Perfect state transfer, graph products and equitable partitions
aa r X i v : . [ qu a n t - ph ] S e p Perfect state transfer, graph products and equitable partitions
Yang Ge ∗ Benjamin Greenberg † Oscar Perez ‡ Christino Tamon § October 24, 2018
Abstract
We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [8, 3, 2] andthe Cartesian graph products (which includes the n -cube Q n ) [11]. Some of our results include: • If G is a graph with perfect state transfer at time t G , where t G Spec( G ) ⊆ Z π , and H is a circulant with odd eigenvalues, their weak product G × H has perfect state transfer.Also, if H is a regular graph with perfect state transfer at time t H and G is a graphwhere t H | V H | Spec( G ) ⊆ Z π , their lexicographic product G [ H ] has perfect state transfer.For example, these imply Q n × H and G [ Q n ] have perfect state transfer, whenever H isany circulant with odd eigenvalues and G is any integral graph, for integer n ≥
2. Thesecomplement constructions of perfect state transfer graphs based on Cartesian products. • The double cone K + G on any connected graph G , has perfect state transfer if the weightsof the cone edges are proportional to the Perron eigenvector of G . This generalizes resultsfor double cone on regular graphs studied in [8, 3, 2]. • For an infinite family G of regular graphs, there is a circulant connection so the graph K + G ◦ G + K has perfect state transfer. In contrast, no perfect state transfer exists if acomplete bipartite connection is used (even in the presence of weights) [2]. Moreover, weshow that the cylindrical cone K + G + K n + G + K has no perfect state transfer, forany family G of regular graphs.We also describe a generalization of the path collapsing argument [10, 11], which reduces ques-tions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitabledistance partitions. Our proofs exploit elementary spectral properties of the underlying graphs. Keywords : perfect state transfer, quantum walk, graph product, equitable partition.
Recently, perfect state transfer in continuous-time quantum walks on graphs has received consider-able attention. This is due to its potential applications for the transmission of quantum informationover quantum networks. It was originally introduced by Bose [7] in the context of quantum walks onlinear spin chains or paths. Another reason for this strong interest is due to the universal property ∗ Department of Mathematics, Harvard University. † Department of Mathematics, Grinnell College. ‡ Centro de Investigaci´on en Matem´aticas, Universidad Aut´onoma del Estado de Hidalgo, Hidalgo, Mexico. § Department of Computer Science, Clarkson University. Contact author: [email protected] P ⊕ P ; (b) P ⊕ P (see Christandl et al. [11]).of quantum walks as a computational model as outlined by Childs [9]. From a graph-theoretic per-spective, the main question is whether there is a spectral characterization of graphs which exhibitperfect state transfer. Strong progress along these lines were given on highly structured graphs byBernasconi et al. [5] for hypercubic graphs and by Baˇsi´c and Petkovi´c [4] for integral circulants.Nevertheless, a general characterization remains elusive (see Godsil [15]).Christandl et al. [12, 11] showed that the n -fold Cartesian product of the one-link P and two-link P graphs admit perfect state transfer. This is simply because P and P have end-to-endperfect state transfer and the Cartesian product operator preserves perfect state transfer. Theyalso drew a crucial connection between hypercubic networks and weighted paths using the so-called path-collapsing argument. This argument was also used by Childs et al. [10] in the context of anexponential algoritmic speedup for a black-box graph search problem via continuous-time quantumwalks. Christandl et al. [11] proved that, although the n -vertex path P n , for n ≥
4, has no end-to-end perfect state transfer, a suitably weighted version of P n has perfect state transfer (via apath-collapsing reduction from the n -cube Q n ). A somewhat critical ingredient of this reductionis that each layer of Q n is an empty graph. We generalize this argument to graphs which haveequitable distance partitions (see Godsil and Royle [17]).Bose et al. [8] observed an interesting phenomena on the complete graph K n . Although K n doesnot exhibit perfect state transfer, they show that by removing an edge between any two vertices,perfect state transfer is created between them. Note that the graph we obtain from removing anedge from K n is the double cone K + K n − (where G + H denotes the join of graphs G and H ).This observation was generalized in Angeles-Canul et al. [3] where perfect state transfer was provedfor double cones { K , K } + G , where G is some regular graph (in place of complete graphs). Theanalyses on these double cones showed that perfect state transfer need not occur between antipodalvertices and that having integer eigenvalues is not a sufficient condition for perfect state transfer(which answered questions raised in [15]).Our goal in this work is to combine and extend both the Cartesian product and the doublecone constructions. The Cartesian product construction (which combines graphs with perfectstate transfer) has the advantage of producing large diameter graphs with antipodal perfect statetransfer. In fact, this construction provides the best upper bound for the order-diameter problem;for a given d , let f ( d ) be the smallest size graph which has perfect state transfer between twovertices of distance d . Then, the best known bounds are d ≤ f ( d ) ≤ α d , where α = 2, if d is odd,and α = √
