SStudy of Quantum Walk over a Square Lattice
Arkaprabha Ghosal ∗ and Prasenjit Deb † Department of Physics and Center for Astroparticle Physics and Space Science,Bose Institute, Bidhan Nagar Kolkata - 700091, India.
Quantum random walk finds application in efficient quantum algorithms as well as in quantumnetwork theory. Here we study the mixing time of a discrete quantum walk over a square latticein presence of percolation and decoherence. We consider bit-flip and phase damping noise, andevaluate the instantaneous mixing time for both the cases. Using numerical analysis we show thatin the case of phase damping noise probability distribution of walker’s position is sufficiently closeto the uniform distribution after infinite time. However, during the action of bit-flip noise, evenafter infinite time the total variational distance between the two probability distributions is largeenough.
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I. INTRODUCTION
Quantum mechanics has puzzled the scientists since itsinception due to the counter-intuitive characteristics. Re-cent experimental and theoretical developments in quan-tum information and computation have revealed a lot ofimportant outcomes which highlight the advantages ofusing quantum mechanical systems and resources in per-forming various tasks. Different fascinating effects havebeen uncovered which are strikingly different from theirclassical counterparts, both from the physical point ofview as well as from a computer science and communica-tion theory perspective. Quantum teleportation[1], densecoding[2], state merging[3], quantum cryptography[4, 5],quantum computation[6] etc are possible iff quantummechanical resources are available. In the classical worldneither any such phenomena will take place nor any suchtask can be accomplished. So, in order to exploit thequantum effects modern technologies are being developedwhich in turn will enable us to make use of quantum re-sources on a large scale.In classical computer science random walks play avital role[7]. So it is plausible to think that the quan-tum counterpart will be equally important in the studyof quantum computation, and infact, recent researchconfirms that the notion of quantum random walk hasemerged as an important element in the development ofquantum algorithms[8, 9]. A lot of algorithms have beendeveloped using quantum walks, which display speed-upscompared to their classical equivalents. One such algo-rithm is the algorithm for element distinctness[10] wherequantum walks appear dramatically to make it the mostefficient.There are two natural variants of basic model ofquantum random walk. One is the continuous modeland another one is the discrete model. The continuousmodel was introduced by Childs et. al [11] and the dis- ∗ Electronic address: [email protected] † Electronic address: [email protected] crete model by Aharonov et. al [12]. Though the discreteand continous time quantum walks have different origin,they can be precisely related to each other. The dy-namics of quantum random walks have been analyzed forexample, on the line[13], circle[14], hyperlattice[15] andhypercube[16] and the theoretical properties of quantumwalks on general graphs have been outlined in Ref.[17]Some important properties of quantum random walk ondirected graphs have been provided in Ref.