-
Decoherence on Szegedy’s Quantum Walk
Raqueline A. M. Santos, Renato Portugal.
Laboratório Nacional de Computação Cient́ıfica (LNCC)
Av. Getúlio Vargas 333, 25651-075, Petrópolis, RJ, Brazil
E-mail: [email protected], [email protected].
Abstract: Quantum walks have been used for developing quantum
algorithms that outperformtheir classical analogues. The quantum
hitting time plays an important role as the stopping timefor
quantum walk based algorithms that search marked elements. It is
known that experimen-tal implementations of quantum systems face
decoherence problems. The interactions with theenvironment will
possibly destroy or reduce the quantum coherence. Thus, it is
important toanalyze the effects of decoherence on quantum walks. In
this work we use a decoherence modelthat is inspired on
percolation. We analyze this decoherence model on Szegedy’s quantum
walk.By performing averages over all possible evolution operators
affected by the decoherence we showthat it is possible to define a
decoherent quantum hitting time.
Keywords: quantum walks, decoherence, quantum hitting time
1 Introduction
In Computer Science, random walks or Markov chains are used in
randomized algorithms, spe-cially in search algorithms that search
a marked vertex in a graph. The expected time to reacha vertex for
the first time, known as hitting time, plays an important role in
those algorithmsas the running time to find a solution. For
instance, we can see applications for the k-SAT andthe graph
connectivity problem [15].
Quantum walks or quantum Markov chains are the quantum analogue
of classical randomwalks. They are obtained through a process of
quantization: the state of the quantum systemis described by a
vector on a Hilbert space and the system’s evolution is governed by
a unitaryoperator if the system is totally isolated from
interactions with the world around it. There arediscrete and
continuous time quantum walks. Both have been used for developing
quantumalgorithms that outperform their classical versions [3, 20,
4, 7]. We can see two different for-malisms for the discrete time
quantum walks. The first one, introduced by Aharonov et al.
[1],adds an additional space that is related to the coin. The
second formalism was developed bySzegedy [21] and it is described
by reflection operators in an associated bipartite graph
obtainedfrom the original one by a process of duplication.
Szegedy [21] showed that the quantum hitting time has a
quadratic improvement over theclassical one to detect a set of
marked vertices for ergodic and symmetric Markov chains. Santosand
Portugal [18] showed that this quadratic improvement remains valid
to finding a vertex, ina set of marked vertices, on the complete
graph. Based on Szegedy’s work, Magniez et al. [14]developed a
quantum algorithm to finding a marked vertex on reversible,
state-transitive Markovchains restricted to the case of only one
marked vertex. Recently, Krovi et al. [11] showed thatthe quadratic
speed-up to finding a marked vertex also holds for any reversible
Markov chainusing a new interpolating algorithm.
When implementing quantum systems, there is no doubt we must
face decoherence problems.As the system may not be completely
isolated, interactions with the environment are possible andcan
destroy or reduce the quantum coherence. This generally undesired
effect might occurs toquantum walks as well. In this way, it is
crucial to understand how the decoherence affects them.
618
ISSN 1984-8218
-
The decoherence is generally modeled as a non-unitary evolution
of the quantum walk. Thiscan be achieved by adding an extra
non-unitary operation (a measure operator, for example),or we can
change the coin or shift operators by non-unitary operators.
In the literature, various works study the influence of
decoherence over the different modelsof quantum walks. Brun et al.
[5] showed how the coined quantum walk behaves as a classicalrandom
walk by the decoherence on the coin operator. Kendon e Tregenna [9]
studied thecomputational consequences of the decoherence on the
coin. Romanelli et al. [17] worked onthe one-dimensional quantum
walk and they considered the possibility of having broken
linksbetween the vertices . This technique was later generalized
for the bidimensional case by Oliveiraet al. [16]. Alagic e Russel
[2] analyzed the effect of making independent measures on
thecontinuos time quantum walk on the hypercube. A review on
decoherence on quantum walks canbe seen in [8]. The decoherence on
Szegedy’s formalism was studied by Chiang and Gomez [6],who
analyzed the sensibility to perturbation due to system’s precision
limitations by adding asymmetric matrix E, representing the noise,
to the transition probability matrix of the graph.The quadratic
speed-up vanishes when the magnitude of the noise ||E|| ≥ Ω(δ(1−
δǫ)), where δis the spectral gap of the transition probability
matrix of the graph and ǫ is the ratio betweenthe number of marked
vertices and the number of vertices of the graph.
