Perfect State Transfer on gcd-graphs

0.17 MB·

by  Hiranmoy Pal

Perfect State Transfer on gcd-graphs
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1 Hiranmoy Pal
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2 Department of Mathematics

v
o Indian Institute of Technology Guwahati
N
Guwahati, India - 781039

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1 Email: [email protected]

]
O
C Bikash Bhattacharjya
.
h
Department of Mathematics
t
a
m Indian Institute of Technology Guwahati
[
Guwahati, India - 781039

2
v Email: [email protected]
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6 November 15, 2016
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0
.
1 Abstract
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6
1
Let G be a graph with adjacency matrix A. The transition matrix of G is denoted by H(t) and
:
v
i it is defined by H(t) := exp(itA), t ∈ R. The graph G has perfect state transfer (PST) from a
X
r vertex u to another vertex v if there exist τ (6= 0) ∈ R such that the uv-th entry of H(τ) has
a
unit modulus. In case when u = v, we say that G is periodic at the vertex u at time τ. The
graph G is said to be periodic if it is periodic at all vertices at the same time. A gcd-graph is
a Cayley graph over a finite abelian group defined by greatest common divisors. We establish
a sufficient condition for a gcd-graph to have periodicity and PST at π. Using this we deduce
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that there exists gcd-graph having PST over an abelian group of order divisible by 4. Also we
find a necessary and sufficient condition for a class of gcd-graphs to be periodic at π. Using this
we characterize a class of gcd-graphs not exhibiting PST at π for all positive integers k.
2k
Keywords: Perfect state transfer, Cayley Graph, Kronecker product of graphs.
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1 Introduction
Perfect state transfer has a great significance in continuous-time quantum walks as it has ap-
plications in quantum information processing. Perfect sate transfer in quantum networks was
introduced by Bose [4]. Quantum networks are modelled on finite graphs and when there is no
external dynamic control over a system then PST depends only on the underlying graph. Main
interest here is to find graphs exhibiting PST.
Christandl et al. [7, 8] found that the paths P and P and their cartesian powers exhibits
2 3
PST. A characterization of PST in NEPS (non-complete extended p-sum) of the path P was
3
given by Pal et al. [13]. Bernasconi et al. [3] showed that PST occurs on certain cubelike graphs
which are actually gcd-graphs over a direct product of finitely many copies of the group Z . A
2
complete characterization of PST in integral circulant networks was found by Baˇsi´c [1, 2]. The
integral circulant graphs are also gcd-graphs over the cyclic group Z .
n
In this article we consider PST on gcd-graphs over general abelian groups. These graphs
are known to have integral spectrum. It is well known that periodicity is a necessary condition
for PST in regular graphs. In [11, 12], Godsil found that a regular graph is periodic if and
only if its eigenvalues are integers. Therefore, if a Cayley graph has PST then it must have
integral spectrum. A characterization of integral Cayley graph over finite abelian groups is
given in [15]. Among the integral Cayley graphs we only consider gcd-graphs. Here we find
some characterizations of periodicity and PST in gcd-graphs. More information regarding PST
and periodicity can be found in [6, 9, 10, 16, 17].
2 Preliminaries
We introduce gcd-graphs over a finite abelian group. Let (Γ,+) be a finite abelian group and
consider S ⊆ Γ with {−s: s ∈ S} = S. Such a set S is called a symmetric subset of Γ. The
Cayley graph over Γ with connection set S is denoted by Cay(Γ,S). The graph has the vertex
set Γ where two vertices a,b ∈ Γ are adjacent if and only if a−b ∈ S. If the additive identity
0 ∈ S then Cay(Γ,S) has loops at each of its vertices. In that case, we use the convention
that each loop contributes one to the corresponding diagonal entry of the adjacency matrix.
However while discussing PST on gcd-graphs, we consider simple graphs, i.e, we consider 06∈ S.
2
The greatest common divisor of two non-negative integers m,n is denoted by gcd(m,n).