[18].Ideally, quantum systems are taken to be isolatedfrom the environment and the evolution of such systemsare described by unitary evolution. But, in the real sce-nario the quantum systems are susceptible to imperfec-tions and interactions with their environment, due towhich decoherence takes place. Sufficient amount of de-coherence can remove any potential benefits from quan-tum dynamics. Therefore, for practical implementationof quantum protocols in communication or computationit is necessary to study the robustness of such walks inpresence of decoherence.Decoherence is unarguably a nemesis of quantum in-formation processing. However, in the context of quan-tum walks it can infact be useful[19]. For example, deco-herence can be used to force a quantum walk to move to-wards a steady state distribution, which has been nicelyillustrated for hypercube[20], line[19] and N-cycles[21].Elegant analysis has been done by Richter[22, 23] show-ing the quantum speedups of classical mixing processesunder a restricted decoherence model. By examining thediscrete-time quantum walk on hypercube Marquezino et. al [24] derived the limiting time-averaged distributionin the coherent case. It has been also shown that thewalk approaches the uniform distribution in the decoher-ent case and there is an optimal decoherence rate whichprovides the fastest convergence[24]. Though the abovementioned literatures show the effect of decoherence onquantum walks, more study is undoubtedly necessary.In this article we aim to investigate whether deco-herence always forces the quantum walk to move towardsuniform distribution or not by analysing the mixing time.For our purpose we consider discrete time quantum walkon a square lattice and assume that the graph structure a r X i v : . [ qu a n t - ph ] M a r changes randomly, i.e., the walk is taking place on a dy-namical percolation graph. The walker is a qubit (two-state quantum system) and the decohering noises are bit-flip and phase damping noise. We find that in presence ofdecoherence quantum walk does not always reach the uni-form distribution. To be more specific, in case of phasedamping noise the probability distribution of walker’s po-sition is sufficiently close to the uniform distribution afterinfinite time, whereas, the walk can not reach the uniformdistribution under the action of bit-flip noise, even afterinfinite time. The article is arranged as follows. In Sec[II] we provide an overview of quantum random walk ondynamical percolation graphs. The main results are pre-sented in Sec[III]. Conclusion about our work is providedin Sec[IV]. II. DISCRETE QUANTUM RANDOM WALKON DYNAMICAL PERCOLATION GRAPHS
A discrete-time quantum walk (DTQW) is modeled asa particle consisting of a two-level coin (a qubit) existingin the Hilbert space H c and a position degree of freedomexisting in the Hilbert space H p . The Hilbert space ofthe quantum walk is therefore the tensor product of H c and H p , i.e., H = H c ⊗H p . The coin space H c is spannedby the basis states of the coin and the position space H p is spanned by the orthonormal basis vectors correspond-ing to the vertices of the graph on which the quantumwalk occurs. The unitary operator that defines the timeevolution of DTQW and acts on the Hilbert space H isgiven by U = S · ( C ⊗ I ) (1)where S , C , I denote shift operator, coin operator andidentity operator respectively. DTQW has been rigor-ously studied for different graphs assuming that the un-derlying graph does not change with time. However, theunderlying graph structure can be changed and it can bedone by keeping an edge with some probability, say, λ or discarding it with probability 1 − λ . In graph theorythis type of structural change is termed as percolation .If the graph changes randomly then the percolation istermed as dynamical percolation . The unitary