In this context, we propose a new model of decoherence on
Szegedy’s quantum walk. Ourdecoherence model is inspired on
percolattion graphs. The Refs. [12, 13] analyzed the behavior ofthe
coined quantum walk on percolation lattices. In this case, we have
edges or vertices randomlymissing on the graph. In our case, the
dynamics acts different from Refs. [12, 13, 17, 16] because,at each
time step, we can introduce defects in the graph whether by the
introduction of newedges or by breaking the links between two
vertices. By performing averages over all possibleevolution
operators affected by the decoherence we are able to define a
decoherent quantumhitting time that comes naturally from the
definition without decoherence.
The paper is organized as follows. In Sec. 2 we review Szegedy’s
quantum walk and thedefinition for the quantum hitting time. In
Sec. 3 we analyze our decoherence model on Szegedy’squantum walk.
In Sec. 4 we draw the conclusions.
2 Szegedy’s Quantum Walk
Szegedy [21] has proposed a quantum walk driven by reflection
operators in an associated bi-partite graph obtained from the
original one by a process of duplication. Let Γ(X,E) be aconnected,
undirected and non-bipartite graph, where X is the set of vertices
and E is the set ofedges. The stochastic matrix P associated with
this graph is defined such that pxy is the inverseof the outdegree
of the vertex x. Define a bipartite graph associated with Γ(X,E)
through aprocess of duplication. X and Y are the sets of vertices
of same cardinality of the bipartitegraph. Each edge {xi, xj} in E
of the original graph Γ(X,E) is converted into two edges in
thebipartite graph {xi, yj} and {yi, xj}.
To define a quantum walk in the bipartite graph, we associate
with the graph a Hilbertspace Hn2 = Hn ⊗ Hn, where n = |X| = |Y |.
The computational basis of the first componentis{∣
∣x〉
: x ∈ X}
and of the second{∣
∣y〉
: y ∈ Y}
. The computational basis of Hn2 is{∣
∣x, y〉
: x ∈ X, y ∈ Y}
. The quantum walk on the bipartite graph is defined by the
evolutionoperator UP given by
UP := RB RA, (1)where
RA = 2AAT − In2 , (2)RB = 2BBT − In2. (3)
619
ISSN 1984-8218
-
The operators A : Hn → Hn2 and B : Hn → Hn2 are defined as
follows
A =∑
x∈X
∣
∣Φx〉〈
x∣
∣, (4)
B =∑
y∈Y
∣
∣Ψy〉〈
y∣
∣, (5)
where
∣
∣Φx〉
=∣
∣x〉
⊗
∑
y∈Y
√pxy∣
∣y〉
, (6)
∣
∣Ψy〉
=
(
∑
x∈X
√pyx∣
∣x〉
)
⊗∣
∣y〉
. (7)
In the bipartite graph, an application of UP corresponds to two
quantum steps of the walk,from X to Y and from Y to X. We have to
take the partial trace over the space associated withY to get the
state on the set X.
2.1 Quantum Hitting Time
Instead of using the stochastic matrix P , Szegedy defined the
quantum hitting time by using amodified evolution operator UP ′
associated with a modified stochastic matrix P
′, that is givenby
p′xy =
{
pxy, x 6∈M ;δxy, x ∈M .
(8)
M is the set of marked vertices. The initial condition of the
quantum walk is
∣
∣ψ(0)〉
=1√n
∑
x∈Xy∈Y
√pxy∣
∣x, y〉
. (9)
Note that∣
∣ψ(0)〉
is an eigenvector of UP with eigenvalue 1. However,∣
∣ψ(0)〉
is not an eigenvectorof UP ′ in general.
