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arXiv:0910.5795v1 [quant-ph] 30 Oct 2009

The eﬀect of large-decoherence on

mixing-time in Continuous-time quantum

walks on long-range interacting cycles

S. Salimi 1, R. Radgohar 2

Faculty of Science, Department of Physics, University of Kurdistan,

Pasdaran Ave., Sanandaj, Iran

Abstract

In this paper, we consider decoherence in continuous-time quantum walks

on long-range interacting cycles (LRICs), which are the extensions of the

cycle graphs. For this purpose, we use Gurvitz’s model and assume that

every node is monitored by the corresponding point contact induced the

decoherence process. Then, we focus on large rates of decoherence and

calculate the probability distribution analytically and obtain the lower and

upper bounds of the mixing time. Our results prove that the mixing time

is proportional to the rate of decoherence and the inverse of the distance

parameter (m) squared. This shows that the mixing time decreases with

increasing the range of interaction. Also, what we obtain for m= 0 is

in agreement with Fedichkin, Solenov and Tamon’s results [48] for cycle,

and see that the mixing time of CTQWs on cycle improves with adding

interacting edges.

1E-mail: shsalimi@uok.ac.ir

2E-mail: r.radgohar@uok.ac.ir

1

1 Introduction

Random walks or Markov chains on graphs have broad applications in var-

ious ﬁelds of mathematics [1], computer sciences [2] and the natural sci-

ences, such as modeling of crystals in solid state physics [3]. The quantum

walk(QW) is a generalization of the classical random walk(CRW) developed

by exploiting the aspects of quantum mechanics, such as superposition and

interference [4, 5]. The QW is widely studied in many distinct ﬁelds, such as

polymer physics, solid state physics, biological physics and quantum com-

putation [6, 7, 8]. There are two distinct variants of QWs: the discrete-time

QWs(DTQWs) [9, 10] and the continuous-time QWs(CTQWs) [11]. In the

CTQW, one can directly deﬁne the walk on the position space, whereas

in the DTQW, it is necessary to introduce a quantum coin operation to

deﬁne the direction in which the particle has to move. Experimental im-

plementations of both QW variants have been reported e.g. on microwave

cavities [12], ground state atoms [13], the orbital angular momentum of pho-

tons [14] , waveguide arrays [15] or Rydberg atoms [16, 17] in optical lattices.

Recently, quantum walks have been studied in numerous publications. For

instance, the DTQWs have been considered on random environments [18],

on quotient graphs [19], in phase space [20] and for single and entangled

particles [21]. Also, the CTQWs have been investigated on the n-cube [22],

on star graphs [23, 24], on quotient graphs [25], on circulant Bunkbeds [26],

on trees [27], on ultrametric spaces [28], on odd graphs [29], on simple one-

dimensional lattices [30], via modules of Bose-Mesner and Terwilliger alge-

bras [31], via spectral distribution associated with adjacency matrix [32] and

by using krylov subspace-Lanczos algorithm [33].

In this paper, we focus on the CTQWs on the one-dimensional networks.

These networks provide the better understanding of the physical systems

dynamics. For examples, they have been used to describe the behavior of

metals in solid state physics [3], to explain the dynamics of atoms in optical

lattices [13, 16], to demonstrate Anderson localization in the systems with

energetic disorder [34] and to address various aspects of normal and anoma-

lous diﬀusion [35]. The simplest structure of the one-dimensional networks

generated by connecting the nearest neighbors can be modeled the physical

systems with short-range interactions. Gravity and electromagnetism are

the only known fundamental interactions extending to a macroscopic dis-

tance. Due to its basic importance, it has been along tradition to search

extra long-range interactions [36]. In some of the above mentioned experi-

ments, e.g. the clouds of ultra-cold Rydberg atoms assembled in a chain over

which an exciton migrates, the trapping of the exciton occurs at the ends of

2

the chain, one ﬁnds long-range interactions have to be considered [17, 37].