evolutionof the quantum walk is disturbed by dynamical perco-lation, leading to a special type of open system causingdecoherence, which can be viewed also as a source ofnoise.From the definition of percolation it is clear thatdifferent graph structures mean different edge configura-tions and the time evolution of quantum walk on eachconfiguration is described by a unitary operator. Theunitary operator that describes the time evolution of aquantum walk on a percolated graph for edge configura-tion K is defined as U K = S K · ( C ⊗ I ) (2) where S K is the shift operator for the configuration K .The dynamic percolation is introduced in this process(quantum walk) by choosing different edge configurations K at each step, and thus random application of differentunitary step operators. The state of the walker is bestdescribed by density matrix. Now if we denote the prob-ability of choosing a configuration K and correspondingunitary step U K by π K then the time evolution of thewalker during the percolated quantum walk is given byΦ( ρ ) ≡ R (cid:88) K =1 π K U K ρU †K (3)where Φ( · ) is a completely positive trace preserving (CPTP) map and R is the total number of edge con-figurations. The shift operator S K for configuration K depends on the graph structure. However, the coin oper-ator C is independent of the graph structure, and it canbe expressed as a function of two angles α and β . Thecoin operator looks like C ( α, β ) = (cid:32) i e − i α sinβ cosβcosβ i e i α sinβ (cid:33) (4)where ‘i’ has its usual meaning. Inserting α = π and β = π we get the Hadamard operator.Along with percolation if the walker (qubit) facesMarkovian noise then the time evolution of the qubit canbe given byΦ( ρ ) = (cid:88) K π K { (1 − p ) U K ρU †K + p (cid:88) m U K M m ρM † m U †K } (5)where M m are the Kraus operators satisfying the condi-tion (cid:80) m M † m M m = 1, and m is the Kraus rank . Here p is the probability with which noise acts on the qubit.Clearly, the above equation describes discrete-time quan-tum walk on a dynamical percolation graph in presenceof external noise. III. QUANTUM WALK OVER A SQUARELATTICE
We consider a discrete-time quantum walk over asquare lattice and assume that dynamic percolation isoccuring during the walk. Moreover, Markovian noise isacting on the walker, which is a qubit here. We analysethe evolution of the walker’s state and study the mixingtime by constructing zone basis.The zone is a boundary starting from the nearestneighour of any choice of vertex, increasing in size as wemove far from our chosen vertex. The number of vertexor nodes which fall on a zone boundary depends on thecoordination number C . In case of square lattice onehave C = 4, for triangular lattice C = 6 and so on. Incase of square lattice, the largest boundary will contain N number of nodes, where N = total number of verticesin the lattice. Thus, for a square lattice number of nodeson a zone boundary increases as a multiple of 4. Toconstruct zone basis we start from the origin and definethe basis vectors as | Z (cid:105) = | , (cid:105)| Z (cid:105) = | , (cid:105) + | , (cid:105) + | , − (cid:105) + |− , (cid:105)| Z (cid:105) = | , (cid:105) + | , (cid:105) + | , (cid:105) + |− , (cid:105) + |− , (cid:105) + |− , − (cid:105) + | , − (cid:105) + | , − (cid:105) (6)and so on. The zone basis | Z M (cid:105) corresponding to theboundary of the M th zone is a linear combination of 4 M no. of vertex states. Having defined the zone basis wenow find out the shift operators which control the ran-dom walk. As we have considered square lattice, two shiftoperators