Definition 2.1 [21] The quantum hitting time HP,M of a quantum
walk with evolution operatorUP given by Eq. (1) and initial
condition
∣
∣ψ(0)〉
is defined as the least number of steps T suchthat
F (T ) ≥ 1 − mn, (10)
where m is the number of marked vertices, n is the number of
vertices of the original graph andF (T ) is
F (T ) =1
T + 1
T∑
t=0
∥
∥
∥UtP ′
∣
∣ψ(0)〉
−∣
∣ψ(0)〉
∥
∥
∥
2, (11)
where U tP ′ is the evolution operator after t steps using the
modified stochastic matrix.
The quantum hitting time was calculated analytically for the
complete graph and the cy-cle [18, 19]. The graphs in Fig. 1 show
the behavior of the function F (T ) for the complete graphand the
cycle when n = 100 and m = 15. F (T ) grows rapidly through the
dashed line 1 − m
n,
then oscillates around its limiting value. The quantum hitting
time can be seen in the graph attime T such that F (T ) = 1 − m
n.
620
ISSN 1984-8218
-
F(T) 1-m/n
T0 10 20 30
0
0,5
1,0
1,5
2,0
(a) Complete graph – HP,M ≃ 1.41.
F(T) 1-m/n
T0 100 200 300
0
0,5
1,0
1,5
2,0
(b) Cycle – HP,M ≃ 21.25.
Figure 1: Graphs of the function F (T ) (solid line) and 1 −
mn
(dashed line) for the completegraph and the cycle when n = 100
and m = 15.
3 Decoherence on the Quantum Hitting Time
According to Kesten [10], percolation is a simple probabilistic
model which exhibits a phasetransition. Consider a 2D lattice, for
example, which we view as a graph with edges betweenneighboring
vertices. All edges are, independently of each other, chosen to be
open (the edgeexists) with probability p and closed (the edge is
missing) with probability 1 − p. A basicquestion in this model is
“What is the probability that there exists an open path, i.e., a
pathall of whose edges are open, from the origin to a destination
vertex in the graph?” Percolationcan be generalized to percolation
on any graph and we can consider site percolation, when wehave
vertices randomly missing, or bond percolation, for the case of
edges randomly missing.
Our decoherence model is inspired on percolation graphs. The
dynamics of the proposedmodel acts different because we consider
that changes on the graph can occur at each time step,due to
insertion or removal of edges between two vertices. The link
between two vertices ofthe graph has a fixed probability, p, of
being removed or created. These modifications on thetopology of the
graph leads to changes on the transition probability matrix
associated to thegraph, which eventually modifies the evolution
operator. Therefore, instead of having an usualwalk evolving as
∣
∣ψ(t)〉
= U tP∣
∣ψ(0)〉
, now we have,
∣
∣ψ(t)〉
= UPtUPt−1 · · ·UP1∣
∣ψ(0)〉
=: U~Pt
∣
∣ψ(0)〉
. (12)
where ~Pt = {P1, . . . , Pt−1, Pt} and U~Pt = UPtUPt−1 · · ·UP1
. Pi’s are not necessarily equal. In thiscontext, it is useful to
define an operator that will gather the behavior of the operators
affectedby the decoherence. Then, let
Ūdec :=∑
P
Pr(P )UP , (13)
be the operator obtained by doing an average over all possible
evolution operators affected ornot by the decoherence. The
following result show that the average over all possible
sequences~P , with size T , according to its probability
distribution, is equal to ŪTdec.