3, if d is even; here, the upper bounds are achieved by P ⊗ n and P ⊗ n . On the otherhand, the double cone construction allows graphs whose quotients (modulo its equitable partition)contain cells which are not independent sets. This can potentially allow for a broader class of2raph family PST Construction Source { K , K } + G yes join [8, 3] { K , K } + ˜ G yes ∗ join this work P n ≥ no path [11] K + G ◦ G + K yes circulant half-join this work K + G + G + K no ∗ half-join [2] K + G + K n + G + K no join this work Q n or P ⊕ n yes Cartesian product [11] { Q n , P ⊕ n } × ODD-CIRC yes weak product this workINT[ Q n ≥ ] yes lexicographic product this workFigure 2: Summary of results on some graphs with perfect state transfer : n is a positive integer; G denotes some family of regular graphs; ˜ G denotes an arbitrary connected graph; P n is the pathon n vertices; Q n is the n -dimensional cube; K n is the complete graph on n vertices; ODD-CIRCis the class of circulant graphs with odd eigenvalues; INT is the class of integral graphs. Asterisksindicate results on weighted graphs.graphs with perfect state transfer (see Bose et al. [8] and Angeles-Canul et al. [3, 2]).In this work, we describe new constructions of families of graphs with perfect state transfer.First, we extend several of the double cone constructions and relax their diameter restrictions. Weshow that the double cone K + G of an arbitrary connected graph G has perfect state transferif we use edge weights proportional to the Perron eigenvector of G . This extends results given in[3] where G is required to be a regular graph. Then, we prove that the glued double cone graph K + G ◦ G + K has perfect state transfer whenever G , G belongs to some class of regular graphsand if they are connected using some matrix C which commutes with the adjacency matrices of G and G . In contrast, Angeles-Canul et al. [2] proved that K + G + G + K has no perfect statetransfer, for any regular graph G , even if weights are allowed.For cones with larger diameter, we consider the graph K + G + H + G + K , where G , G belong to the same class of regular graphs and H is another regular graph. This symmetry is anecessary condition for perfect state transfer as shown by Kay [18]. Nevertheless, in contrast tothe previous positive results, we show there is no perfect state transfer whenever H is the emptygraph. The 4-dimensional cube Q (which has perfect state transfer) is an example of such a graphbut without the join (or complete bipartite) connection.Our other contribution involves constructions of perfect state transfer graphs using alternativegraph products, namely the weak and lexicographic products. An interesting property of theseproducts is that they can create perfect state transfer graphs by combining graphs with perfectstate transfer and ones which lack the property. For example, we show that Q n × K m has perfectstate transfer, for any integers n and m . Recall that the complete graph has no perfect statetransfer (as observed by Bose et al. [8]). In comparison, the Cartesian product requires both ofits graph arguments to have perfect state transfer (with the same perfect state transfer times).We also consider the lexicographic graph product (or graph composition) and its generalizations.Our generalized lexicographic product of G and H using a connection matrix (or graph) C is agraph whose adjacency matrix is A G ⊗ C + I ⊗ A H . Note we recover the Cartesian product byletting C = I and the standard lexicographic product by letting C = J . So, this generalization3nterpolates between these two known graph products. For example, we show that G [ Q n ] hasperfect state transfer for any integral graph G and n ≥ Let [ n ] denote the set { , , . . . , n − } . For a tuple of binary numbers ( a, b ) ∈ { , } \ { (0 , } , let Q a,b denote the set of rational numbers of the form p/q , with gcd ( p, q ) = 1, where p ≡ a (mod 2)and q ≡ b (mod 2). These denote rational numbers (in lowest terms) that are ratios of two oddintegers or of an odd integer and an even integer, or vice versa. We denote the even and oddintegers as 2 Z and 2 Z + 1, respectively.The graphs G = ( V, E ) we study are finite, simple, undirected, connected, and mostly un-weighted. The adjacency matrix A G of a graph G is defined as A G [ u, v ] = 1 if ( u, v ) ∈ E and 0otherwise; we also use u ∼ v to mean u is adjacent to v . The spectrum Spec( G ) of G is the setof eigenvalues of A G . The graph G is called integral if all of its eigenvalues are integers. A graph G = ( V, E ) is called k -regular if each vertex u ∈ V has exactly k adjacent neighbors. For integers n ≥ ≤ k < n , let G n,k be the set of all n -vertex k -regular graphs. The distance d ( a, b )between vertices a and b is the length of the shortest path connecting them.Some standard graphs we consider include the complete graphs K n , paths P n , and circulantsgraphs. An n -vertex circulant graph G on is a graph whose adjacency matrix is an n × n circulantmatrix; that is, there is a sequence ( a , . . . , a n − ) so that A G [ j, k ] = a k − j , where arithmetic on theindices is done modulo n . Alternatively, we may define a circulant graph G on [ n ] through a subset S ⊆ [ n ] where j is adjacent to k if and only if k − j ∈ S ; we denote such a circulant as Circ ( n, S ).Known examples of circulants include the complete graphs K n and cycles C n .Let G and H be two graphs with adjacency matrices A G and A H , respectively. The complementof G = ( V, E ), denoted G = ( V, E ), is a graph where ( u, v ) ∈ E if and only if ( u, v ) E , for u = v .Some relevant binary graph operations are defined in the following: • The
