Boundary-induced coherence in the staggered quantum walk on different topologies
BBoundary-induced coherence in the staggered quantum walk on different topologies
J. Khatibi Moqadam and A. T. Rezakhani Department of Physics, Sharif University of Technology, Tehran 14588, Iran (Dated: April 13, 2018)The staggered quantum walk is a type of discrete-time quantum walk model without a coin which can begenerated on a graph using particular partitions of the graph nodes. We design Hamiltonians for potentialrealization of the staggered dynamics on a two-dimensional lattice composed of superconducting microwaveresonators connected with tunable couplings. The naive generalization of the one-dimensional staggered dy-namics generates two uncoupled one-dimensional quantum walks thus more complex partitions need to beemployed. However, by analyzing the coherence of the dynamics, we show that the quantumness of the evolu-tion corresponding to two independent one-dimensional quantum walks can be elevated to the level of a singletwo-dimensional quantum walk, only by modifying the boundary conditions. In fact, by changing the latticeboundary conditions (or topology), we explore the walk on different surfaces such as torus, Klein bottle, realprojective plane and sphere. The coherence and the entropy reach different levels depending on the topology ofthe surface. We observe that the entropy captures similar information as coherence, thus we use it to explore theeffects of boundaries on the dynamics of the continuous-time quantum walk and the classical random walk.
PACS numbers: 03.67.-a, 05.40.Fb, 02.40.Pc, 85.25.Hv
I. INTRODUCTION
The quantization of the classical random walks can beachieved in different ways, among which the coined discrete-time quantum walk and the continuous-time quantum walk(CTQW) are widely known [1]. In particular, the staggeredquantum walk (SQW) [2], a discrete-time quantum walkmodel without a coin, has attracted growing attention re-cently. This model has various interesting features, forexample, it enables efficient quantum search algorithms intwo dimensions—providing approximately quadratic speedupwith respect to the classical case [3]. The two-dimensional(2D) CTQW model (including no coin either), however, failsto speed up the search algorithms, unless additional degreesof freedom as extra sites are embedded into the lattice [4].Interestingly, the SQW somehow has already taken into ac-count this idea, where, in fact, the coin degrees of freedomhas been converted to extra nodes in the corresponding lat-tice. Besides, in terms of physical implementations, the SQWis more favorable than the coined discrete-time quantum walkdue to the very absence of the coin. Although removing thecoin operator comes with the price of demanding a dynamicalgraph for the walk evolution, by employing the state of theart superconducting circuits technology, the dynamics can berealized in a lattice of superconducting microwave resonatorswith controllable couplings [5].To generate the staggered dynamics on a graph, distinctpartitions of the set of graph nodes are considered. The el-ements of each partition contains only the first-neighbor ver-tices, namely the nodes which are all connected by edges.Such partitions are called tessellations, and associated to eachtessellation a unitary operator is constructed. The conditionthat the set of graph edges should be covered in the union of alltessellations determines the number of required unitary opera-tors. The SQW evolution operator is obtained by multiplyingall those operators. Being devised principally for constructingthe staggered operators, the partitioning process can also beused to obtain the staggered Hamiltonians [6]. The connection between the SQW and other quantum walkmodels has been explored in Refs. [7–10]. Different discrete-time quantum walk models including the 2-tessellable SQWcan be analyzed under a common framework called the two-partition model [11], inspired by the staggered dynamics.The 1D coined quantum walk model was used to exploretopological phases in condensed matter systems [12–16]. In-cluding a position-dependent phase shift in each step of thewalk, the dynamics of a charged particle in the presence ofexternal fields can be simulated [17, 18]. Using such Blochoscillating quantum walks, topological invariants correspond-ing to the split-steps quantum walks [12] can be directlymeasured, employing a superconducting microwave resonatorcavity coupled to a transmon qubit [19, 20]. The 1D SQW,on the other hand, is related the Su-Schrieffer-Heeger model[21, 22] which was also used in simulating topological invari-ants [23–25].The effects of the boundary conditions determining thetopological