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Journal of Magnetism and Magnetic Materials 310 (2007) e468–e470

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Quantum annealing effect on entropic slowing downin frustrated decorated bond system

Shu Tanakaa,�, Seiji Miyashitaa,b

aDepartment of Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, JapanbCREST, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan

Available online 7 November 2006

Abstract

We propose that the importance of the quantum annealing procedure to find the ground state of frustrated decorated bond systems

where ‘entropic slowing down’ happens due to peculiar density of states. Here, we use the time dependent Schrodinger equation to analyze

the real time dynamics of the process. It is found that the quantum annealing is very efficient comparing to the thermal annealing for

searching the ground state of the systems. We analyze the mechanism of quantum annealing from a view point of adiabatic process.

r 2006 Elsevier B.V. All rights reserved.

PACS: 02.60.Cb; 05.30.�d; 75.10.Hk

Keywords: Quantum annealing; Entropic slowing down; Decorated bond system; Adiabatic process

1. Introduction

Optimization problem is an important topic in vast areaof science. The most typical example is searching theground state of random spin system such as the spin glass.The free energy landscape of random spin system is verycomplex. Therefore, we encounter the difficulty that thesystem does not reach the ground state but stays in themetastable one. There are two types of slowing down. Oneis the case where the state is trapped at a energeticallymetastable state. An energy barrier prevents the state fromescaping to the equilibrium state. This is ‘energetic slowing

down’. On the other hand, though there is no energybarrier, the system cannot reach the equilibrium state in ashort time which has been found in the regularly decoratedbond system [1,2]. We called this situation ‘entropic slowing

down’. In the latter case, we have found that effective timescale t0 of the ordering process becomes extremely longand the relaxation process is practically frozen. In this caseit is difficult to find the energy minimum structure using thethermal annealing process.

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/j.jmmm.2006.10.417

onding author.

ddress: shu-t@spin.phys.s.u-tokyo.ac.jp (S. Tanaka).

Kadowaki and Nishimori [3] have proposed the quan-tum annealing method for the energetic slowing downsystem. Afterward, the efficiency of the quantum annealingmethod has been confirmed in the energetic slowing downsystem [4,5]. Our aim of this study is to demonstrate theefficiency of the quantum annealing method in the entropicslowing down system.

2. Model

We study the decorated bond system depicted in Fig. 1 asan example of systems with ‘entropically slowing down’.The Hamiltonian of this system is

H0 ¼ �J 0sz1s

z2 � Jsz

1

XN

i¼1

Szi � Jsz

2

XN=2

i¼1

Szi �

XN

i¼N=2þ1

Szi

0@

1A,

where N is the number of the decorated spin fSig. Theeffective coupling Jeff between sz

1 and sz2 is defined

hsz1s

z2i ¼ tanh bJeff . In the case of Fig. 1, the contributions

through the decoration spins are canceled out each otherand Jeff ¼ J 0 due to the direct interaction. However, thedynamics is not simple as the case of the single bond. If thesystem is trapped at the state depicted in Fig. 1(b), it does

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σ1

σ2

J

JJ'

a b

Fig. 1. The thick and thin lines denote the ferromagnetic coupling J 040

and J40, respectively. The dotted lines denote the antiferromagnetic

coupling �Jo0. The black triangles denote + spin and the gray spins are

not fixed (+ or -) spin. (a) The ground state with the correlation function

sz1s

z2 ¼ þ1. (b) The ‘entropically metastable state’ with sz

1sz2 ¼ �1. In

order to relax to (a) from this configuration, all the gray spins must be

align to cancel the interaction from the black spins. But the probability of

happening of this situation is very small. This is entropic slowing down.

0 0.5 1 1.5

−8

−6

−4

transverse field

energ

y

Fig. 2. The energy diagram of decorated bond system (N ¼ 6) as a

function of transverse field.

0 200 400

0.6

0.8

1

τ

pro

babili

ty P

0

Fig. 3. The probability of reaching the ground state of G ¼ 0 as a function

of t.

S. Tanaka, S. Miyashita / Journal of Magnetism and Magnetic Materials 310 (2007) e468–e470 e469

not easily reach the equilibrium state due to the entropyeffect [1]. In this case, the thermal annealing is not efficientand the relaxation time is about 2N=2 for this system toreach the ground state [2]. Here, we introduce quantumannealing using the time dependent transverse fieldHtðtÞ ¼ �GðtÞðsx

1 þ sx2 þ

PiS

xi Þ, and we change the trans-

verse field as GðtÞ ¼ G0ð1� t=tÞ.

3. Real time dynamics of the decorated bond system by

Schrodinger equation

We study the real time dynamics of the decorated bondsystem with the time dependent transverse field. Here, thetime evolution of the state vector jFðtÞi is determined bythe Schrodinger equation. The initial state jFð0Þi is theground state of the initial Hamiltonian Hð0Þ. Fig. 2 showsthe energy as a function of the transverse field, whereJ ¼ 1, J 0 ¼ 0:1, G0 ¼ 1:5, and N ¼ 6. It should be notedthat there are two-fold evident degeneracy because of theup-down symmetry of spin. In Fig. 2, we find that theground state at G ¼ G0 is connected to that of G ¼ 0. Thus,the adiabatic motion leads the system to the true ground

state of G ¼ 0. In this sense, the success of quantumannealing is insured.Now we study properties of the quantum annealing.

First, the t dependence of the probability P0 to reach theground state of H0. Fig. 3 shows P0 as a function of t.There we find that the slower the speed of the sweep, thelarger P0. In the quantum mechanical system, we haveseveral method to obtain the ground state such as thepower method, Lanczos method, etc. These methods arealso possible processes to obtain the ground state. We willcompare these methods as the annealing process and bereported elsewhere.

4. Conclusion

Here we assume ferromagnetic interaction between s1and s2 thus Fig. 1(a) gives the ground state. However, if s1and s2 are antiparallel initially, the decorated spinsdenoted by rightward triangles align upwards and thedecorated spins denoted as the right half of the trianglesremain disorder. In order to change s1 and s2 antiparallelto parallel, all of the left half of the triangle must align,which is difficult. This is called ‘entropically slowing down’.Here we treated an ‘easy’ example to find the ground stateand to demonstrate the efficiency of the quantum anneal-ing. The decorated bond systems with larger number of N

(e.g. N ¼ 20) have been studied to research the temperaturedependent structure [1]. When N increases the energy levelsare much closer and the annealing becomes more difficult,which will be reported elsewhere [2].

Acknowledgments

The present work is partially supported by Grand-in-Aidfrom the Ministry of Education, Culture, Sports, Science,and Technology, and also by NAREGI NanoscienceProject, Ministry of Education Culture, Sports, Science,and Technology, Japan.

ARTICLE IN PRESSS. Tanaka, S. Miyashita / Journal of Magnetism and Magnetic Materials 310 (2007) e468–e470e470

References

[1] S. Tanaka, S. Miyashita, Prog. Theor. Phys. 157 (Suppl.) (2005) 34.

[2] S. Tanaka, S. Miyashita, in preparation.

[3] T. Kadowaki, H. Nishimori, Phys. Rev. E 58 (1998) 5355.

[4] G.E. Santoro, R. Martonak, E. Tosatti, R. Car, Science 295 (2002)

2427.

[5] S. Suzuki, M. Okada, J. Phys. Soc. Japan 74 (2005) 1649.