Cartesian product G ⊕ H is a graph defined on V ( G ) × V ( H ) where ( g , h ) is adjacentto ( g , h ) if either g = g and ( h , h ) ∈ E H , or ( g , g ) ∈ E G and h = h . The adjacencymatrix of G ⊕ H is A G ⊗ I + I ⊗ A H . • The weak product G × H is a graph defined on V ( G ) × V ( H ) where ( g , h ) is adjacent to( g , h ) if ( g , g ) ∈ E G and ( h , h ) ∈ E H . The adjacency matrix of G × H is A G ⊗ A H . • The lexicographic product G [ H ] is a graph defined on V ( G ) × V ( H ) where ( g , h ) is adjacentto ( g , h ) if either ( g , g ) ∈ E G or g = g and ( h , h ) ∈ E H . The adjacency matrix of G [ H ] is A G ⊗ J + I ⊗ A H . • The join G + H is a graph defined on V ( G ) ∪ V ( H ) obtained by taking two disjoint copies of G and H and by connecting all vertices of G to all vertices of H . The adjacency matrix of G + H is (cid:20) A G JJ A H (cid:21) .We assume appropriate dimensions on the identity I and all-one J matrices used above. The n -dimensional hypercube Q n may be defined recursively as Q = K and Q n = K ⊕ Q n − , for n ≥ K + K ◦ K ◦ K ◦ K + K .The cone of a graph G is defined as K + G . The double cone of G is K + G , whereas the connected double cone is K + G .A partition π of a graph G = ( V, E ) given by V = U mj =1 V j is called equitable if the number ofneighbors in V k of a vertex u in V j is a constant d j,k , independent of u (see [17, 16]). The quotient graph of G over π , denoted by G/π , is the directed graph with the m cells of π as its vertices and d j,k edges from the j th to the k th cells of π . The adjacency matrix of G/π is given by A G/π [ j, k ] = d j,k .A graph G has an equitable distance partition π with respect to a vertex a if π = U mj =0 V j issuch that G/π is a path and V j = { x ∈ V : d ( x, a ) = j } where V = { a } ; typically, we also requirethat there is a vertex b , antipodal to a , so that V m = { b } . We also call a graph a cylindrical cone (see Figure 3) if it has an equitable distance partition and is denoted K ◦ G ◦ . . . ◦ G m ◦ K , where G j are regular graphs and ◦ denote (semi-)regular bipartite connections (induced by the equitablepartition π ).Further background on algebraic graph theory may be found in the comprehensive texts ofBiggs [6], Godsil and Royle [17], and Godsil [16].Next, we describe the continuous-time quantum walk as defined originally by Farhi and Gut-mann [13]. For a graph G = ( V, E ), let | ψ (0) i ∈ C | V | be an initial amplitude vector of unit length.Using Schr¨odinger’s equation, the amplitude vector of the quantum walk at time t is | ψ ( t ) i = e − itA G | ψ (0) i . (1)Note since A G is Hermitian (in our case, symmetric), e − itA G is unitary (hence, an isometry). Moredetailed discussion of quantum walks on graphs can be found in the excellent surveys by Kempe[19] and Kendon [20]. The instantaneous probability of vertex a at time t is p a ( t ) = |h a | ψ ( t ) i| . Wesay G has perfect state transfer from vertex a to vertex b at time t if a continuous-time quantumwalk on G from a to b has unit fidelity or |h b | e − itA G | a i| = 1 , (2)where | a i , | b i denote the unit vectors corresponding to the vertices a and b , respectively. The graph G has perfect state transfer if there exist vertices a and b in G and time t so that (2) is true. In this section, we describe constructions of perfect state transfer graphs using the weak andlexicographic products. These complement the well-known Cartesian product constructions [11].5igure 4: Graph products with perfect state transfer: (a) the weak product K × K m , for m ≥ m = 1); (b) the lexicographic product (or composition) K m [ Q n ] (shown here with m = n = 2). An interesting property of the weak product graph operator is that it can create graphs with perfectstate transfer by combining ones with perfect state transfer and ones which lack the property. Incontrast, the Cartesian graph product can only create perfect state transfer graphs from ones whichhave the property. We start with the following simple observation.
Fact 1
Let G be an n -vertex graph and H be an m -vertex graph whose eigenvalues and eigenvectorsare given by A G | u k i = λ k | u k i , for k ∈ [ n ] , and A H | v ℓ i = µ ℓ | v ℓ i , for ℓ ∈ [ m ] , respectively. Let g , g ∈ G and h , h ∈ H . Then, the fidelity of a quantum walk on their weak product G × H between ( g , h ) and ( g , h ) is given by h g , h | e − itA G × H | g , h i = h g | "X k (X ℓ h h | v ℓ ih v ℓ | h i e − itλ k µ ℓ ) | u k ih u k | | g i . (3) Proof