properties of the system are indispensable in thedynamics associated with topological insulators [26, 27] andtopological quantum computing [28, 29]. The properties ofdiscrete- and continuous-time quantum walks on differenttopologies have already been studied in Refs. [30–36]. For ex-ample, the quadratic speed up of quantum-walk-based searchalgorithms was obtained on lattices with periodic boundaryconditions, that is with a torus topology [1, 3, 4].In this paper, we address the implementation of the 2DSQW referring to a lattice of superconducting microwave res-onators coupled with adjustable devices. Two types of Hamil-tonians are constructed, one of which generates two 1D SQWsand the other one generates a 2D SQW. The quantumness[37, 38] of those dynamics—quantified by the coherence ofthe walker during the evolution—are different. However,by analyzing the SQW dynamics on a lattice with differentboundary conditions, resembling 2D manifolds such as torus,Klein bottle, real projective plane and sphere [39], we showthat the coherence can be increased by using twisted bound-ary conditions. We also analyze the dynamics of the entropy, a r X i v : . [ qu a n t - ph ] A p r which appears to be qualitatively similar to the behavior ofcoherence in the dynamics; hence we use it to quantify thequantumness of the CTQW. The boundary-independence ofthe entropy associated with the classical random walk dynam-ics supports that the boundary-induced coherence in quantumwalk dynamics can be reflected by the entropy.The paper is organized as follows. In Sec. II, we describehow to construct 2D SQW Hamiltonians and to realize the cor-responding dynamics. In Sec. III, we use different boundaryconditions to change the topology of the lattice and find theircorresponding Hamiltonians. The properties of the walk dy-namics on these topologies are analyzed in Sec. IV. In Secs.Vand VI, the properties of the CTQW and the classical randomwalk are explored. Summary and further discussions are givenin Sec. VII. II. STAGGERED QUANTUM WALK DYNAMICS
Consider a lattice of harmonic resonators whose dynamicsis described by the time-dependent tight-binding Hamiltonianwith switchable couplings ( (cid:126) ≡ ), H ( t ) = (cid:88) n ω n a † n a n − (cid:88) (cid:104) n,m (cid:105) κ nm ( t )( a † n a m + a † m a n ) , (1)where ω n are the resonators frequencies, a † n and a n are thecreation and annihilation operators satisfying [ a n , a † m ] = δ nm , and κ nm ( t ) are the switchable couplings betweennearest-neighbor resonators (denoted by (cid:104) n, m (cid:105) ). We restrictthe system to the “single-photon” regime (cid:80) n (cid:104) a † n a n (cid:105) = 1 ,where (cid:104) a † n a n (cid:105) is the average of the operator in the systemstate. In this regime, the canonical basis for the lattice Hilbertspace is associated with the presence of photon at each singlesite (resonator) of the lattice. The physical implementations ofsuch Hamiltonian, with arrays of superconducting microwaveresonators coupled through superconducting quantum inter-ference devices ( SQUID s), have already been investigated inRefs. [5, 40–42]. The tunable couplings in those systems areachieved by adjusting the magnetic flux threading the
SQUID loop. The typical frequencies are of the order of GHz for theresonators and MHz for the couplings.In the following, we describe how to adjust the couplings inthe system Hamiltonian (1) to realize the SQW dynamics ona 2D lattice. The rigorous mathematical framework for con-structing the SQW on a generic graph has been put forwardin Refs. [2, 8] (and a SQW-based quantum search algorithmswas presented in Ref. [3]). Starting with the 1D SQW im-plementation of Ref. [5], we give explicit forms for the 2DSQW Hamiltonians using the tight-binding model with tun-able couplings. The SQW includes at least two unitary opera-tors which are obtained according to independent partitions ofthe graph vertices. The elements in each partition should con-tain only the neighboring vertices—those that are connectedby some edges (a single vertex is also accepted). Whereaseach partition contains all the graph vertices, it includes onlypart of the graph edges. New partitions are then considered toinclude the edges not covered. Partitioning the graph vertices