For every non-negative integer n we use the convention that gcd(0,n) = gcd(n,0) = n. Let
us consider two r-tuples of non-negative integers m = (m ,...,m ) and n = (n ,...,n ). For
1 r 1 r
i = 1,...,r, suppose gcd(m ,n ) = d and we write d = (d ,...,d ). We define gcd of m,n to
i i i 1 r
be d and write gcd(m,n) = d.
Let Z be the cyclic group of order n. Every finite abelian group (Γ,+) has a cyclic group
n
decomposition
Γ =Z ⊕...⊕Z , where r ≥ 1 and m ≥ 1 for i = 1,...,r.
m1 mr i
For each i = 1,...,r, assume that d is a divisor of m with 1 ≤ d ≤ m . For the divisor tuple
i i i i
d = (d ,...,d ) of m= (m ,...,m ), define
1 r 1 r
S (d) = {x ∈Γ : gcd(x,m) = d}.
Γ
Let D be a set of divisor tuples of m and define
S (D) = S (d).
Γ [ Γ
d∈D
Note that the union is actually a disjoint union. The sets S (D) are called gcd-sets of Γ. A
Γ
Cayley graph over a finite abelian group whose connection set is a gcd-set is called a gcd-graph.
More information regarding gcd-graphs can be found in [14, 15].
ConsidertwographsG andG withthesetofverticesU andV,respectively. TheKronecker
1 2
product [5] of G and G is denoted by G ×G which has the vertex set U ×V. Two vertices
1 2 1 2
(u ,u ) and (v ,v ) are adjacent in G × G whenever u is adjacent to v in G and u is
1 2 1 2 1 2 1 1 1 2
adjacent to v in G . If G and G have the adjacency matrices A and A , respectively, then
2 2 1 2 1 2
G ×G has the adjacency matrix A ⊗A , the tensor product of A and A . The transition
1 2 1 2 1 2
matrix of Kronecker product of two graphs can be calculated as follows.
Proposition 2.1. [11, 13] Let G and G be two graphs with adjacency matrices A and A ,
1 2 1 2
respectively. Suppose the transition matrix of G is H(t). If the spectral decomposition of A is
1 2
q q
µ F then G ×G has the transition matrix H(µ t)⊗F .
s s 1 2 s s
P P
s=1 s=1
More information regarding PST on Kronecker products can be found in [11].
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3 PST on cubelike graphs
AcubelikegraphX(C)isaCayleygraphoverZn withaconnectionsetC⊂ Zn. PSToncubelike
2 2
graphs (simple) has already been discussed in [3, 6]. Here we discuss some relevant results from
[6], which remain valid for looped cubelike graphs. Recall that each loop contributes one to the
adjacency matrix. Finally we deduce a simple result that will be used later to characterize PST
in gcd-graphs over general abelian groups.
For each x ∈ Zn, the map P :Zn −→ Zn defined by P (y) = x+y is a permutation of the
2 x 2 2 x
elements of Zn and hence it can be realized as a permutation matrix of appropriate order. It is
2
easy to see that P = I and P P = P which imply P2 = I. The following result finds the
0 x y x+y x
adjacency matrix of a cubelike graph.
Lemma 3.1. [6] If C ⊆ Zn and X(C) is the cubelike graph with connection set C then X(C)
2
has the adjacency matrix A = P .
x
P
x∈C
Note that P P = P = P P for all x,y ∈ Z . Thereforeby usingthe propertyof matrix
x y x+y y x n
exponential we find that
exp(it(P +P )) = exp(itP )exp(itP ).
x y x y
Using this the transition matrix of a cubelike graph can be calculated as follows.
Lemma 3.2. [6] If H(t) is the transition matrix of the cubelike graph X(C) then
H(t) = exp(itP ).
Y x
x∈C
We already have P2 = I and this implies that
x
t2 t3 t4
exp(itP ) = I +itP − I −i P + I +...
x x x
2! 3! 4!
= cos(t)I +isin(t)P .
x
Observe that if σ is the sum of the elements in C ⊆ Zn then P = P . Therefore by using
2 x σ
Q
x∈C
Lemma 3.2, we deduce that
π
H = iP = i|C|P .
(cid:16)2(cid:17) Y x σ
x∈C
The next result determines periodicity and PST in cubelike graphs at t = π.