Recall that the adjacency matrix of A G × H is A G ⊗ A H . Thus, the eigenvalues and eigenvectorsof the weak product G × H are A G × H ( | u k i ⊗ | v ℓ i ) = λ k µ ℓ ( | u k i ⊗ | v ℓ i ) , where k ∈ [ n ] and ℓ ∈ [ m ]. (4)So, the quantum walk on G × H from ( g , h ) to ( g , h ) is given by h g , h | e − itA G × H | g , h i = X k,ℓ h g | u k ih u k | g ih h | v ℓ ih v ℓ | h i e − itλ k µ ℓ . (5)After rearranging summations, we obtain the claim. Proposition 2
Let G be a graph with perfect state transfer at time t G so that t G Spec( G ) ⊆ Z π. (6) Then, G × H has perfect state transfer if H is a circulant graph with odd eigenvalues. Proof
Suppose G is an n -vertex graph whose eigenvalues and eigenvectors are given by A G | u k i = λ k | u k i , for k ∈ [ n ]. Assume that G has perfect state transfer at time t G from vertex g to g . Also,suppose H be an m -vertex graph whose eigenvalues and eigenvevtors are given by A H | v ℓ i = µ ℓ | v ℓ i ,6or ℓ ∈ [ m ]. In Equation (3), if H is circulant on m vertices, we have h | v ℓ ih v ℓ | i = 1 /m . Moreover,if each eigenvalue of H is odd, say µ ℓ = 2 m ℓ + 1, with m ℓ ∈ Z , then h g , | exp( − it G A G × H ) | g , i = 1 m X k,ℓ h g | u k ih u k | g i e − it G λ k µ ℓ (7)= 1 m X k h g | u k ih u k | g i X ℓ e − it G λ k (2 m ℓ +1) (8)= X k h g | u k ih u k | g i e − it G λ k , since t G λ k ∈ Z π . (9)The last expression equals to h g | e − it G A G | g i , by the spectral theorem. This proves the claim. Remark : Note Q n has eigenvalues λ k = 2 n − k , for k = 0 , . . . , n , and perfect state transfer time t = π/
2. Also, P ⊗ n has eigenvalues from λ k ∈ Z √ t = π/ √
2. Inboth cases, we have tλ k ∈ Z π , for all k . Thus, by Proposition 2, we get that { Q n , P ⊗ n } × H hasperfect state transfer for any circulant H with odd eigenvalues. For example, we may let H = K m be the complete graph of order m , for an even integer m . The generalized lexicographic product G C [ H ] between a graph G and two graphs H and C , with V H = V C , is a graph on V G × V H where ( g , h ) is adjacent to ( g , h ) if and only if either ( g , g ) ∈ E G and ( h , h ) ∈ E C , or, g = g and ( h , h ) ∈ E H . In terms of adjacency matrices, we have A G C [ H ] = A G ⊗ A C + I ⊗ A H . (10)We describe constructions of perfect state transfer graphs using generalized lexicographic products.Again, we start with the following simple observation. Fact 3
Let G be an n -vertex graph whose eigenvalues and eigenvectors are given by A G | u k i = λ k | u k i , for k ∈ [ n ] . Let H and C be m -vertex graphs whose adjacency matrices commute, that is [ A H , A C ] = 0 , and whose eigenvalues and eigenvectors are given by A H | v ℓ i = µ ℓ | v ℓ i , and A C | v ℓ i = γ ℓ | v ℓ i , for ℓ ∈ [ m ] , respectively. Suppose g , g ∈ G and h , h ∈ H . Then, the fidelity of a quantumwalk on the generalized lexicographic product G C [ H ] between ( g , h ) and ( g , h ) is given by h g , h | exp( − itA G C [ H ] ) | g , h i = X k h g | u k ih u k | g i X ℓ h h | v ℓ ih v ℓ | h i e − it ( λ k γ ℓ + µ ℓ ) (11) Proof
The eigenvalues and eigenvectors of G C [ H ] are given by A G C [ H ] ( | u k i ⊗ | v ℓ i ) = ( λ k γ ℓ + µ ℓ )( | u k i ⊗ | v ℓ i ) , k ∈ [ n ] and ℓ ∈ [ m ] . (12)So, the quantum walk on G C [ H ] from ( g , h ) to ( g , h ) is given by h g , h | e − itA GC [ H ] | g , h i = X k,ℓ h g | u k ih u k | g ih h | v ℓ ih v ℓ | h i e − it ( λ k γ ℓ + µ ℓ ) , (13)which proves the claim. 7n the following, we show a closure property of perfect state transfer graphs using a generalizedlexicographic product with the complete graph as a connection matrix. This is similar to the weakproduct construction from Proposition 2. Proposition 4
Let G and H be perfect state transfer graphs with a common time t . Assume H isa m -vertex graph which commutes with K m . Suppose that t | V H | Spec( G ) ⊆ Z π. (14) Then, the lexicographic product G K m [ H ] has perfect state transfer at time t . Proof
Suppose G has perfect state transfer from g to g at time t , where g , g ∈ V G . Let theeigenvalues and eigenvectors of G be given by A G | u k i = λ k | u k i , for k ∈ [ n ]. Also, suppose H is acirculant with perfect state transfer from h to h at time t , where h , h ∈ V H . Let the eigenvaluesand eigenvectors of H be given by A H | v ℓ i = µ ℓ | µ ℓ i , for ℓ ∈ [ m ]. Thus, Equation (11) becomes h g , h | e − itA GKm [ H ] | g , h i (15)= X k h g | u k ih u k | g i e − it ( λ k ( m − µ ) h h | v ih v | h i + X ℓ =0 e − it ( − λ k + µ ℓ ) h h | v ℓ ih v ℓ | h i (16)= h g | e itA G | g ih h | e − itA H | h i , (17)since e − it ( m − λ k = e itλ k , for all k . This shows that G K m [ H ] has perfect state transfer from ( g , h )to ( g , h ) at time t .The standard lexicographic product G [ H ] is obtained when we let C = J in Equation (10). In thiscase, Equation (11) decouples nicely and we have a similar result to Proposition 4 but withoutrequiring G to have perfect state transfer. Lemma 5