FIG. 1. Configuration of the turned-on couplings in the Hamiltonian(1) to realize the SQW Hamiltonians for the 1D lattice in (a) andthe 2D lattice in (b), with d = 6 . The ending sites specified withthe same numbers inside the diamonds are connected. To generateeach Hamiltonian all the sites are contributed. However, the set ofall possible edges is divided into two subsets in the 1D case and foursubsets in the 2D case. is performed as many times as required so that all edges of thegraph are covered in the union of the partitions. Note that theintersection of the partitions should contain no edges. Asso-ciated to each partition, a unitary operator is defined and themultiplication of all such unitaries is the SQW operator. Inthe extended version of the SQW [6] the graph partitions areused to construct the SQW Hamiltonians which then generatethe walk operators.Realizing the SQW on a 1D lattice with d sites requirestwo unitary operators corresponding to two time-independentHamiltonians [5]. Each Hamiltonian is comprised of a col-lection of disjoint pairs of interacting resonators. Turning onthose couplings in the Hamiltonian (1) which couple odd-evenresonators generates H = d/ ⊗ ( ω − κσ x ) , (2)where d/ is the ( d/ × ( d/ identity operator with even d , σ x = ( ) is the x -Pauli matrix, all the resonators areconsidered at resonance at frequency ω , and all the nonzerocouplings are taken to be κ . Moreover, the interaction (hop-ping) term in the Hamiltonian (1) for any pair of resonatorsis written in terms of σ x . The Hamiltonian (2) describes thecollection of disjoint odd-even pairs of interacting resonators.The configuration of the turned-on couplings for Hamiltonian(2), with d = 6 , can be seen in Fig. 1 (a).In the same manner, the couplings between even-odd res-onators in the Hamiltonian (1) can be turned on to generate H = ω − κ ( d − / ⊗ ( ω − κσ x ) − κ ω , (3)which describes a collection of disjoint even-odd pairs of res-onators. The periodic boundary conditions are used here. InFig. 1 (a), the configuration of the turned-on couplings for thiscase can also be seen. Note that the Hamiltonians (2) and (3)do not commute.The SQW is implemented by repeatedly switching theHamiltonian (1) between H and H . In fact, each stepof the walk consists of generating H in the time interval [0 , τ ) followed by generating H in [ τ, τ ) . More details forthe walk on a 1D array can be found in Ref. [5].For a 2D lattice, each step of the SQW is realized byapplying four unitary operators corresponding to four time-independent Hamiltonians [3]. Suppose that the lattice has N = d × d sites with even d and periodic boundary con-ditions. This lattice contains d edges (hence couplings).Again, each staggered Hamiltonian is constructed by turningon only parts of the couplings in the form (1) such that thelattice comprises a collection of disjoint pairs of interactingresonators. The pairs can be selected row-by-row or column-by-column (bearing in mind that only neighboring sites canbe paired). The first SQW Hamiltonian can be constructed byswitching on those couplings in the Hamiltonian form (1) thatgenerate odd-even pairs in each row of the 2D lattice, H = d ⊗ H . (4)The second Hamiltonian is obtained by generating even-oddpairs in each row of the lattice, H = d ⊗ H . (5)Finally, two more Hamiltonians are constructed by creatingpairs of interacting resonators in the columns of the lattice, H = H ⊗ d , (6) H = H ⊗ d . (7)The configuration of the turned-on couplings for the Hamilto-nians (4) - (7) (with d = 6 ) are depicted in Fig. 1 (b).The Hamiltonians (4) and (5) commute with those in Eqs.(6) and (7), which implies the dynamics on the rows is inde-pendent of the dynamics on the columns. In fact, such Hamil-tonians generate two 1D SQWs in the horizontal and verticaldirections. One way to couple the dynamics in these two di-rections to obtain a 2D dynamics is to change the way thepairs of the coupled resonators are selected in the above con-structions. For example, we can turn on those couplings in Eq.(1) that generate the Hamiltonian (2) for the odd rows and theHamiltonian (3) for the even rows. The turned-on couplingsin this case are shown in Fig. 2 below H (cid:48) . The Hamiltoniantakes the form H (cid:48) = d ⊗ H + d ⊗ H , (8) FIG. 2. Configuration of the turned-on couplings in the Hamiltonian(1) to realize the non-commutating SQW Hamiltonians (cf. Fig. 1),with d = 6 . In each diagram the ending sites specified with the samenumbers inside the diamonds are connected. where d = d/ ⊗ (cid:18) (cid:19) , d = d/ ⊗ (cid:18) (cid:19) . The second Hamiltonian is constructed similarly but by in-terchanging the place of the 1D terms in the Hamiltonian (8).This is achieved by adjusting the couplings in Eq. (1) suchthat the Hamiltonians (2) and (3) are generated for the evenand odd rows, respectively, and hence H (cid:48) = d ⊗ H + d ⊗ H , (9)which corresponds to Fig. 2 below H (cid:48) .The next two Hamiltonians are constructed using the sameidea; however, the couplings in the Hamiltonian (1) are ad-justed such that the 1D array Hamiltonians (2) and (3) aregenerated for the columns of the