are needed to describe the walk. For K th real-ization of graph structure these two operators are givenby[25] S x K = (cid:88) { x,y } (cid:88) c =0 ( (cid:88) { ( X,y ) , ( x,y ) }∈ κ | c (cid:105)(cid:104) c |⊗| X, y (cid:105)(cid:104) x, y | + (cid:88) { ( X,y ) , ( x,y ) } / ∈ κ ( σ x ⊗ I ) | c (cid:105)(cid:104) c |⊗| x, y (cid:105)(cid:104) x, y | ) S y K = (cid:88) { x,y } (cid:88) c =0 ( (cid:88) { ( x,Y ) , ( x,y ) }∈ κ | c (cid:105)(cid:104) c |⊗| x, Y (cid:105)(cid:104) x, y | + (cid:88) { ( x,Y ) , ( x,y ) } / ∈ κ ( σ x ⊗ I ) | c (cid:105)(cid:104) c |⊗| x, y (cid:105)(cid:104) x, y | ) (7)where X = x ⊕ c and Y = y ⊕ c . The unitary op-erator defined in Eq.(2) can, therefore, be written as U ( x,y ) κ = { S yκ . ( C ( α,β ) ⊗ I ) }{ S xκ . ( C ( α,β ) ⊗ I ) } . The shiftoperators determine the evolution of the walker’s posi-tion with time. However, in case of an empty graphthe walker has no option to shift anywhere, only thespin state will change. Therefore, for an empty graphthe resulting effect on the walker can be expressed as ξ κ =0 ( ρ ) = ( σ x C ( α,β ) ⊗ I ) ρ { ( σ x C ( α,β ) ⊗ I ) } † where K = 0 denotes empty graph realization. Thus by choos-ing an empty graph one can easily discard vertex sys-tem H N . If Markoian noise simultaneously affects walkerthen operation on H will be [26, 27] ξ ( ρ c ) = (1 − p )( σ x C ( α,β ) ) ρ c { ( σ x C ( α,β ) ) } † + p (cid:88) l ( σ x C ( α,β ) ) K l ρ c K † l { ( σ x C ( α,β ) ) } † (8)where, ρ c ∈ L ( H ), ξ : L ( H ) → L ( H ) is the mappingand K l are the Krauss operators satisfying the condition (cid:80) l K † l K l = 1. The superscript c stands for coin state.Since empty graph operation acts only on the spin state,we can use it to construct the walker’s state after arbi-trary iteration. For that purpose, we first need to findout the position probabilities of the walker in differentzones. Starting from the origin the walker will traversethe entire lattice through succesive iterations, and hence we can define the position probabilities of the walker indifferent zones as P = Tr[( I ⊗ | Z (cid:105) (cid:104) Z | ) ρ ] P = Tr[( I ⊗ | Z (cid:105) (cid:104) Z | ) ρ ] P = Tr[( I ⊗ | Z (cid:105) (cid:104) Z | ) ρ ]... P M = M Tr[( I ⊗ | Z M (cid:105) (cid:104) Z M | ) ρ ] (9)Using the above defined probability distribution we canfinally write the state of the walker when it reached M th zone as ρ = P ξ n e ( ρ c ) ⊗ | Z (cid:105) (cid:104) Z | + M (cid:88) m =1 P m ρ cm ⊗ | Z m (cid:105)(cid:104) Z m | (10)where, ρ c = λξ n e ( ρ c ) + (1 − λ ) ξ n e − ( ρ c ), ρ c = λξ n e − ( ρ c ) + (1 − λ ) ξ n e − , ρ cM = ξ n e − m ( ρ c ) and P >P > P > ......... > P M holds until the walker reachesthe uniform distribution. From the above equation it isclear that the state of the walker depends explicitly onthe weight of edges λ ∈ [0 ,
1] but not on any specific re-alization occuring randomly. The operation ξ acts onlyon the coin space (qubit space) and changes the direc-tion of the walk. The total number of effective iterationswhich a qubit has undergone while being in a particularzone is given by n e . In our case n e = 3 m + n , where m and n denotes zone and iteration number respectively,0 ≤ m ≤ M = √ N and 0 ≤ n ≤ N is the total numberof vertices in the lattice.Now, depending on the different values of the perco-lation parameter λ the graph structure changes in severalways. For λ = , edges will fall or break randomly, andtherefore, the walk will be fully unbiased if one choosesHadamard coin. When λ →