Let G be an arbitrary graph and let H be a regular graph with perfect state transferat time t H from h to h , for h , h ∈ V H . Then, G [ H ] has perfect state transfer from ( g, h ) to ( g, h ) , for any g ∈ V G , if t H | V H | Spec( G ) ⊆ Z π. (18) Proof If H is an m -vertex regular graph, then [ A H , J m ] = 0. The all-one matrix J m has eigenvalues m (with multiplicity one) and 0 (with multiplicity m − h g, h | e − it H A G [ H ] | g, h i (19)= X k h g | u k ih u k | g i e − it H ( λ k m + µ ) h h | v ih v | h i + X ℓ =0 e − it H µ ℓ h h | v ℓ ih v ℓ | h i (20)= h h | e − it H A H | h i , (21)since e − it H mλ k ( G ) = 1, for all k , and P k | u k ih u k | = I . This proves the claim. Remark : We will adopt the convention of scaling quantum walk time with respect to the size of theunderlying graphs. Moore and Russell [22] proved that a continuous-time quantum walk on the n -cube Q n has a uniform mixing time of (2 Z + 1) π n (which shows the time scaling with respect tothe dimension of the n -cube). They used H = n A Q n as their Hamiltonian – which is the probabilitytransition matrix of the simple random walk on Q n .8 orollary 6 Suppose H is a k H -regular graph with perfect state transfer at time t H = π k H and G is an integral graph (all of its eigenvalues are integers). Then, G [ H ] has perfect state transferprovided k H | V H | Spec( G ) ⊆ Z . Proof
Apply Lemma 5 by noting that e − it H | V H | λ k ( G ) = 1, since t H = π k H and λ k ( G ) k H | V H | isdivisible by 4, for all k .The n -cube Q n is a n -regular graph on 2 n vertices which has perfect state transfer at time n π (withtime scaling) (see [5]). Thus, for any integral graph G , the composition graph G [ Q n ] has perfectstate transfer if n ≥ In this section, we explore some constructions of perfect state transfer graphs which generalize thedouble cones studied by Bose et al. [8] and Angeles-Canul et al. [3, 2]. The goal behind theseconstructions is to understand the types of intermediate graphs which allow perfect state transferbetween the two antipodal vertices. For the double cones { K , K } + G n,k , the intermediate graphsare n -vertex k -regular graphs and sufficient conditions for perfect state transfer on n and k werederived in [3].Here, we consider more complex cones by allowing irregular graphs (on double cones), byincreasing the number of intermediate layers, and by varying the connectivity structure (usingsemi-regular bipartite connections). We show new perfect state transfer graphs for irregular doublecones and for double half-cones with circulant connections, and also prove negative results for longerdiameter cones on join connections. We recall the Perron-Frobenius theory of nonnegative matrices. A matrix is called nonnegative if it has no negative entries. The spectral radius of a matrix A , denoted ρ ( A ), is the maximumeigenvalue of A (in absolute value). The Perron-Frobenius theorem for nonnegative matrices statesthat if A is a real nonnegative n × n matrix whose underlying directed graph G is strongly connected,then ρ = ρ ( A ) is a simple eigenvalue of A ; moreover, the unique eigenvector corresponding to ρ hasno zero entries and all entries have the same sign.In what follows, we denote K b as the two-vertex graph which equals K if b = 1, and equals K if b = 0. Theorem 7
Let G be any connected graph whose maximum (simple) eigenvalue is λ with a cor-responding positive (normalized) eigenvector | x i . Consider the double cone G = K b + G , for b ∈ { , } , where the edges adjacent to the vertices of K b , say A and B , are weighted proportionalto α | x i . Then, the fidelity between A and B is given by h B | e − itA G | A i = 12 (cid:26) e − it ˜ λ +0 (cid:20) cos( t ∆) + i λ − ∆ sin( t ∆) (cid:21) − (cid:27) , (22) where ˜ λ ± = ( λ ± b ) / and ∆ = q (˜ λ − ) + 2 α . Thus, perfect state transfer is achieved if ˜ λ +0 / ∆ ∈ Q , ∪ Q , . K + P ; (b) K + P . Proof
Let A G be the adjacency matrix of G . The adjacency matrix of G is A G = b α h x | b α h x | α | x i α | x i A G . (23)For 1 ≤ k ≤ n −
1, let λ k and | x k i be the other eigenvalues and eigenvectors of A G . Next, we definethe following quantities:˜ λ ± = λ ± b , ∆ = q (˜ λ − ) + 2 α , λ ± = ˜ λ +0 ± ∆ . κ ± = ˜ λ − ± ∆ . (24)The eigenvalues of A G are given by λ = 0, λ ± , and λ k , 1 ≤ k ≤ n −
1, with correspondingeigenvectors | z i = 1 √ +1 − | n i , | z ± i = 1 L ± α/κ ± α/κ ± | x i , | z k i = | x k i (25)where L ± = 2 α /κ ± + 1. Note ( κ ± L ± ) = 2 α + κ ± = 2∆(∆ ± ˜ λ − ). The fidelity between A and B , namely h B | e − itA G | A i , is given by X ± α e − itλ ± ( κ ± L ± ) −
12 = 12 ( (∆ − ˜ λ − ) e − it (˜ λ +0 +∆) + (∆ + ˜ λ − ) e − it (˜ λ +0 − ∆) − ) (26)= 12 ( e − it ˜ λ +0 " cos( t ∆) + i ˜ λ − ∆ sin( t ∆) − ) . (27)For perfect state transfer to occur, it is sufficient to have ˜ λ +0 / ∆ ∈ Q , ∪ Q , . Corollary 8
Let G be any connected graph whose maximum (simple) eigenvalue is λ with corre-sponding positive eigenvector | x i . Consider the double cone G = K + G where the edges adjacentto the two vertices of K , say A and B , are weighted according to √ n | x i . Then, perfect statetransfer exists from A to B if λ p λ + 8 n ∈ Q , ∪ Q , . (28) Proof
In Theorem 7 with b = 0, let α = √ n and note ˜ λ ± = λ /
2. Thus, ˜ λ +0 / ∆ = λ / p λ + 8 n ,which proves the claim. 10 emark : Given n , we may choose λ = p n/ λ / p λ + 8 n = 1 / G so that p n/ K + G hasperfect state transfer. Analogous to the construction of glued-(binary)trees in Childs et al. [10], we consider gluing twodouble cones using a semi-regular bipartite connection to obtain a perfect state transfer graph. Incontrast, gluing two double cones using the join (full bipartite) connection yields no perfect statetransfer (even with weights) as proved in [2].