lattice, alternately. In Fig.2 below H (cid:48) and H (cid:48) , the desired couplings in these cases areshown where the corresponding Hamiltonians can be writtenas H (cid:48) = H ⊗ d + H ⊗ d , (10) H (cid:48) = H ⊗ d + H ⊗ d . (11)We note that the Hamiltonians (8) and (9) do not commutewith those in Eqs. (10) and (11); thus, these Hamiltoniansgenerate a genuinely 2D SQW.The Hamiltonian form (1) can be controlled such that theHamiltonians (4) - (7) or (8) - (11) are generated in a sequence.Such sequence generates the SQW operator. For instance, thesequence H (cid:48) H (cid:48) H (cid:48) H (cid:48) , each for the period τ , within the timeinterval [0 , τ ) , corresponds to the time evolution U (0 , τ ) = e − iτH (cid:48) e − iτH (cid:48) e − iτH (cid:48) e − iτH (cid:48) . (12)Calculating such evolution is reduced to obtaining the evolu-tions of the 1D Hamiltonians (2) and (3), which is achievedby noting e − iτ ( ω − κσ x ) = e − iωτ (cid:18) cos κτ i sin κτi sin κτ cos κτ (cid:19) . (13)Recalling that all resonators are in resonance and setting κτ =2 π(cid:96) + π/ , for an integer (cid:96) , the evolution (12) implements theSQW dynamics that was used in Ref. [3] for designing anefficient quantum search algorithm.To realize the SQW dynamics the system is initialized bygenerating a photon in one of the resonators, e.g., in the mid-dle of the lattice, | ψ (cid:105) = (cid:12)(cid:12) ( d − d ) / (cid:11) . (14)After applying the evolution (12) l times, for the total timeperiod τ l , the system evolves to the final state U l | ψ (cid:105) . Thesystem can then be measured to find the photon in one of theresonators. For a system composed of superconducting mi-crowave resonators coupled through SQUID elements, similarprotocols as suggested in Ref. [5] can be used to initializeand measure the quantum walk. The probability distributionof finding the photon in the lattice is given by P l ( n ) = (cid:12)(cid:12) (cid:104) n | U l | ψ (cid:105) (cid:12)(cid:12) , (15)where (cid:8) | n (cid:105) ; n = 1 , . . . , N = d (cid:9) is the canonical basis forthe lattice Hilbert space. III. WALK ON 2D MANIFOLDS
We described how to control the couplings in the systemHamiltonian (1) in order to realize the SQW dynamics on a2D lattice with the periodic boundary conditions in both di-rections. In this section, we explore the quantum walk dynam-ics on various 2D manifolds or surfaces which are obtainedthrough modifying the boundary conditions.A 2D manifold is a topological space that locally has thestructure of the Euclidean plane R . Basic 2D manifolds canbe obtained by appropriately identifying the boundaries of asquare [39]. For instance, by identifying two opposite sidesof a square, a cylinder is obtained; identification the other twosides creates a torus. For the 2D square lattice, identificationof the boundaries can be performed by connecting the bound-ary sites by edges, i.e., by coupling any two sites that are sup-posed to be identified.In this manner, by using the periodic boundary condi-tions for the 2D lattice, a torus is obtained. As it is im-plied by Figs. 1 (b) and 2, − κ − ( (cid:80) i =1 H i − ω d ) and − κ − ( (cid:80) i =1 H (cid:48) i − ω d ) correspond to the adjacency ma-trices for the 2D lattice with the torus topology. However, de-pending on the way the SQW Hamiltonians are constructed, FIG. 3. Configuration of the turned-on couplings in the Hamiltonian(1) to realize the twisted boundary conditions, for d = 6 . In eachdiagram, the end sites specified with the same numbers inside thediamonds are connected. the choices of the boundary conditions may not affect all SQWHamiltonians. This can be seen in Fig. 1 (b), where theHamiltonians H and H do not feel the boundary conditions.The repeated application of the dynamics (12) on the initialstate (14) gives then the evolution of a quantum walker on atorus.Another possibility for coupling two opposite sides of the2D lattice is to twist one of the edges and then perform theidentification. In this case, the sites on the boundaries arecoupled in the opposite directions—Fig. 3. The other twosides of the lattice can be coupled as before. By doing suchidentification a 2D nonorientable surface, called Klein bottle,is obtained. The SQW Hamiltonians generating the walk onthe Klein bottle can be constructed similarly to the Hamilto-nians (4) - (7) for the torus, depicted in Fig. 1 (b). Let uschoose the Klein bottle boundary conditions in the horizontaldirection, as seen in the left plot of Fig. 3, and keep the ver-tical boundary conditions as before. Thus, the correspondingHamiltonian H KB takes a different form compared with H ,given by H KB = d ⊗ (cid:101) H − κ Σ hor , (16)where (cid:101) H is obtained from