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Theorem 3.3. [6] Let C ⊆ Zn and let σ be the sum of the elements of C. If σ 6= 0 then PST
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occurs in X(C) from x to x+σ at π. If σ = 0 then X(C) is periodic with period π.
2 2
Now we find a sufficient condition for periodicity in a class of graphs constructed from
cubelike graphs. Consider the following result.
Proposition 3.4. Let C ⊆ Zn and let X(C) be a cubelike graph with the connection set C.
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Assume that the sum of the elements of C is 0 and |C| ≡ 0 (mod 4). Then for every integral
graph G, the transition matrix of X(C)×G at π is the identity operator.
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Proof. If H(t) is the transition matrix of the cubelike graph X(C) then we have
π
H = iP = i|C|P ,
(cid:16)2(cid:17) Y x σ
x∈C
where σ is the sum of the elements in C. Now σ = 0 implies that P = I and therefore if
σ
|C| ≡ 0 (mod 4), then H π = I. For every integer µ, we find that H πµ = H π µ = I.
(cid:0)2(cid:1) (cid:0) 2 (cid:1) (cid:0) (cid:0)2(cid:1)(cid:1)
q
Now consider µ F to be the spectral decomposition of adjacency matrix of G. As the graph
s s
P
s=1
G is assumed to be integral, the eigenvalues µ of G are integers. By Proposition 2.1, the
s
transition matrix of the graph X(C)×G at π is obtained as
2
q q
πµ
s
H ⊗F = I ⊗ F = I ⊗I = I,
X (cid:16) 2 (cid:17) s X s
s=1 s=1
where the identity matrices have appropriate orders. This proves our claim.
4 Periodicity and PST on gcd-Graphs
Now we investigate gcd-graphs for periodicity and PST. First we construct some periodic gcd-
graphs which are not necessarily connected. Also we find some gcd-graphs exhibiting PST and
then we club them to obtain connected gcd-graph having PST. First consider the following
characterization of PST in vertex-transitive graphs as given in [11].
Theorem 4.1. [11] Suppose G is a connected vertex-transitive graph with vertices u and v, and
perfect state transfer from u to v occurs at time τ. Then the transition matrix of G is a scalar
multiple of a permutation matrix of order two and no fixed points, and it lies in the center of
the automorphism group of G.
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Consequently, if a vertex-transitive graph admits PST then it must have an even number of
vertices. We therefore have the following obvious characterization for PST in gcd-graphs.
Corollary 4.2. A gcd-graph over a group of odd order does not exhibit perfect state transfer.
The eigenvalues and the corresponding eigenvectors of a Cayley graph over an abelian group
are well known. In [15], it is shown that the eigenvectors are independent of the connection set.
ConsidertwosymmetricsubsetsS ,S inΓ. Sothesetofeigenvectors ofbothgraphsCay(Γ,S )
1 2 1
and Cay(Γ,S ) can be chosen to be equal. Hence we have the following result.
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Proposition 4.3. If S and S are symmetric subsets of an abelian group Γ then adjacency
1 2
matrices of the Cayley garphs Cay(Γ,S ) and Cay(Γ,S ) commute.
1 2
The following result allows us to find the transition matrix of a Cayley graph, which is union
of two edge disjoint Cayley graphs over an abelian group.
Proposition 4.4. Let Γ be a finite abelian group and consider two disjoint and symmetric
subsets S,T ⊂ Γ. Suppose the transition matrices of Cay(Γ,S) and Cay(Γ,T) are H (t) and
S
H (t), respectively. Then Cay(Γ,S ∪T) has the transition matrix H (t)H (t).
T S T
Proof. The group Γ is a finite abelian group. By Proposition 4.3, the adjacency matrices of the
graphsCay(Γ,S)andCay(Γ,T)commute. Weknowthatforanytwosquarematrices A,B with
AB = BA, we have exp(A+B)= exp(A)exp(B). Using this we get the desired result.
Suppose the prime factorization of an integer n(≥ 2) is n = p ...p , where the primes are
1 k
not necessarily distinct. In [14], the authors showed that every gcd-graph with n vertices is
isomorphic to a gcd-graph over Γ = Z ⊕...⊕Z . If n is a power of 2 then the gcd-graph
p1 pk
is actually a cubelike graph. The next theorem is thus a special case to that result. Still we
include the result so as to have a definite structure of the connection set of the cubelike graph,
which will be used to characterize PST on gcd-graphs. We make use of some techniques from
[14] and consider looped graphs.
