED

Parameter name Value used

Number of pebbles, [pebbles] Number of ants, [ants]

Strength of , , , Moving distance of a pebble [dots/strength]

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International Journal of Emerging Technology and Advanced Engineering

[image:5.612.38.296.130.314.2]

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Figure 3: Experimental result in the case of .

From the experimental analysis, we have confirmed that the three quantum-inspired ants imitating quantum entan-glement state, i.e., three entangled ants, can push a pebble up to thrice relative to the three classical ants in independ-ent relation, i.e., three independindepend-ent ants.

VI. CONCLUSIONS

We have novelly applied the quantum entanglement state of three qubits, which is called GHZ (Greenberger-Horne-Zeilinger) state, to cooperative behavior of three ants pushing a pebble which may be too heavy for one ant or two ants. From the experimental analysis, we have con-firmed that the three entangled ants can push a pebble up to thrice relative to the three independent ants.

In the near future, we plan to analyze the cooperative behavior of three entangled ants in more detail. Finally, we expect that these results can be utilized for studying the optimal modeling of cooperative relations in society by means of a simulation and designing algorithms of games, etc.

REFERENCES

[1] Narayanan, A. and Moore, M. 1996. Quantum-inspired genetic algo-rithms. Proc. IEEE Int. Conf. Evolutionary Computation, 61-66. [2] Han, K.-H. and Kim, J.-H. 2002. Quantum-inspired evolutionary

algorithm for a class of combinatorial optimization. IEEE Trans. Evolutionary Computation, 6(6), 580-593.

[3] Summhammer, J. 2006. Quantum cooperation of two insects. arXiv:quant-ph/0503136v2.

[4] Nakayama, S. and Iimura, I. 2010. Experimental study on quantum-entangled cooperative behavior of two ants. Proc. 2nd World Cong. Nature and Biologically Inspired Computing, 573-578.

[5] Nakayama, S. and Iimura, I. 2011. Cooperative action in two ants inspired by quantum entanglement state and an interpretation in col-lective decision making. IPSJ Journal, 52(8), 2467-2473.

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