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: raqueline@lncc.br, portugal@lncc.br.

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.

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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)

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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.

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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)

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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.

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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.

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