Assume that X(C ) and X(C ) are two cubelike graphs. It is easy to see that there is a
1 2
natural isomorphism between X(C )×X(C ) and X(C ×C ).
1 2 1 2
Theorem 4.5. [14] A gcd-graph over an abelian group of order 2n is isomorphic to a cubelike
graph.
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Proof. Consider an abelian group Γ = Z2n1 ⊕...⊕Z2nr. For each set of divisor tuples D, we
show that Cay(Γ,S (D)) is isomorphic to a cubelike graph. Let d = (d ,...,d ,...,d ) ∈ D.
Γ 1 k r
For i = 1,...,r we have di = 2ki for some ki ≤ ni. If x = (x1,...,xr),y = (y1,...,yr) ∈ Γ
and n = (2n1,...,2nr) then gcd(x − y,n) = d if and only if gcd(xi − yi,2ni) = di. That is
x ∼ y in Cay(Γ,SΓ(d)) if and only if xi ∼ yi in Cay(Z2ni,SZ2ni(di)). Note that, if di = 2ni
then Cay(Z2ni,SZ2ni(di)) is the graph with loops at each of its vertices and no other edges. We
therefore have the following:
Cay(Γ,SΓ(d)) ∼= Cay(Z2n1,SZ2n1(d1))×...×Cay(Z2nr,SZ2nr(dr)).
Now for a fixed i, if z ∈ Z2ni then there exists a unique 2-adic representation
ni−1
z = z 2j, where z ∈ {0,1} for j = 0,1,...,n −1.
X j j i
j=0
We write z˜ = (z ,...,z ) ∈ Zni. We show that the map z 7→ z˜ gives an isomorphism of
0 ni−1 2
Cay(Z2ni,SZ2ni(di)) to the cubelike graph X(Cdi) over Zn2i where
C = {(c ,c ,...,c ) ∈ Zni : c = 0 for every j < k and c = 1}, where d = 2ki.
di 0 1 ni−1 2 j i ki i
Note that for j > k , the value of c can be either 0 or 1. It is enough to show that u ∼ v in
i j
Cay(Z2ni,SZ2ni(di)) if and only if u˜ −v˜ ∈ Cdi. Now u ∼ v in Cay(Z2ni,SZ2ni(di)) if and only
if gcd(u−v,2ni) = 2ki. Observe that gcd(u−v,2ni) = 2ki if and only if uj −vj = 0 for every
j < k and u −v = 1. Thus we have
i ki ki
Cay(Z2ni,SZ2ni(di)) ∼= X(Cdi), for each i = 1,...,r.
Therefore for d ∈D we find that
Cay(Γ,S (d)) ∼= X(C )×...×X(C ) ∼= X(C ×...×C ).
Γ d1 dr d1 dr
TheisomorphismthatweexhibitedbetweentheverticesofCay(Γ,S (d))andX(C ×...×C )
Γ d1 dr
will work for all divisors d ∈ D. Hence we have
Cay(Γ,S (D)) ∼= X(C), where C = C ×...×C .
Γ [ d1 dr
d∈D
This completes the proof.
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Assume that Γ is a finite abelian group. We write Γ = Γ ⊕ Γ , where Γ is an abelian
1 2 1
group of order 2n and Γ is an abelian group of odd order. Also, consider the cyclic group
2
decomposition of Γ and Γ as follows:
1 2
Γ1 = Z2n1 ⊕...⊕Z2nr, Γ2 = Zpm11 ⊕...⊕Zpmss.
We write m := (2n1,...,2nr,pm1,...,pms) associated to the group Γ. In what follows, we will
Γ 1 s
always consider a finite abelian group Γ in the form Γ ⊕Γ as described above. We have the
1 2
following lemma on the structure of certain gcd-graphs.
