The effect of large decoherence on mixing time in continuous-time quantum walks on long-range interacting cycles

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 ($\emph{m}$) squared. This shows that the mixing time decreases with increasing the range of interaction. Also, what we obtain for $\emph{m}=0$ is in agreement with Fedichkin, Solenov and Tamon's results \cite{FST} for cycle, and see that the mixing time of CTQWs on cycle improves with adding interacting edges. Comment: 16 Pages, 2 Figures

Full text

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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