Theorem 9
Let G ∈ G n,k and let C be a symmetric Boolean matrix which commutes with theadjacency matrix of G . Suppose that C | n i = γ | n i . Let k ± = ( k ± γ ) and ∆ ± = p k ± + n .Then, the graph G = K + G ◦ G + K , formed by taking two copies of K + G and connecting thecopies of G using C , has perfect state transfer if ∆ + / ∆ − ∈ Q , ∪ Q , and at least one of γ/ ∆ + or γ/ ∆ − is in Q , . Proof
Suppose the eigenvalues and eigenvectors of G are λ k and | v k i , respectively, where k = λ >λ ≥ . . . ≥ λ n − . The adjacency matrix of G is given by A G = h n | h n | h n | h n || n i | n i A G C | n i | n i C A G . (29)Let k ± = ( k ± γ ) and ∆ ± = k ± + n . Let α ± = k + ± ∆ + and β ± = k − ± ∆ − . The eigenvalues of A G are given by α ± , β ± , and ± λ k , for k = 0, with corresponding eigenvectors: | α ± i = 1 L ± n α ± | n i n α ± | n i , | β ± i = 1 M ± +1 − n β ± | n i− n β ± | n i , | λ k i = 1 √ | v k i±| v k i , (30)where L ± = n ( n + α ± ) and M ± = n ( n + β ± ) are the normalization constants. The quantum walksbetween involving the cone vertices, say A and B , are given by h B | e − itA G | A i = X ± e − itα ± L ± − X ± e − itβ ± M ± (31) h A | e − itA G | A i = X ± e − itα ± L ± + X ± e − itβ ± M ± (32)At time t = 0, the second equation yields 1 = P ± L − ± + P ± M − ± . To achieve perfect state transfer,it suffices to require e − itα ± = +1 , e − itβ ± = − , e − itγ/ = ± . (33)We may restate these conditions as ∆ + / ∆ − = Q , ∪ Q , and { γ/ ∆ + , γ/ ∆ − } ∩ Q , = ∅ .11igure 6: Glued cones (a) K + G ◦ G + K has perfect state transfer, with (b) G = Circ (15 , { , , } );(c) C = Circ (15 , { , , , } ). The connection ◦ is defined by C . Remark : In Theorem 9, the result also holds if we replace G with two distinct graphs G and G from the same family G n.k .In the following corollary, we describe an explicit family of glued double cones which exhibitperfect state transfer. The construction uses a pair of circulant families of graphs (see Figure 6). Corollary 10
For a ≥ , let n = 15 × a − , k = 3 × a − , and γ = 4 × a − . Consider twocirculant graphs G = Circ ( n, [ k/ and C = Circ ( n, [ γ/ . Then, the graph G = K + G ◦ G + K has perfect state transfer, where the connection ◦ is specified by C . Proof
Note we have k ± = ( k ± γ ) = 2 a − (3 ±
4) and ∆ ± = 2 a − ((3 ± + 15) ∈ a − { , } .Thus, ∆ + / ∆ − = 2 ∈ Q , and γ/ ∆ − = 2 ∈ Q , , which satisfy the sufficiency conditions for perfectstate transfer in Theorem 9. In this section, we consider graphs of the form K + G + H + G + K , where G , G ∈ G n,k and H ∈ G m,ℓ . We show a negative result for perfect state transfer whenever H is the empty graph.This generalizes known negative results on P and K + G + G + K (see [11, 2]). Theorem 11
For any integers n, k, m where n ≥ , ≤ k < n , and m ≥ , the graph K + G + K m + G + K has no perfect state transfer, whenever G , G ∈ G n,k . Proof
Let G be the graph K + G + H + G + K , where G , G ∈ G n,k and H ∈ G m,ℓ . Let A G be the adjacency matrix of G with eigenvalues α r and eigenvectors | u r i ; similarly, let A G be theadjacency matrix of G with eigenvalues β r and eigenvectors | v r i , for r ∈ [ n ]. Note k = α = β are the simple maximum eigenvalues of both G and G . Let A H be the adjacency matrix of H with eigenvalues ρ s and eigenvectors | w s i , where ℓ = ρ is the simple maximum eigenvalue of H .Thus, the adjacency matrix of G is given by A G = h n | h n | h n | h n | h n | h n || n i | n i A G J n,m O n,n | m i | m i J m,n A H J m,n | n i | n i O n,n J n,m A G . (34)12n our case, we have A H = O m,m is the zero m × m matrix and ℓ = 0.Let λ ± be the roots of quadratic polynomial λ − kλ − n = 0; thus λ ± = ˜ k ± ∆, where ˜ k = k/ = ˜ k + n . Consider roots of the cubic polynomial ( µ − ℓ )( µ − kµ − (2 m + 1) n ) − ℓmn =0. For ℓ = 0, zero is a root of this cubic along with the two roots of the quadratic equation µ − kµ − (2 m + 1) n = 0. Let µ ± = ˜ k ± Γ, where Γ = ˜ k + (2 m + 1) n . The eigenvalues of A G aregiven by λ ± , µ ± , 0, and λ (1) r , λ (3) r , for r = 0, and λ (2) s , for s = 0, with corresponding eigenvectors: | λ ± i = 1 L ± +1 − n λ ± | n i| n i− n λ ± | n i , | µ ± i = 1 M ± n µ ± | n i | n i n µ ± | n i , | v i = 1 N | n i− /m | n i| n i , (35)and | λ (1) r i = | u r i| m i| n i , | λ (2) s i = | n i| w s i| n i , | λ (3) r i = | n i| m i| v r i , (36)where 1 ≤ r < n and 1 ≤ s < m . Here L ± , M ± and N are normalization factors. We have thefollowing fidelities: h B | e − itA G | A i = − X ± e − itλ ± L ± + X ± e − itµ ± M ± + 1 N (37) h A | e − itA G | A i = X ± e − itλ ± L ± + X ± e − itµ ± M ± + 1 N . (38)At time t = 0, Equation (38) yields1 = X ± L ± + X ± M ± + 1 N . (39)So, to achieve perfect state transfer in Equation (37), we require that e − itλ ± = − , e − itµ ± = +1 . (40)This implies t (˜ k ± ∆) ∈ (2 Z + 1) π and t (˜ k ± Γ) ∈ (2 Z ) π . We restate these conditions as˜ k ± ∆˜ k ± Γ ∈ Q , , ˜ k ± ∆˜ k ∓ Γ ∈ Q , . (41)Clearly it is necessary to have ∆ , Γ ∈ Z , else the above quotients are not even rational.Observe that if both ˜ k and n are odd, both quotients lie in Q , . If ˜ k is even and n is odd, thesame is true. If ˜ k is odd and n is even, then the numerator and denominator of at least one of thequotients must be congruent to 2 modulo 4, and so one lies in Q , . If both ˜ k and n are even, than13igure 7: Cylindrical cones of diameter five with no perfect state transfer: (a) K + K + K + K + K ; (b) K + K + K + K + K .we can divide the numerator and denominator of each quotient by 2 (clearly, then, 4 divides n aswell), rewriting the conditions as:˜ k ′ ± ∆ ′ ˜ k ′ ± Γ ′ ∈ Q , , ˜ k ′ ± ∆ ′ ˜ k ′ ∓ Γ ′ ∈ Q , , (42)where ˜ k ′ = ˜ k/