Lemma 4.6. Let Γ1 = Z2n1 ⊕... ⊕Z2nr and Γ2 = Zpm11 ⊕ ... ⊕ Zpmss, pi > 2, be such that
Γ = Γ ⊕Γ . Assumethatd ,...,d are fixeddivisorsof pm1,...,pms, respectively. Consider
1 2 r+1 r+s 1 s
a set of divisor tuples D such that m ∈/ D. If the last s components of each d ∈ D are
Γ
d ,...,d then there exist a cubelike graph X(C) such that
r+1 r+s
Cay(Γ,S (D)) ∼= X(C)×Cay(Γ ,S ({(d ,...,d )})).
Γ 2 Γ2 r+1 r+s
Proof. Assume that D∗ = {(d ,...,d ) :d = (d ,...,d ,...,d ) ∈D}. Therefore
1 r 1 r r+s
Cay(Γ,S (D)) ∼= Cay(Γ ,S (D∗))×Cay(Γ ,S ({(d ,...,d )})).
Γ 1 Γ1 2 Γ2 r+1 r+s
By Theoren 4.5, there exist a cubelike graph X(C) such that Cay(Γ ,S (D∗)) ∼= X(C). Hence
1 Γ1
we have the desired result.
Now we find a sufficient condition for a gcd-graph to exhibit periodicity at π. Let D′ be the
2
collection of all divisors sets D satisfying the conditions of Lemma 4.6 as well as the following
two conditions:
1. the sum of the elements in C is zero; and
2. |C| ≡0 (mod 4) whenever d < pmi for some i= 1,...,s.
r+i i
Now suppose D is the collection of all disjoint union of members of D′. The following result
determines a class of periodic gcd-graphs.
Theorem 4.7. If Γ is a finite abelian group and D∈ D then Cay(Γ,S (D)) is periodic at π.
Γ 2
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Proof. Let D = D ∪ ... ∪ D ∈ D, where D ∈ D′ for all l = 1,...,k. Suppose d is a
1 k l r+j
fixed divisor of pmj for j = 1,...,s. For a fixed l assume that the last s components of each
j
d ∈ D are d ,...,d . By using Lemma 4.6 we find that Cay(Γ,S (D )) is isomorphic to
l r+1 r+s Γ l
X(C)×Cay(Γ ,S ({(d ,...,d )})). Consider the following two cases.
2 Γ2 r+1 r+s
Case I: (d < pmj for some j) By our assumption on D, the connection set C in X(C) in
r+j j
this case is such that |C| ≡ 0 (mod 4) and the sum of the elements in C is zero. Therefore by
Proposition 3.4, the transition matrix of Cay(Γ,S (D )) is the identity matrix at π.
Γ l 2
Case II: (d = pmj for all j) In this case the sum of the elements of C in X(C) is zero.
r+j j
Note that Cay(Γ ,S ({(d ,...,d )})) has loops at each of its vertices and no more edges.
2 Γ2 r+1 r+s
Recall that each loop contributes 1 to the adjacency matrix according to our convention. If A
is the adjacency matrix of X(C) then by using Lemma 4.6 we find that the adjacency matrix
of Cay(Γ,S (D )) is A⊗I. The transition matrix of Cay(Γ,S (D )) can be calculated as
Γ l Γ l
exp(it(A⊗I)) = exp(itA)⊗I.
Observe that exp(itA) is the transition matrix of X(C). By Theorem 3.3, the graph X(C) is
periodic at π and therefore Cay(Γ,S (D )) is also periodic at π.
2 Γ l 2
In both the cases the graph Cay(Γ,S (D )) is periodic at π. Finally, for D ∈ D, we apply
Γ l 2
Proposition 4.4 to have the desired result.
We illustrate this by the following example. Here we finda periodicgraph over Z ⊕Z ⊕Z .
4 2 3
Example 4.1. Consider Γ = Z ⊕Z ⊕Z and the set of divisor tuples D= {(1,1,1),(1,2,1)}
4 2 3
of (4,2,3). We show that the graph Cay(Γ,S (D)) is periodic at π. Suppose Γ′ = Z ⊕Z and
Γ 2 4 2
consider D∗ = {(1,1),(1,2)}. Now we follow Theorem 4.5 to find the cubelike graph isomorphic
to Cay(Γ′,SΓ′(D∗)). For d = (d1,d2) = (1,1), we set Cd1 = {(1,0),(1,1)} and Cd2 = {(1)}.