0, the spreading of walk willbe much much slower. For λ = 1, there will be perfectanti-correlation between spin states situated at two an-tipodal points of any particular zone. When λ < < λ ≤ . The reason behind this choice isthat in this limit the walker can not find any large con-nected path between two widely separated zones. There-fore, upto two steps of iteration the walk will be boundedbetween two consecutive zones.In the next we will analyse the mixing time of thequantum walk considering two types of noises: continu-ous phase damping and bit-flip. A. Percolation with continuous phase dampingnoise
Let us consider that phase damping noise is acting con-tinuously on the walker (qubit) and the noise parameter p = Γ dep dt , where Γ dep denotes dephasing rate. For con-tinuous dephasing (1 − p dep ) n = e − Γ dep t . Now, if wechoose the initial state of the walker as ρ c = { a | ψ θ,φ (cid:105) (cid:104) ψ θ,φ | + (1 − a ) I } ⊗ | , (cid:105) (cid:104) , | (11)then the empty graph operation looks like ξ ( ρ c ) = 12 (cid:18) − a cos θ a sin θe − iφ (1 − p dep ) n a sin θe iφ (1 − p dep ) n a cos θ (cid:19) = 12 (cid:18) − a cos θ a sin θe − iφ e − Γ dep t a sin θe iφ e − Γ dep t a cos θ (cid:19) Therefore, the state of the walker will be ρ dep ( t ) = P ξ n e ( ρ c ) ⊗ | Z (cid:105) (cid:104) Z | + P ρ c ⊗ | Z (cid:105) (cid:104) Z | + P ρ c ⊗ | Z (cid:105) (cid:104) Z | + .......... + P M ρ cn ⊗ | Z M (cid:105) (cid:104) Z M | (12)where, ρ c = λξ n e ( ρ c ) + (1 − λ ) ξ n e − ( ρ c ), ρ c = λξ n e − ( ρ c ) + (1 − λ ) ξ n e − , ρ cM = ξ n e − m ( ρ c ). The condi-tion Tr[ ρ dep ] = 1 implies that the walk is bounded within M th zone.For an N vertex lattice, the trace norm between ρ dep ( t ) and the final asymptotic state ρ dep ( ∞ )[28] is givenby D [ ρ dep ( t ) , ρ dep ( ∞ )] = 12 || ρ dep ( t ) − ρ dep ( ∞ ) || (13)If we define, ρ c ( t ) = Tr p [ ρ ( t )] then it can be shown that D [ ρ dep ( t ) , ρ dep ( ∞ )] ≥ D [ ρ cdep ( t ) , ρ cdep ( ∞ )] (14)Thus for an N vertex square lattice, trace norm can becalculated as2 D [ ρ cdep ( t ) , ρ cdep ( ∞ ) ≤ || P ξ n e ( ρ c ) − I N || + 4 || P ξ n e − ( ρ c ) − I N || + 8 || P ξ n e − ( ρ c ) − I N || + ........ + 4 M || P M ξ n e − M ( ρ c ) − I N || (15)Due to percolation and the dephasing noise the coin statewill become maximally mixed for n e = (3 m + n ) → ∞ .Now, position probability distributions of the walker ateach zone will be nearly equal at mixing time ( t (cid:39) t mix < ∞ ). To study the asymptotic behaviour we choose N (cid:39)O (10 ) or higher. For t (cid:39) t mix one can say | P ( t ) − P u | (cid:39) | P ( t ) − P u |(cid:39) | P ( t ) − P u |(cid:39) ............ (cid:39) | P M ( t ) − P u | (cid:39) δ dep where, P u = ( N ) is the uniform distribution of the pos-tion of the walker inside ( M × M ) sublattice. Choosinglarger and larger lattice will also increase the size of thissublattice and thus we can cover a larger set. Therefore,for t (cid:39) t mix we can write P ( t ) (cid:39) P ( t ) (cid:39) P ( t ) (cid:39) ... (cid:39) P M ( t ) (cid:39) ( δ dep + 2 N ) (16)Substituting the values of the probabilities from Eq.() inEq.