Eq. (3) by replacing two − κ swith s in the first and the last rows, decoupling the first andthe last resonators in each row of the lattice. The horizontaltwisted boundary conditions are imposed by the matrix Σ hor = J d ⊗ (cid:0) σ + d + σ − d (cid:1) , (17)where J d is the row reversed version of the d -dimensionalidentity matrix (a matrix with s on the main anti-diagonaland s elsewhere) and σ + d ( σ − d ) is equal to the d -dimensionalzero matrix except for the top-right (down-left) corner entrywhich is replaced with .The new boundary condition, however, does not modify H KB with respect to H . Using the periodic boundary con-ditions for the columns of the lattice, as before, the verticalSQW Hamiltonians for the Klein bottle ( H KB and H KB ) re-main equal to H and H (respectively).If the twisted boundary conditions are applied for both di-rections (see Fig. 3), namely the opposite edges of the 2Dlattice are twisted and then identified, the resulting 2D mani-fold is the real projective plane. The SQW dynamics on sucha surface is generated by the Hamiltonians H RP i ( i = 1 , , ,equal to the Hamiltonians for the walk on the Klein bottle, and H RP = (cid:101) H ⊗ d − κ Σ ver , (18)where the vertical twisted boundary conditions are given by Σ ver = (cid:0) σ + d + σ − d (cid:1) ⊗ J d . (19)Finally, by identification of the adjacent sides (rather thanthe opposite sides) of the 2D lattice a sphere is obtained. Thestaggered Hamiltonians for the walk on the sphere, H S i ( i =1 , , , , can be constructed such that the first and the thirdHamiltonians remain the same as H and H , similarly tothe previous cases. The turned-on coupling for the other twoHamiltonians are shown in Fig. 4. The second Hamiltonian isgiven by H S = ω d − κP − κP − κP T d − ⊗ (cid:101) H − κQ T − κP T − κQ ω d , (20)which is the sum of a block diagonal matrix, obtained from theHamiltonian (5) by decoupling the sites on all four sides of thelattice, and the desired boundary conditions, shown in the leftdiagram in Fig. 4. In this equation, the boundary conditionsare incorporated to P d × d − d = ( d − × d − ) T ⊗ (1 × d − ) ,Q d × d − d = ( d − × d − d − × ) T ⊗ ( × d − . where we have represented P by [ P P ] and “ T ” denotestransposition. The fourth Hamiltonian (see the right diagramin Fig. 4) takes the form H S = (cid:101) H H ( d − ⊗ d (cid:101) H , (21)where H ( d − is similar to the 1D Hamiltonian (2) but fora chain of d − sites, and all other elements are . Note thatthe adjacency matrix of the lattice with the sphere topology isgiven by − κ − ( (cid:80) i H S i − ω d ) , which includes d − edges— edges less than the previous cases.For the Klein bottle, the real projective plane, and thesphere, the more complex version of the SQW Hamiltonians,based on Fig. 2, can also be constructed. However, since ourobjective in this paper is to study effects of the boundary con-ditions, we skip those cases.We remark that among the surfaces considered in this sec-tion, the sphere and the torus are orientable surfaces withgenus and , respectively. In contrast, the real projectiveplane and the Klein bottle are non-orientable surfaces withnon-orientable genus equal to and , respectively. FIG. 4. Configuration of the turned-on couplings in the Hamiltonian(1) to realize the sphere boundary conditions, for d = 6 . In eachdiagram the ending sites specified with the same numbers inside thesame boxes (diamonds or circles) are connected. IV. BOUNDARY-INDUCED COHERENCE
Having considered a 2D manifold and the correspondingstaggered Hamiltonians, as described in the previous section,the SQW dynamics U can be constructed according to Eq.(12). After applying the U operator l times on the localizedinitial state of the walker (photon) [Eq. (14)], the density ma-trix of the photon is obtained as (cid:37) l = U l | ψ (cid:105)(cid:104) ψ | U l . (22)The coherence of the (pure) state of the photon at a given step l can be quantified by [37] C l = N (cid:88) n,m =1 (cid:12)(cid:12) [ (cid:37) l ] nm (cid:12)(cid:12) − . (23)The diagonal elements of the density matrix ( [ (cid:37) l ] nn ) give theprobability distribution of finding the photon in different sites,as computed by Eq. (15). The Shannon entropy of the systemat step l is then calculated by E l = − N (cid:88) n =1 [ (cid:37) l ] nn log [ (cid:37) l ] nn . (24)The coherence and the entropy are appropriate quantitiesfor addressing quantumness in behavior of the quantum walkon different topologies [37, 38, 43]. In fact, the von Neumannentropy cannot be used here, since it is identically for theisolated system.Figure 5 shows C l /C l max in terms of the number of stepsfor quantum walks on the torus with the staggered Hamilto-nians (4) - (7), the Klein