Recently, it has been shown, from the view point of Quantum Electrodynam-

ics, the CTQW for all long-range interactions in a linear system decaying

as R−ν(where Ris the distance between two nodes of the network) belong

to the same universality class for ν > 2, while for classical continuous-time

random walk (CTRW) universality only holds for ν > 3 [38]. It has been

shown the long-range interaction lead to a slowing-down of the decay of the

average survival probability, which is counter intuitive since for the corre-

sponding classical process one observes a speed-up of the decay [39]. In [40],

authors studied coherent exciton transport on a new network model, namely

long-range interacting cycles. LRICs are constructed by connecting all the

two nodes of distance mon the cycle graphs. Since all the LRICs have the

same value of connectivity k= 4 (the number of edges which exit from ev-

ery node), thus LRICs provide a good facility to study the inﬂuence of long

range interaction on the transport dynamics on various coupled dynami-

cal systems, including Josephson junction arrays [41], synchronization [42],

small-world networks [43] and many other self-organizing systems. All of the

mentioned articles have focused on a closed quantum system without any

interaction with its environment. Recently, more realistic analysis of quan-

tum walks by using decoherence concept has been provided on line [44],

on circulant [45, 46] and on hypercube [47]. In [48], the eﬀect of decoher-

ence in the CTQWs on cycles has been considered analytically. The results

showed that, for small rates of decoherence, the mixing time improves lin-

early with decoherence, whereas for large rates of decoherence, the mixing

time deteriorates linearly towards the classical limit. In [49], the authors

have studied the inﬂuence of small decoherence in the CTQWs on long-

range interacting cycles(LRICs). The authors proved that the mixing time

is inversely proportional to the decoherence rate and also independent of the

distance parameter m. Moreover, they showed the mixing time upper bound

for CTQWs on LRIC, remain close to its value in the absence of shortcut

links(cycle). Now, we want to investigate the eﬀect of large decoherence

on the mixing time in the CTQWs on LRICs. For this end, we use of an

analytical model developed by Gurvitz [50]. In this model, every vertex is

monitored by an individual point contact induced the decoherence process.

3

Figure 1: Long-range interaction cycles G(8,3) and G(10,4)

We calculate the probability distribution analytically then, for large rates

of decoherence, obtain the lower and upper bounds of the mixing time. Our

analytical results prove the bounds of the mixing time are proportional to

the decoherence rate that in agreement with [48]’s results . Moreover, we

prove that these bounds are inversely proportional to square of the distance

parameter m, i.e. the mixing time decreases with increasing m.

This paper is organized as follows: In Sec. 2, we brieﬂy describe the network

structure LRIC. The CTQWs on LRICs are considered in Sec. 3. In Sec. 4,

we study the eﬀect of decoherence in CTQWs on LRICs, then we focus on

the large rates of decoherence and calculate the probability distribution in

Sec. 5. In Sec. 6, we deﬁne the mixing time and obtain its lower and upper

bounds. We conclude with a summery of results in Sec. 7.

2 Structure of LRIC

Long-range interacting cycle (LRIC) can be generated as follows [40]: First,

the network be composed of a cycle graph. Second, two nodes of distance

mon the cycle graph are linked by an additional bond. We continue the

second step until all the two nodes of distance mhave been connected. LRIC

denoted by G(N, m), is characterized by the network size Nand the long-

range interaction parameter m. Fig. 1 shows the sketches of G(8,3) and

G(10,4).

3 CTQWs on LRICs

In general, every network is characterized by a graph whose bonds connect

nodes with a wide distribution of mutual distances. Algebraically, every

graph corresponds with a discrete Laplace operator A. We introduce the

states |jiwhich are localized at the nodes jof the graph and take the set

{|ji} to be orthonormal. We assume that transition rates γbetween all

nodes are equal and γ≡1. The non-diagonal element of matrix A(Aij )

equals 1 if nodes iand jare connected by a bond and 0 otherwise. The

diagonal element Aii equals −kithat kiis the number of bonds which exit

from node i[11, 51]. Since the states |jispan the whole Hilbert space,

the time evolution of a state |jistarting at time 0 is determined by the

system Hamiltonian H=Aas |j, ti=U(t)|ji, where U(t) = exp[−iHt]

4

is the quantum mechanical time evolution operator [11, 51]. Thus the

Hamiltonian matrix Hof G(N, m) (m≥2) can be written as

Hij =hi|H|ji=

−4,if i=j;

1,if i=j±1;

1,if i=j±m;

0,Otherwise.

(1)

The Hamiltonian acting on the state |jiis given by

H|ji=−4|ji+|j−1i+|j+ 1i+|j−mi+|j+mi.(2)

The above equation is the discrete version of the Hamiltonian for a free

particle moving on the cycle. Using Bloch function [3, 52] in solid state

physics, the time independent Schr¨odinger equation can be written as

H|ψni=En|ψni.(3)

The Bloch states |ψnican be expanded as a linear combination of states |ji

|ψni=1

√N

N

X

j=1

e−iθnj|ji.(4)

Substituting Eqs. (2) and (4) into Eq. (3), we obtain the eigenvalues of the

system as

En=−4 + 2 cos(θn) + 2 cos(mθn).(5)

The Bloch relation can be obtained by projecting |ψnion the state |jisuch

that ψn(j)≡ hj|ψni=e−i(θnj)/N. It follows that θn= 2nπ/N with n

integer and n∈[0, N). From Schr¨odinger equation, we have

i~d

dt|ψn(t)i=H|ψn(t)i.(6)

By assuming ~= 1 and |ψn(0)i=|0i, the solution of the above equation is

|ψn(t)i=e−iHt|0i.