() one gets for t = t mix ,2 D [ ρ cdep ( t ) , ρ cdep ( ∞ )] ≤ || ( δ dep + 2 N ) ξ n e ( ρ c ) − I N || +4 || ( δ dep + 2 N ) ξ n e − ( ρ c ) − I N || + ... +4 M || ( δ dep + 2 N ) ξ n e − M ( ρ c ) − I N || (17)By putting λ = in the above equation we get2 D [ ρ dep ( t ) , ρ dep ( ∞ )] ≤ δ dep + 4 M A + 4 M − (cid:88) k =1 kB (18)where, A = [ δ dep + a ( δ dep + N ) e − dep t ] , B = [ δ dep +( δ dep + N ) a sin θcos φe − dep t ] . Now, by taking theapproximation a (1 + Nδ dep ) e − dep t (cid:28) N (cid:29) D [ ρ cdep ( t ) , ρ cdep ( ∞ )] ≤ N δ dep + a M X + a M Y (19)where X = δ dep (1 + Nδ dep ) e − dep t and Y = δ dep (1 + Nδ dep ) sin θcos φ { e − t mix +3)Γ dep (1 − e − M Γ dep )(1 − e − dep ) . Now,the coin state that we have considered has Bloch sphererepresentation. Therefore, taking average over all possi-ble coin states we have the average trace distance as D [ ρ cdep ( t ) , ρ cdep ( ∞ )] = 14 π (cid:90) (cid:90) ( θ,φ ) D ( θ, φ ) sinθdθdφ = 14 N δ dep + a M X × (cid:26) f ( M, Γ dep )3 (cid:27) (20)where, f ( M, Γ dep ) = e − dep (1 − e − M Γ dep )(1 − e − dep ) is a functionof the size of lattice and dephasing rate. Hence, we haveevaluated the trace norm nearly arround the mixingtime. Now, there comes a question. Using the noise ratecan we estimate trace norm for all t < t mix , such thatthe trace norm can be written as D [ ρ cdep ( t ) , ρ cdep ( ∞ )] = (cid:104) D ( t = ∞ ) (cid:105) + (cid:104) D [ ρ ( t = 0) , ρ ( N, t = ∞ )] (cid:105) e − dep t − (cid:104) D ( ∞ ) (cid:105) e − dep t (21) FIG. 1: (Color on-line) Tunable dephasing rate for N ∼ or higher order lattice size. We choose δ dep = N √ N as the closeness withthe uniform distribution P u inside ( M × M ) sublat-tice. Choosing δ dep (cid:39) O ( N − ) makes the quantity M δ dep (cid:16) Nδ dep (cid:17) (cid:39) O (1) and then we can write a (cid:26) f ( M, Γ dep )3 (cid:27) = 12 (cid:114) (1 − N ) + a and hence f ( M, Γ dep ) = 3 (cid:113) (1 − N ) + a a − (22)Therefore, for a fixed coin parameter a and fixed vertexnumber N , we get a specific value of f ( M, Γ dep ). Plot-ting f ( M, Γ dep ) vs . Γ dep we can estimate those tunabledephasing rates for which the following equation holds inthe range 0 ≤ t ≤ ∞ : D [ ρ cdep ( t )) , ρ cdep ( ∞ )] = 14 √ N { − e − t Γ dep } + (cid:113)(cid:0) − N (cid:1) + a × e − t Γ dep t (23)From the above equation one can imply an interest-ing phenomenon. Suppose the walker has passed severalzones without knowing how many different or similar re-alizations has occured during its walk. If anyone is nowinterested to know that the distance between walker’scurrent state and its asymptotic state, then he must tuneat a particular dephasing rate keeping all other parame-ter fixed. Tuning at that particular rate will enable himto know trace norms at all previous times. From the FIG. 2: (Color on-line) Time evolution of average trace dis-tance ( D ( t, δ )) under continuous dephasing. trace norm, one can estimate the mixing time in case ofΓ dep → t mix = log e D [ ρ cdep ( t ) ,ρ cdep ( ∞ )] − √ N √ (1 − N )2+ a − √ N log e (1 − dep ) (24) B. Percolation with continuous bit-flip noise
Now we analyse the mixing time of the quantum walkby considering that bit-flip noise is acting on the qubitcontinuously. The initial state of the walker is the sameas given in Eq.(). The empty graph operation in this casewill be ξ ( ρ c ) = 12 (cid:18) acosθe − Γ bit t AasinθBasinθ − acosθe − Γ bit t (cid:19) (25)where A = cosφ − isinφe − Γ bit t and B = cosφ + isinφe − Γ bit t Using the above equations and proceedingsimilar to previous calculations we can find out the tracenorm D [ ρ cbit ( t ) , ρ cbit ( ∞ )] at t = t mix . For that we assume | P ( t ) − P u | (cid:39) | P ( t ) − P u |(cid:39) ................... (cid:39) | P M ( t ) − P u | (cid:39) δ bit (26)Now, for φ = π , the coin state will become maximallymixed in the asymptotic limit. Therefore, by taking φ = π and λ = we get the average trace norm at t (cid:39) t mix as FIG. 3: (Color on-line) Tunable bit flip rate for three differentlattice size. D [ ρ cbit ( t ) , ρ cbit ( ∞ )] ≤ N δ bit (cid:40) M