Therefore C × C = {(1,0,1),(1,1,1)}. Similarly, for d′ = (d′,d′) = (1,2), we find that
d1 d2 1 2
Cd′1 ×Cd′2 = {(1,0,0),(1,1,0)}. Hence Cay(Γ′,SΓ′(D∗)) is isomorphic to X(C) which has the
connection set C = (Cd1 ×Cd2)∪(cid:16)Cd′1 ×Cd′2(cid:17) = {(1,0,1),(1,1,1),(1,0,0),(1,1,0)}. Clearly
|C| ≡ 0 (mod 4) and the sum of the elements in C is 0 in Z3. Therefore D ∈ D and hence by
2
Theorem 4.7, the graph Cay(Γ,S (D)) is periodic at π.
Γ 2
In this way wecan construct many gcd-graphs which areperiodic. Ournextmotive is to add
extra edges to these graphs so that the graphs exhibit PST. Generally periodicity and PST are
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considered for connected graphs. Observe that whenever D generates the whole group Γ then
the associated gcd-graph is connected. In the following result we find a sufficient condition for
a gcd-graph to exhibit PST.
Consider a set of divisor tuples D of m with m ∈/ D. Also, assume that for each tuple
Γ Γ
d = (d ,...,d ,...,d ) in D, the divisors d = pmj for j = 1,...,s. Let X(C) be the
1 r r+s r+j j
cubelike graph as in Lemma 4.6 associated to Cay(Γ,S (D)). We denote the set of all D such
Γ
that sum of the elements in C is non-zero by D˜.
Theorem 4.8. Let Γ be a finite abelian group. Suppose D ∈ D and D ∈ D˜ and D ∩D = ∅.
1 2 1 2
If D = D ∪D so that D generates Γ then Cay(Γ,S (D)) is connected and admits perfect state
1 2 Γ
transfer at π.
2
Proof. If D generates the group Γ then the graph Cay(Γ,S (D)) is clearly connected. By
Γ
Theorem 4.7, the graph Cay(Γ,S (D )) is periodic at π. Thus by Proposition 4.4, it is enough
Γ 1 2
to show that Cay(Γ,S (D )) exhibits PST at π.
Γ 2 2
Using Lemma 4.6 we find that
Cay(Γ,S (D )) ∼= X(C)×Cay(Γ ,S ({(pm1,...,pms)})).
Γ 2 2 Γ2 1 s
Since G = Cay(Γ ,S ({(pm1,...,pms)})) has loops at each of its vertices and no more edges,
2 Γ2 1 s
the adjacency matrix of G is I. Suppose X(C) has the adjacency matrix A. The adjacency
matrix of Cay(Γ,S (D )) is therefore A×I. Now the transition matrix of Cay(Γ,S (D )) can
Γ 2 Γ 2
be calculated as exp(itA×I) = exp(itA)×I. Observe that exp(itA) is the transition matrix
of X(C). Since D ∈D˜, the sum of the elements in C is non-zero. Therefore, by Theorem 3.3,
2
the cubelike graph X(C) exhibits PST at π. Hence Cay(Γ,S (D )) also exhibits PST at π.
2 Γ 2 2
We illustrate Theorem 4.8 by the following example.
Example 4.2. Suppose Γ = Z ⊕Z ⊕Z and let D = {(1,1,1),(1,2,1),(2,2,3),(4,1,3)}. We
4 2 3
show that the graph Cay(Γ,S (D)) admits PST at π. Let D = {(1,1,1),(1,2,1)}. We already
Γ 2 1
have in Example 4.1 that D ∈ D. Now consider D = {(2,2,3),(4,1,3)}. Here the connection
1 2
set C in X(C) associated to Cay(Γ,S (D )) can be evaluated as C= {(0,1,0),(0,0,1)}. Thus
Γ 2
we have D ∈ D˜. Note here that D generates Γ. Hence, by using Theorem 4.8, we find that
2
Cay(Γ,S (D)) is connected and exhibits PST at π.
Γ 2
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