Lemma 3.1 Consider t ≤ T , then∑
~PT
Pr(~PT )U~Pt = Ūtdec. (14)
621
ISSN 1984-8218
-
Proof Since Pr(~PT ) =∏T
i=1 Pr(Pi), we have,
∑
~PT
Pr(~PT )UPtUPt−1 · · ·UP1 =∑
~PT
T∏
i=1
Pr(Pi)UPtUPt−1 · · ·UP1
=∑
PT
∑
PT−1
· · ·∑
P2
(
T∏
i=2
Pr(Pi)
)
UPtUPt−1 · · ·UP2
∑
P1
Pr(P1)UP1
=∑
PT
∑
PT−1
· · ·∑
Pt+1
Pr(PT )Pr(PT−1) · · ·Pr(Pt+1)Ū tdec
= Ū tdec
In order to define the quantum hitting time for the evolution
with decoherence we shouldmake an average over all possible
sequences ~P . Define,
Fdec(T ) :=∑
~PT
Pr(~PT )
(
1
T + 1
T∑
t=0
∥
∥
∥U~Pt
∣
∣ψ(0)〉
−∣
∣ψ(0)〉
∥
∥
∥
2)
. (15)
Now we prove that Fdec(T ) is equivalent to F (T ) of Eq. (11)
when the evolution operator isŪdec.
Theorem 3.2
Fdec(T ) =1
T + 1
T∑
t=0
∥
∥
∥Ū tdec
∣
∣ψ(0)〉
−∣
∣ψ(0)〉
∥
∥
∥
2. (16)
Proof
Fdec(T ) =∑
~PT
Pr(~PT )
(
1
T + 1
T∑
t=0
∥
∥
∥U~Pt
∣
∣ψ(0)〉
−∣
∣ψ(0)〉
∥
∥
∥
2)
=∑
~PT
Pr(~PT )
(
1
T + 1
T∑
t=0
(
2 − 2〈
ψ(0)∣
∣U~Pt
∣
∣ψ(0)〉
)
)
=1
T + 1
T∑
t=0
2 − 2〈
ψ(0)∣
∣
∑
~PT
Pr(~PT )U~Pt
∣
∣ψ(0)〉
(17)
By Lemma 3.1, we have
Fdec(T ) =1
T + 1
T∑
t=0
(
2 − 2〈
ψ(0)∣
∣Ū tdec∣
∣ψ(0)〉)
=1
T + 1
T∑
t=0
∥
∥
∥Ūtdec
∣
∣ψ(0)〉
−∣
∣ψ(0)〉
∥
∥
∥
2.
(18)
The occurrence probability of a given Pi is determined as
follows. If 0 < p < 1, thenPr(Pi) = (1 − p)ac−adpad , where
ac = n(n−1)2 is the number of edges of the complete graphwith n
vertices and ad is the number of edges removed plus the number of
edges included toobtain Pi from P . If p = 0, Pr(Pi = P
′) = 1, and Pr(Pi 6= P ′) = 0. And, if p = 1, wehave Pr(Pi = P̄
′) = 1, and Pr(Pi 6= P̄ ′) = 0, where P̄ ′ is the complement of P
′. Now, wecan naturally define the quantum hitting time with
decoherence, using the expression of Fdecobtained in Theorem
3.2.
622
ISSN 1984-8218
-
Definition 3.3 The quantum hitting time HdecP,M of a quantum
walk with evolution operator UPgiven by Eq. (1) and initial
condition
∣
∣ψ(0)〉
is defined as the least number of steps T such that
Fdec(T ) ≥ 1 −m
n. (19)
We notice that when p = 0, we have the original definition,
since Ūdec = UP ′ .
4 Conclusions
We have proposed a new model of decoherence on quantum walks
inspired on percolation graphs.This model is characterized by the
possibility of insertion or removal of edges at each time step.By
applying this model on Szegedy’s quantum walk we can notice that
the probability matrix ofthe graph can be changed at each time step
and, consequently, the evolution operator. Thus, thestate of the
walker on a given instant of time is obtained by the application of
possible differentevolution operators to the initial state. We were
able to define a decoherent hitting time by usinga new operator
that is obtained by performing an average over all possible
evolution operatorsaffected or not by the decoherence. Future works
might analyze the behavior of the decoherentquantum hitting time,
whether numerically or algebraically, verifying the consequences on
thespeed-up obtained by the version without decoherence.