δ bit a (cid:18) N δ bit (cid:19) (cid:41) × { f bit ( M, Γ bit ) } e − bit t mix (27)where f bit ( M, Γ bit ) = e − bit ( e − bit ) ( − e − M Γ bit ) ( − e − bit ) is afunction that depends on the noise rate as well as onlattice size. Now, analytically one can calculate the tracedistance of initial and the asymptotic state as D [ ρ cbit ( t ) , ρ cbit ( ∞ )] = 12 (cid:115)(cid:26)(cid:18) − N (cid:19) + acosθ (cid:27) + a (28)Using Simpson’s rd rule, one can numerically calculatethe average trace norm over θ ∈ [ − π, π ] which gives dif-ferent values of it against different values of coin param-eter a . The values are given in table 1. (cid:10) D φ = π [ ρ ( t = 0 , θ ) , ρ ( N, ∞ )] (cid:11) a FIG. 4: (Color on-line) Time evolution of average trace dis-tance ( D ( t, δ )) under continuous bit flip channel. Now, if we want to calculate the time evolution oftrace norm for 0 ≤ t ≤ ∞ , analytically one can say that D [ ρ cbit ( t ) , ρ cbit ( ∞ )] = D [ ρ cbit ( t ) , ρ cbit ( ∞ )]+ D [ ρ cbit ( t = 0) , ρ cbit ( ∞ )] e − bit t − D [ ρ cbit ( t ) , ρ cbit ( ∞ )] e − bit t (29)So, the exponentially converging form of trace norm to-wards the asympotitc state is possible iff one can say that M δ bit (cid:16) Nδ bit (cid:17) a { f bit ( M, Γ bit ) } = D [ ρ cbit ( t = 0) , ρ cbit ( ∞ )] − D [ ρ cbit ( t ) , ρ cbit ( ∞ )]The above equality will satisfy iff we choose δ bit (cid:39) O (cid:0) N (cid:1) ,this value of δ bit increases the separation from uniformdistribution over ( M × M ) sublattice. Putting the chosenvalue of δ bit in the above equation gives94 √ N a { f bit ( M, Γ bit ) } + 14 = D [ ρ cbit ( t = 0) , ρ cbit ( ∞ )]Thus evolution of trace norm simply becomes D [ ρ cbit ( t ) , ρ cbit ( ∞ )] = 14 (1 − e − bit t )+ D [ ρ cbit ( t = 0) , ρ cbit ( ∞ )] e − bit t (30)The above equation implies an exponential convergenceof trace norm towards a fixed number 0 .
25. So spreadingof walk inside the largest square sublattice remains veryfar from the uniform distribution. Whereas under contin-uous dephasing this spreading becomes more and moreclose to the uniform distribution with increasing size oflattice. From the time evolution of trace norm one canestimate roughly the mixing time t bitmix as t bitmix (cid:39) log e (cid:16) D [ ρ cbit ( t ) ,ρ cbit ( ∞ )] − D [ ρ cbit ( t ) ,ρ cbit ( ∞ )] − (cid:17) log e (1 − bit ) (31) IV. CONCLUSION
In this paper we have studied quantum walk over asquare lattice in a scenario where dynamic percolationis changing the graph structure randomly and Marko-vian noise is acting continuously on the walker (qubit).We have considered two types of noises, phase damp-ing and bit-flip, and investigated the quantum walk byconstructing zone basis on the square lattice. The ini-tial coin state chosen here is a generic qubit state. Theamount of mixedness included in the coin state has beenparametrized by the variable a . It is obvious that theasymptotic state of the walker in C ⊗ C N will be ⊗ N N ,which implies an uniform distribution. Our result recon-firms the fact that quantum walk is faster than the clas-sical one. Thus, for a → a → dep or Γ bit ) of Markovian noise. Under de-phasing noise the probability distribution of walker’s po-sition becomes close to the uniform distribution with thecloseness factor δ dep ≡ O ( N − ). Whereas, in case of bitflip noise the closeness parameter δ bit ≡ O ( N n ) impliesthat bit flip noise resists the walk to spread uniformly.Finally, this work can be extended for higher dimensionswith multiple walker. Acknowledgement
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