2, ∆ ′ = ∆ /
2, and Γ ′ = Γ /
2. Since this is in essence the same set of conditions asbefore, we argue by infinite descent that there can be no solutions of this form. Since we have ruledout all parity combinations for ˜ k and n , there can be no solutions and no perfect state transfer inthis case. The path-collapsing argument was used by Christandl et al. [11] to show that weighted paths haveperfect state transfer. This follows because the (unweighted) n -dimensional hypercube Q n hasperfect state transfer and it can be collapsed to a weighted path. On the other hand, this argumentwas used in the opposite direction by Childs et al. [10] to show that a continuous-time quantumwalk on an unweighted layered graph has polynomial hitting time by observing its behavior on acorresponding weighted path.A natural way to view this reduction is by using equitable distance partitions and their quotientgraphs (for example, see [16, 21]). But, most quotient graphs derived this way are directed andhence not suitable for quantum walks. The path-collapsing reduction offers a way to symmetrize these directed quotient graphs into undirected graphs. In what follows, we formalize and generalizethis argument using the theory of equitable partitions (see Godsil [16]). Lemma 12
Let G = ( V, E ) be a graph with an equitable distance partition π = U m − j =0 V j withrespect to vertices a and b . Then, the fidelity of a quantum walk on G between vertices a and b isequivalent to the fidelity of a quantum walk on a symmetrized quotient graph G/π between π ( a ) = V and π ( b ) = V m − ; namely, if B G/π [ j, k ] = p d j,k d k,j , for all j, k ∈ [ m ] , then |h b | e − itA G | a i| = |h π ( b ) | e − itB G/π | π ( a ) i| . (43) Proof
For j, k ∈ [ m ], let d j,k be the number of vertices in V k adjacent to each vertex x in V j . Let P be the characteristic partition n × m matrix of π ; namely, P [ j, ℓ ] = 1 if vertex j belongs to partition14 ℓ , and 0 otherwise. Suppose Q be the matrix P after we normalize each column; so Q T Q = I n .Then, we have A G Q = QB G/π , (44)where B G/π [ j, k ] = p d j,k d k,j . (45)The matrix B G/π is defined implicitly in [11] through the columns of Q (viewed as basis states ina new graph) . The following spectral correspondences between A G and B G/π can be shown: • If A G | y i = λ | y i , then B G/π | x i = λ | x i , where | x i = Q T | y i , provided Q T | y i 6 = 0. • If B G/π | x i = λ | x i , then A G | y i = λ | y i , where | y i = Q | x i .Suppose that E ( A G ) = {| y k i : k ∈ [ n ] } is the (orthonormal) set of eigenvectors of A G ; similarly, let E ( B G/π ) = {| x k i : k ∈ [ m ] } be the (orthonormal) set of eigenvectors of B G/π . Since π ( a ) = { a } and π ( b ) = { b } are singleton partitions, we have Q T | a i = | π ( a ) i and Q T | b i = | π ( b ) i . Thus, we have h π ( b ) | e − itB G/π | π ( a ) i = h π ( b ) | m − X k =0 (cid:16) e − itλ k | x k ih x k | (cid:17) | π ( a ) i (46)= h b | m − X k =0 (cid:16) e − itλ k Q | x k ih x k | Q T (cid:17) | a i (47)= h b | e − itA G | a i . (48)The last step holds since the orthonormal eigenvectors of A G can be divided into two types: thosethat are constant on cells of π (the ones of the form | y k i = Q | x k i , for some eigenvector | x k i of B G/π ) and those that sum to zero on each cell of π . The eigenvectors of the latter type do notcontribute to the quantum walk between the antipodal vertices a and b . Remark : Lemma 12 shows that the double cones K + G , for regular graphs G ∈ G n,k , which havediameter two, are equivalent (in the sense of the fidelity of quantum walks between the antipodalvertices) to a weighted P with adjacency matrix ˜ A (shown below).˜ A = √ n √ n k √ n √ n ˜ A = √ n √ n k √ n √ n (49)The case of the connected double cone K + G , where G ∈ G n,k , can also be shown to be equivalentto the weighted graph with adjacency matrix ˜ A (shown above). This simplifies the analyses onvalues of n and k which allows perfect state transfer (see [3]).In what follows, we use the generalized path-collapsing argument above to revisit (unweighted)graphs of diameter three and compare them to (weighted) paths of length four. Then, we comparea family of symmetrically weighted paths P (without self-loops) with a construction based on weakproducts. This symmetry restriction on the weights can be made without loss of generality; seeKay [18]. Note B G/π is different from A G/π (as defined in [16]), since B G/π is symmetric and represents an undirectedweighted graph whereas A G/π represents a directed graph. emma 13 Let P ( γ ; κ ) denote a weighted path whose middle edge has weight γ while the othertwo edges have unit weights and whose two internal vertices have self-loops with weight κ each. Let ∆ ± = p ( κ ± γ ) + 4 . Then, P ( γ ; κ ) has perfect state transfer if:1. Case κ = 0 : ∆ + / ∆ − ∈ Q , ∪ Q , and { γ/ ∆ + , γ/ ∆ − } ∩ ( Q , ∪ Q , ) = ∅ ; or2. Case κ = 0 : { γ/ ∆ + , γ/ ∆ − } ⊆ Q , or { γ/ ∆ + , γ/ ∆ − } ⊆ Q , . Proof