Acknowledgments
We thank F. Marquezino for fruitful discussions. R.A.M. Santos
acknowledges a CAPES’ fel-lowship and R. Portugal acknowledges
CNPq.
References
[1] Y. Aharonov, L. Davidovich, and N. Zagury. Quantum random
walks. Physical Review A,48(2):1687-1690, 1993.
[2] G. Alagic and A. Russell. Decoherence in quantum walks on
the hypercube. Physical ReviewA, 2005.
[3] A. Ambainis. Quantum walk algorithm for element
distinctness. In Proceedings of the 45thAnnual IEEE Symposium on
Foundations of Computer Science, 2004.
[4] A. Ambainis, J. Kempe, and A. Rivosh. Coins make quantum
walks faster. In Proceedingsof the 16th ACM-SIAM Symposium on
Discrete Algorithms, pages 1099-1108, 2005.
[5] T. A. Brun, H. A. Carteret, and A. Ambainis. Quantum to
classical transition for randomwalks. Physical Review Letters,
91(130602), 2003.
[6] C.-F. Chiang and G. Gomez. Hitting time of quantum walks
with perturbation. QuantumInformation Processing, 2012,
http://dx.doi.org/10.1007/s11128-012-0368-9.
[7] A. Childs and J. Goldstone. Spatial search by quantum walk.
Physical Review A, 70(022314),2004.
[8] V. Kendon. Decoherence in quantum walks - a review.
Mathematical Structures in ComputerScience, 17(6):1169-1220,
2007.
[9] V. Kendon and B. Tregenna. Decoherence can be useful in
quantum walks. Physical ReviewA, 67(042315), 2003.
623
ISSN 1984-8218
-
[10] H. Kesten. What is... percolation? Notices of the American
Mathematical Society,53(5):572-573, 2006.
[11] H. Krovi, F. Magniez, M. Ozols, and J. Roland. Finding is
as easy as detecting for quan-tum walks. In Proceedings of the 37th
International Colloquium Conference on Automata,Languages and
Programming, pages 540-551, 2010.
[12] G. Leung, P. Knott, J. Bailey, and V. Kendon. Coined
quantum walks on percolationgraphs. New J. Phys., 12(123018),
2010.
[13] N. B. Lovett, M. Everitt, R. M. Heath, and V. Kendon. The
quantum walk search algorithm:Factors affecting efficiency, 2011.
Available in: arXiv:quant-ph/1110.4366v2.
[14] F. Magniez, A. Nayak, P. C. Richter, and M. Santha. On the
hitting times of quantumversus random walks. In Proceedings of the
Nineteenth Annual ACM-SIAM Symposium onDiscrete Algorithms, pages
86-95, 2009.
[15] R. Motwani and P. Raghavan. Randomized Algorithms.
Cambridge University Press, 1995.
[16] A. C. Oliveira, R. Portugal, and R. Donangelo. Decoherence
in two-dimensional quantumwalks. Physical Review A, 74(012312),
2006.
[17] A. Romanelli, R. Siri, G. Abal, A. Auyuanet, and R.
Donangelo. Decoherence in the quan-tum walk on the line. Physica A,
347(C):137-152, 2005.
[18] R. A. M. Santos and R. Portugal. Quantum hitting time on
the complete graph. Interna-tional Journal of Quantum Information,
8(5):881-894, 2010.
[19] R. A. M. Santos and R. Portugal. Quantum hitting time on
the cycle. In III WECIQ -Workshop-School of Computation and Quantum
Information, Petrópolis, Brazil, 2010.
[20] N. Shenvi, J. Kempe, and K. B. Whaley. A quantum random
walk search algorithm. PhysicalReview A, 67(052307), 2003.
[21] M. Szegedy. Quantum speed-up of markov chain based
algorithms. In Proceedings of the45th Symposium on Foundations of
Computer Science, pages 32-41, 2004.
624
ISSN 1984-8218