The probability to ﬁnd the walker at node jat time tis given by

Pj(t) = |hj|ψn(t)i|2.

5

4 The Decoherent CTQWs on LRICs

Here, we want to study the appearance of decoherence in the CTQWs and

obtain the probability distribution P(t). For this end, we make use of an

analytical model developed by Gurvitz [50, 53]. In this model, a ballistic

point-contact is placed near each node of network that is taken as non-

invasive detector. Gurvitz demonstrated that measurement process is fully

described by the Bloch-type equations applied to whole system. These equa-

tions led to the collapse of the density-matrix into the statistical mixture in

the course of the measurement process.

According to Gurvitz model, the time evolution of density matrix for our

network (LRIC), has the following form

d

dt ρj,k(t) = i

4[−ρj−1,k −ρj+1,k −ρj−m,k −ρj+m,k +ρj,k−1

+ρj,k+1 +ρj,k−m+ρj,k+m]−Γ(1 −δj,k )ρj,k ,

(7)

that density matrix ρ(t) is as ρ(t) = |ψ(t)ihψ(t)|and Γ is the decoherence

parameter. Also, we observe that Pj(t) = ρj,j(t).

We deﬁne the variable Sj,k as [48]

Sj,k =ik−jρj,k (8)

and by substituting it into Eq. (7), we obtain

d

dt Sj,k =1

4[−Sj−1,k +Sj+1,k −i−m+1Sj−m,k −im+1Sj+m,k −Sj,k−1

+Sj,k+1 +im+1Sj,k−m+i−m+1Sj,k+m]−Γ(1 −δj,k )Sj,k .

(9)

Note that by assuming m= 0, we achieve the relations mentioned in [48].

5 Large Decoherence

Firstly, we review the results obtained in [49] for the decoherent CTQWs

on LRICs with small rate of decoherence. The mixing time upper bound for

the odd values of mis as Tmix(ǫ)<1

Γln(N

ǫ)[ N

N−2] and for the even ms is as

Tmix(ǫ)≤1

Γln(N

ǫ)[ N

N−1]. Thus, the mixing time upper bound for odd mis

larger than the mixing time upper bound for even m. In addition to these

relations show the upper bound of the mixing time is inversely proportional

to decoherence rate Γ and independent of the distance parameter m.

In the following, we assume that the rate decoherence Γ is large (Γ ≫1)

6

and consider its eﬀect in CTQWs on LRICs.

Firstly, as mentioned in [48], we deﬁne the diagonal sum Dkas

Dk=

N−1

X

j=0

Sj,j+k(modN ),(10)

where the indices are treated as integers modulo N.

From Eq. (9), one can achieve the following form

d

dtDk=−Γ(1 −δk,0)Dk.(11)

Thus the minor diagonal sums (Dkfor k6= 0) decay with characteristic time

of order 1/Γ.

Also, by applying Eq. (9) for the elements on the two minor diagonals

nearest to the major diagonal, we observe that these elements make limit to

small values of order 1/Γ.

Using the above way for the other diagonals, it is yielded the secondary set

of matrix elements in terms of nearness to major diagonal will be of the

order of 1/Γ2, etc.

Now, we consider only matrix elements of order of 1/Γ and get

S′

j,j =1

4[−Sj−1,j +Sj+1,j −i−m+1Sj−m,j −im+1 Sj+m,j

−Sj,j−1+Sj,j +1 +im+1 Sj,j−m+i−m+1Sj,j +m],

(12)

S′

j,j+1 =1

4[Sj+1,j+1 −Sj,j ]−ΓSj,j+1,(13)

S′

j,j−1=1

4[−Sj−1,j−1+Sj,j ]−ΓSj,j−1,(14)

S′

j,j+m=1

4[−im+1Sj+m,j+m+im+1Sj,j]−ΓSj,j+m(15)

S′

j,j−m=1

4[−i−m+1Sj−m,j−m+i−m+1Sj,j]−ΓSj,j−m.(16)

7

For simplicity’s sake, we deﬁne

aj=Sj,j, dj=Sj,j+1 +Sj+1,j , fj=i−m+1Sj,j+m−im+1 Sj+m,j .(17)