bottle with H KB i , the real projectiveplane with H RP i , the torus with the Hamiltonians (8) - (11),and the sphere with H S i . The corresponding dynamics arelabeled by T , KB , RP , T (cid:48) , and S , respectively. In gen-erating those plots, the system frequencies are set such that κτ = 2 π(cid:96) + π/ and ωτ = 2 π(cid:96) (cid:48) , for some integers (cid:96) and (cid:96) (cid:48) .The lattice size is fixed to N = 100 × sites.The coherence takes its minimum C l min = 0 for the initiallocalized state given in Eq. (14). It increases, then, for all the S T 'RP KBT C l / C l m a x l FIG. 5. The normalized coherence [Eq. (23)] for a 2D lattice withdifferent boundary conditions. The plots are generated by using thestate (14) as the initial state of the walker and setting d = 100 , n =1000 , κτ = π/ , and ωτ = 2 π . The inset shows the coherence forthe first steps. boundary conditions until the photon-wave function populatesthe boundary sites. It can be seen that within that interval theenvelope function of the coherence is a convex function (seethe inset in Fig. 5) but later it changes to a concave function.Except for the case T (cid:48) , the other dynamics are indistinguish-able until the boundaries are met. A single populated site canaffect up to the second neighbors, at each step of the staggeredevolution. Hence the effects of the boundary conditions, forthe photon initially located at the center, appears around thestep l = d/ . For the case T (cid:48) , due to the interference, it takesmore steps until the boundaries are sufficiently populated andthe interference between different parts of the wave-functionis started.The walk on the torus (green plot labeled with T ) has thelowest level of coherence during the whole dynamics. Thiscan be justified by recalling the staggered Hamiltonians cor-responding to this case generate two independent
1D quantumwalks. The coherence increases by applying twisted boundaryconditions which make the horizontal and vertical dynamicscorrelated. The plots of KB and RP lie above the plot of T . The dynamics on S has the highest coherence. The 2Dquantum walk on the torus (black plot labeled with T (cid:48) ) gen-erates the coherence comparable with the coherence for thewalk on the real projective plane ( RP ). The oscillatory be-havior of the coherence plots is considerably decreased for thedynamics corresponding to the case T (cid:48) .The maximum of the coherence, corresponds to the diago-nal state, the maximally coherent state [37] | ψ diag (cid:105) = N (cid:88) n =1 | n (cid:105) / √ N , (25)and its value is C l max = d − . The plots in Fig. 5 are farfrom the maximum value C l /C l max = 1 associated with the S T 'RP KBT E l / E l m a x l
010 50
FIG. 6. The normalized entropy [Eq. (24)] for a 2D lattice withdifferent boundary conditions. The plots are generated by using (14)as the initial state of the walker and setting d = 100 , n = 1000 , κτ = π/ , and ωτ = 2 π . The inset shows the entropy for the first steps. state (25) and apparently the state of the photon never con-verges to that diagonal state.Figure 6 shows E l /E l max for quantum walks on a latticewith different boundary conditions. The entropy is for thelocalized initial state (14) and then increases having a con-cave envelope function in terms of the walk steps (see also theinset in Fig. 6). The walk on the torus (green plot labeledwith T ) has the lowest level for the entropy. The next levelscorrespond to the walk on the Klein bottle and the real projec-tive plane, consecutively, which have twisted boundary condi-tions. The entropy of the walk on the torus labeled with T (cid:48) isabout the same level as the walk on the real projective plane.The walk on the sphere generates the largest level for the en-tropy. The maximum value of the entropy, E l max = 2 log d ,seems not to be achieved by the SQW dynamics. In this man-ner, the entropy (related to the the diagonal elements of thedensity matrix) represents qualitatively a similar informationas the coherence implies.It is seen that the choice of the boundary conditions affectsthe quantum walk dynamics by modifying the interferencepattern. The dynamics of the coherence and the entropy aredistinct for quantum walks on different manifolds. In partic-ular, the coherence associated with the 2D SQW on the toruscan be reached by generating independent 1D SQWs in twodimensions but modifying the boundary conditions. This im-plies that, the coherence is indeed induced by the boundaries.The observation that the dynamics of the coherence and theentropy qualitatively represent similar information suggeststhat the entropy can also quantify the coherence in quantumwalks. In the following sections, we investigate the dynamicsof the entropy for the CTQW on different topologies. Thisanalysis gives an indication of the boundary-induced coher-ence in the walk. We then explore the dynamics of the entropyfor the random walk to see the boundary effects in the absence FIG. 7. The normalized entropy [Eq. (24)] in terms of the numberof steps approximated for the CTQW on a 2D lattice with differentboundary conditions. The plots are generated by using an SQW withthe initial state (14) and setting d = 100 , n = 16 × , κτ = 10 − ,and ωτ = 2 π . of the quantum coherence. V. CTQW