Let k ± = ( κ ± γ ) /
2, ∆ = k + 1, and ∆ − = k − + 1. The adjacency matrix A of P ( γ ; κ ),whose eigenvalues are α ± = k + ± ∆ + and β ± = k − ± ∆ − , and its corresponding eigenvectors | α ± i and | β ± i are given by: A = κ γ γ κ
10 0 1 0 , | α ± i = 1 L ± α ± α ± , | β ± i = 1 M ± +1 − β ± − β ± , (50)where L ± = 4∆ + (∆ + ± k + ) and M ± = 4∆ − (∆ − ± k − ). The end-to-end fidelity of the quantumwalk on P ( γ ; κ ) is given by h | e − itP ( γ ; κ ) | i = X ± e − itα ± L ± − X ± e − itβ ± M ± (51)At time t = 0, h | e − itP ( γ ; κ ) | i equals P ± /L ± + P ± /M ± = 1. Thus, to achieve unit fidelitywhen κ = 0, it suffices to have e − itγ/ = ±
1, cos( t ∆ + ) = ±
1, and cos( t ∆ − ) = ∓ t ∆ + and t ∆ − differ in their parities (as a multiple of π ) while tγ is of even parity. But, when κ = 0, andthus ∆ + = ∆ − , it suffices to simply have tγ be of odd parity.Note Theorem 9 forms a special case of Lemma 13 when κ = 0. The fact that the analyses areequivalent follows from Lemma 12. Remark : Let P ( γ ) denote P ( γ ; 0); that is, a weighted path with no self-loops. In this case,∆ + = ∆ − and a sufficient perfect state transfer condition is ∆ + /γ ∈ Q , ∪ Q , . So, P ( γ ) hasend-to-end perfect state transfer if either: • for odd integer K and even integer L , with L > K , we have γ = 2 p K / ( L − K ); or • for odd integers K and L , with 2 L > K , we have γ = 2 p K / (4 L − K ).The weak product K × K k , for k ≥
1, has perfect state transfer if m is divisible by 4, by Proposition2 (see Figure 4(a)). The path collapsing argument shows K × K k is equivalent to P ( τ ) where τ = (4 k − / √ k − >
1. Thus, for perfect state transfer, the weak product construction K × K k yields edge weights greater than 1, whereas P ( γ ) can yield edge weights smaller than 1 (with longerPST times). We are not aware of unweighted constructions which can emulate the latter property.16 Conclusions
Using the Cartesian graph product, Christandl et al. [11] constructed two families of perfect statetransfer graphs with large diameter, namely, Q n and P ⊕ n . They also showed that weighted pathshave perfect state transfer by a path-collapsing reduction from Q n . This argument was usedspecifically on graphs with equitable distance partitions whose cells are empty graphs. Our originalmotivation was to generalize the Cartesian product construction and extend the path-collapsingargument to larger classes of graphs.In this work, we described new families of graphs with perfect state transfer using the weakgraph product and a generalized lexicographic product (which includes the Cartesian graph productas a special case). We also considered constructions involving double cones which allow the cellpartitions to be non-empty graphs (unlike the Cartesian product graphs). Here, we prove perfectstate transfer on double cones of irregular graphs and on double half-cones connected by circulants.These generalized results in [8, 3] on double cones of regular graphs and complement the negativeresult on double half-cones in [2]. Although these cone constructions involve small diameter graphs,they provided insights into which intermediate graphs allow antipodal perfect state transfer. Non-antipodal perfect state transfer can also be derived from certain cones (as shown in [3]).We also generalized the path-collapsing argument using the theory of equitable partitions. Thiscan be used to show that certain weighted paths with self-loops have perfect state transfer. Apossible interesting direction is to study random graphs with equitable distance partitions (as inthe Anderson model [1]). A weighted path-collapsing argument would also be interesting since itcan be used to analyze graphs produced in Feder’s intriguing construction [14]. The most elusivegraph not covered by this framework is P ⊕ n since none of the path-collapsing arguments apply.This is because the connections are irregular (see Figure 1(b)). We leave these as open questions. Acknowledgments
The research was supported in part by the National Science Foundation grant DMS-1004531 andalso by the National Security Agency grant H98230-09-1-0098. We thank Richard Cleve, DavidFeder, and Michael Underwood for their helpful comments on perfect state transfer.
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