After some algebra, one can get

a′

j=1

4[−dj−1+dj+fj−fj−m],(18)

d′

j=1

2[aj+1 −aj]−Γdj,(19)

f′

j=1

2[aj+m−aj]−Γfj.(20)

Diﬀerentiation of the above equation gives

a′′

j=1

4[−d′

j−1+d′

j+f′

j−f′

j−m],(21)

d′′

j=1

2[a′

j+1 −a′

j]−Γd′

j,(22)

f′′

j=1

2[a′

j+m−a′

j]−Γf′

j.(23)

Let us now assume that we have the solutions of Eqs. (21), (22) and

(23) as

aj=

N−1

X

k=0

Ake2πjk

Ne−γkt, dj=

N−1

X

k=0

Dke2πjk

Ne−γkt, fj=

N−1

X

k=0

Fke2πjk

Ne−γkt,(24)

that γk,Ak,Dkand Fkare unknown.

To obtain γk, we substitute these solutions into the above mentioned equa-

tions.

Aγ2

k+D(γk

4(−e

−2πik

N+ 1)) + F(γk

4(−e

−2πikm

N+ 1)) = 0,

A(γk

2(e2πik

N−1)) + D(γ2

k−γkΓ) = 0,

A(γk

2(e2πikm

N−1)) + F(γ2

k−γkΓ) = 0.

(25)

8

Note that for having nontrivial solutions, the determinant of the coeﬃ-

cients matrix must be zero. Thus, we get

γ2

k(γ2

k−γkΓ)[γ2

k−γkΓ + 1

2(sin2(πk

N) + sin2(πkm

N))] = 0,(26)

and achieve

γk=

γk,0=1

2Γ (sin2(πk

N) + sin2(πkm

N)),

γk,1= Γ −1

2Γ (sin2(πk

N) + sin2(πkm

N)),

γk,2= 0,

γk,3= Γ.

(27)

The general solutions of Eq. (21), (22) and (23) are as

aj=1

N

N−1

X

k=0 {Ak,0e−γk,0t+Ak,1e−γk,1t+Ak,2e−γk,2t+Ak,3e−γk,3t}ωj k,(28)

dj=1

N

N−1

X

k=0 {Dk,0e−γk,0t+Dk,1e−γk,1t+Dk,2e−γk,2t+Dk,3e−γk,3t}ωj k ,(29)

fj=1

N

N−1

X

k=0 {Fk,0e−γk,0t+Fk,1e−γk,1t+Fk,2e−γk,2t+Fk,3e−γk,3t}ωjk,(30)

where ω=e2πi

N. We replace γks in Eqs. (25) and use of the initial conditions

aj=δj,0and dj= 0 for j= 0, ..., N −1. We set the constant coeﬃcients

into the others and obtain

Ak,0≃1, Ak,1≃−1

2Γ2(sin2(πk

N) + sin2(πkm

N)), Ak,3= 0,(31)

Dk,0≃i

Γsin(πk

N) exp( iπk

N), Dk,1≃−i

Γsin(πk

N) exp( iπk

N),(32)

Dk,3≃i

Γ2sin(πk

N) exp( iπk

N),

Fk,0≃i

Γsin(πkm

N) exp( iπkm

N), Fk,1≃−i

Γsin(πk m

N) exp( iπkm

N),(33)

Fk,3≃i

Γ2sin(πkm

N) exp( iπkm

N).

9

The full solution for S(t) has the form as

Sj,k(t) =

aj,if j=k;

dj/2,if j=k±1;

fj/2,if j=k±m;

0,otherwise.

(34)

Thus, the probability distribution Pj(t) is the same aj. At large rates of

decoherence Γ, Eq. (28) reduce to

aj(t) = 1

N

N−1

X

k=0 {exp[−1

2Γ (sin2(πk

N) + sin2(πkm

N))t]}ωjk (35)

One can see the above distribution do not converge to any stationary

distribution. The reason being that the evolution of the QW, as mentioned

in Sec. 3, is given by the unitary operator U=e−itH . This is the fact

that unitary operators preserve the norm of states, and hence the distance

between the states describing the system at subsequent times does not con-

verge to zero[9]. This implies that the probability distribution of CTQW

does not converge to CRW.

6 The bounds of mixing time

Mixing time is the time it takes for the walk to approximate the uniform

distribution [48, 9], i.e.