The SQW formalism can be employed to approximateCTQW dynamics. The approximation error is given by thegeneralized decomposition relation [44, 45] (cid:13)(cid:13)(cid:13)(cid:13) e − (cid:80) j =1 itH j − (cid:20) (cid:89) j =1 e − i tL H j (cid:21) L (cid:13)(cid:13)(cid:13)(cid:13) (cid:54) t L (cid:88) j>k (cid:13)(cid:13) [ H j , H k ] (cid:13)(cid:13) , (26)where { H j ; j = 1 , , , } is any set of the staggered Hamil-tonians discussed in the previous sections, t is a given periodof time, L is an integer, and (cid:107) X (cid:107) = sup (cid:107) v (cid:107) =1 |(cid:104) v | X | v (cid:105)| (withthe Euclidean vector norm (cid:107) v (cid:107) = (cid:112) (cid:104) v | v (cid:105) ) is the standard op-erator norm. The left-hand side of the inequality correspondsto the difference between a CTQW evolving for the total time t and an L -step SQW dynamics. The difference, as given inthe right-hand side of the inequality, is bounded and can bedecreased by increasing L . Note that the diagonal elementsof the CTQW Hamiltonian simulated by the SQW generatorsare times larger than the diagonal elements (resonator fre-quencies) that appear in the direct construction of the CTQWdynamics. However, since it is supposed that all the the res-onators are in resonance, this modification has no effect on thedynamics.To obtain an upper bound in Eq. (26), first we write thestaggered Hamiltonians as H j = ω d − κ G j , in order toseparate the contribution of the resonator and the couplingfrequencies. In fact, G j are the adjacency matrices of the lat-tices associated with H j . In this manner, the terms containing the resonator frequency are canceled out in the commutatorbrackets (cid:107) [ H j , H k ] (cid:107) (cid:54) κ (cid:107) G j (cid:107) (cid:107) G k (cid:107) . (27)The matrices G j are orthogonal reflections [8], namely Her-mitian and unitary. This yields an upper bound error as ε (cid:54) κ t /L. (28)As an example, to simulate a CTQW dynamics on a latticewith the fixed coupling frequency κ = 1 MHz and for the totaltime evolution t = (1 / × − s, we can set the total numberof steps for the SQW to L ≈ (1 / × . This leads to κτ = κt/L = 10 − and the error is bounded by . The total timefor realizing the SQW dynamics, however, is t . The totaltime here is comparable with the total time considered in theprevious section for realizing steps of the SQW dynamicswith κτ = π/ , namely . × s.Figure 7 shows the approximated normalized entropy forthe CTQW, using the SQW dynamics. The CTQW dynam-ics on the tori correspond to the summations (cid:80) i =1 H i and (cid:80) i =1 H (cid:48) i which are identical; thus, the plots related to thecases T and T (cid:48) coincide in this figure. It is also seen thatthe plots associated with the walk on the tori lie lower thanthe plots with the twisted boundary conditions, and in this re-spect there is no qualitative difference between the CTQWand the SQW dynamics. The dynamics associated with thesphere (red plot labeled with S ) almost coincides with theplot for the Klein bottle (blue plot labeled with KB ), for theCTQW. Moreover, the effects of the boundary conditions ap-pear around step l = 10 ( d/ , when the boundary sites aresufficiently populated.The entropy dynamics are then discernible for the CTQWevolutions on different 2D manifolds. In fact, the CTQW dy-namics can reveal the topology of the underlying surface. Itshould be remarked that the direct calculation of the CTQWdynamics on a lattice of the size N = 100 × requires ob-taining the evolution of Hamiltonians of the size × the computational cost of which is relatively high. How-ever, the above approximation provides a means to calculatethe CTQW dynamics with significantly less computational re-sources. Of course, it is still relatively costly to calculate thecoherence for × steps of the SQW, and hence we haveresorted to calculate the entropy. The above analysis, how-ever, indicates that the boundary-induced coherence can bereflected by the dynamics of the entropy too. VI. RANDOM WALK
To compare the quantum walk dynamics with the classicalrandom walk behavior, we analyze the random walk evolutionon the 2D manifolds. The desired classical dynamics can begenerated by modifying the staggered evolutions. The SQWoperators are block-diagonal and the blocks are given by Eq.(13). Substituting each block with (cid:18) cos κτ sin κτ sin κτ cos κτ (cid:19) , (29) FIG. 8. The normalized entropy [Eq. (24)] in terms of the numberof steps for the random walk on a 2D lattice with different boundaryconditions. The plots are generated by using (14) as the initial stateof the walker and setting d = 100 , n = 16 × , and κτ = π/ inEq. (29). All plots coincide. we obtain different sets of doubly-stochastic matrices { U cl i ; i = 1 , , , } corresponding to different boundary con-ditions. The discrete-time 2D random walk dynamics can begenerated by applying the doubly-stochastic matrix U cl = 14 (cid:88) i =1 U cl i , (30)on the (classical) initial state (14). Indeed, the diagonal ele-ments are the probabilities that the walker stays at each siteand the off diagonal terms are the probabilities that the walkerjumps to the neighboring sites.Figure 8 shows the normalized entropy for the randomwalks on different 2D manifolds discussed in the previoussections. It can be seen that the entropy behavior is indepen-dent of the choice of boundary conditions. Comparing Fig.8 with Figs. 6 and 7 reveals a sharp difference between theclassical and the quantum dynamics on 2D manifolds. Theinterference causes the quantum walk dynamics to be sensi-tive to the boundary conditions which is manifested in the en-tropy evolution. However, for the random walk, there is nointerference and the entropy dynamics is identical for all themanifolds. In fact, the insensitivity of the entropy dynam-ics to the boundary conditions in random walks supports thatthe different levels of the entropy value in quantum walks canreflect the quantumness of the walks modified by the bound-aries. Note that, as expected, the maximum value of the en-tropy E l max = 2 log d is achieved for the random walk, infinite time steps. VII. SUMMARY AND DISCUSSIONS
We have designed the required Hamiltonians for the re-alization of quantum walks on 2D manifolds. The coinlessdiscrete-time SQW model has been considered, which canbe implemented on a lattice of superconducting microwaveresonators interacting with tunable couplings. Using peri-odic boundary conditions, we have devised two sets of SQWHamiltonians. One set generates two uncoupled 1D walks andthe other set corresponds to a 2D quantum walk. By changingthe lattice boundary conditions, surfaces with different topolo-gies can be obtained. We have given the explicit forms of theSQW Hamiltonians for quantum walk on various surfaces andinvestigated the properties of the corresponding dynamics.We have also explored the coherence and entropy for thewalk on some specific 2D manifolds, such as torus, Klein bot-tle, real projective plane and sphere. We have shown thatboth functions are sensitive to the boundary conditions hav-ing distinct, discernible behaviors for the walk on differentmanifolds. It has been shown that these quantities have largeraverage values for lattices with twisted boundary conditions.We have also considered the behavior of the CTQW andthe corresponding classical random walk on the 2D manifoldsand compared the results with the SQW dynamics. It has beenobserved that whereas the entropy is resolved for the quantumwalks on different manifolds, it takes a fixed value for the clas-sical random walk on all of those 2D surfaces.For the analysis of the SQW in this paper, we fixed the ini-tial place of the walker to the center of the lattice, the fre-quency of each step to κτ = π/ , and the size of the lattice to N = 100 × . Further numerical simulations (not reportedhere) have indicated that the general picture is fairly stableversus variations in the frequency and some translations ofthe initial state. Increasing the size of the lattice will decreasethe amplitude of the oscillations in the coherence and the en-tropy, but changing the order of the SQW Hamiltonians doesnot change the general picture.The dynamics explored in this paper can be used to simu-late topological insulators in two dimensions. It may providea tool to investigate the electron dynamics on Fermi surfaces,which are 2D manifolds embedded in the Brillouin zone ofa crystal. Moreover, it may provide a way to study the effi-ciency of quantum-walk-based algorithms on databases withtopological structures. Acknowledgments. —This work was supported by the IranNational Elites Foundation under Grant No. 7000/2000-1396/03/08 (to J.K.M.) and Sharif University of Technology’sOffice of Vice President for Research (to A.T.R.). [1] R. Portugal,
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