Tmix =min{T:

N−1

X

j=0 |Pj(t)−1

N| ≤ ǫ}.(36)

From Eq. (35), we obtain

N−1

X

j=0 |aj(t)−1

N|=

N−1

X

j=0 |1

N

N−1

X

k=0

exp[−1

2Γ (sin2(πk

N) + sin2(πkm

N))t]e2πijk

N−1

N|,(37)

that simpliﬁes to

N−1

X

j=0 |aj(t)−1

N|=1

N

N−1

X

j=0 |

N−1

X

k=1

exp[−1

2Γ (sin2(πk

N) + sin2(πkm

N))t] cos( 2πkj

N)|.(38)

10

Lower bound

To obtain a lower bound, we apply the following way:

We retain only the terms j= 0 and k= 1, N −1.

N−1

X

j=0 |aj(t)−1

N| ≥ |a0(t)−1

N|=1

N

N−1

X

k=1

exp(−(sin2(πk

N) + sin2(πkm

N))t

2Γ )

≥2

Nexp(−(sin2(π

N) + sin2(πm

N))t

2Γ ).

(39)

According to the mixing time deﬁnition, we have

Tlower =2Γ

sin2(π

N) + sin2(πm

N)ln( 2

Nǫ ),(40)

and for N≫1

Tlower ≃2ΓN2

π2(1 + m2)ln( 2

Nǫ ).(41)

Note that for m= 0,

Tlower ≃2ΓN2

π2ln( 2

Nǫ ),(42)

that is the same [48]’s result. One observe the mixing time lower bound in

CTQWs on cycle decreases with adding interacting links.

Upper bound

Now, we want to obtain upper bound as following

N−1

X

j=0 |aj(t)−1

N|=1

N

N−1

X

j=0 |

N−1

X

k=1

exp[−1

2Γ (sin2(πk

N) + sin2(πkm

N))t] cos( 2πkj

N)|

≤1

N

N−1

X

j=0

N−1

X

k=1

exp(−(sin2(πk

N) + sin2(πkm

N))

2Γ )

(43)

11

Since for 0 < x < π

2, there is sin x > 2x

π[54], we get

N−1

X

j=0 |aj(t)−1

N| ≤ 2

N

N−1

X

j=0

[N/2]

X

k=1

exp(−1

Γ(sin2(πk

N) + sin2(πkm

N))t)

≤2

N

N−1

X

j=0

[N/2]

X

k=1

exp(−1

Γ(2k2+ 2k2m2

N2)t)

≤2

N

N−1

X

j=0

[N/2]

X

k=1

exp(−1

Γ(2k+ 2km

N2)t)

≤2

N

N−1

X

j=0

∞

X

k=1

exp(−1

Γ(2k+ 2km

N2)t)

(44)

that in the third inequality, we used of the relation k2≥kfor k≥1.

N−1

X

j=0 |aj(t)−1

N| ≤ 2

exp[−1

Γ(2

N2+2m2

N2)] −1(45)

Thus,

Tupper =ΓN2

2(1 + m2)ln(2 + ǫ

ǫ).(46)

Note that for m= 0, we get

Tupper =ΓN2

2ln(2 + ǫ

ǫ).(47)

which is in agreement with [48]’s result. One can see that the mixing time

upper bound for LRIC is smaller than the mixing time upper bound for cycle.

Based on the above analysis, the mixing time in CTQWs on cycle decreases

with adding interacting links. Moreover, we observe that the mixing time is

proportional to the square of distance parameter msuch that it decreases

with increasing the interaction range. Now, we want to compare the lower

bound with the upper bound. Since Eq. (41) was obtained by assuming

N≫1 and ǫ≪1, we have ln( 2

N ǫ )<ln(2+ǫ

ǫ). Thus, we can write

2ΓN2

π2(1 + m2)ln( 2

Nǫ )< Tmix <ΓN2

2(1 + m2)ln(2 + ǫ

ǫ).(48)

12

7 Conclusion

We studied the continuous-time quantum walk on long-range interacting

cycles (LRICs) under large decoherence Γ ≫1. We obtained the probability

distribution analytically and found the mixing time is bounded as

2ΓN2

π2(1 + m2)ln( 2

Nǫ )< Tmix <ΓN2

2(1 + m2)ln(2 + ǫ

ǫ).(49)

We proved that the Tmix is proportional to decoherence rate Γ that

this result accord with the conclusion is obtained in [48]. In [49] has been

proved that Tmix, for small rates of decoherence, is independent of distance

parameter m, while in this paper we showed that Tmix, for large rates of

decoherence, is inversely proportional to square of m. In other words, the

mixing time decreases with increasing the range of interaction. Also, since

LRIC is the same generalized cycle, by replacing m= 0 in all the above

relations, one achieves [48]’s results. Moreover, we proved the mixing times

improve with adding interacting links.

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