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Quantum walks and algebraic graph theory Krystal Guo Universit´ e Libre de Bruxelles Symmetry and Regularity 2018, Pilsen
Quantum walks and algebraic graph theory
Krystal Guo
Universite Libre de Bruxelles
Symmetry and Regularity 2018, Pilsen
Continuous-time quantum walk
G a graph
Continuous-time quantum walk
G a graph
A = A(G) is the adjacency matrix of G
Continuous-time quantum walk
G a graph
A = A(G) is the adjacency matrix of G
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
Continuous-time quantum walk
G a graph
A = A(G) is the adjacency matrix of G
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA
Continuous-time quantum walk
G a graph
A = A(G) is the adjacency matrix of G
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA = I + itA− 12 t
2A2 + 16 it
3A3 · · ·
Continuous-time quantum walk
G a graph
A = A(G) is the adjacency matrix of G
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA = I + itA− 12 t
2A2 + 16 it
3A3 · · ·
Say, G = K2.
Continuous-time quantum walk
G a graph
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA = I + itA− 12 t
2A2 + 16 it
3A3 · · ·
Say, G = K2.
A =
(0 11 0
)
Continuous-time quantum walk
G a graph
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA = I + itA− 12 t
2A2 + 16 it
3A3 · · ·
Say, G = K2.
A =
(0 11 0
)A2 = I and A3 = A.
Continuous-time quantum walk
G a graph
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA
Say, G = K2.
A =
(0 11 0
)A2 = I and A3 = A.
= I + itA− 12 t
2I + 16 it
3A · · ·
Continuous-time quantum walk
G a graph
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA
Say, G = K2.
A =
(0 11 0
)A2 = I and A3 = A.
= I + itA− 12 t
2I + 16 it
3A · · ·
=
(cos(t) i sin(t)i sin(t) cos(t)
)
Continuous-time quantum walk
G a graph
We study the continuous-time quantum walk,whose behaviour is governed by its transitionmatrix:
U(t) = eitA
Say, G = K2.
A =
(0 11 0
)A2 = I and A3 = A.
= I + itA− 12 t
2I + 16 it
3A · · ·
=
(cos(t) i sin(t)i sin(t) cos(t)
)U(t) is unitary and symmetric matrix.
Special behaviour of K2
There are some times when U(t) has a special form:
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
U(π/4) = 1√2
11i
i
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
U(π/4) = 1√2
11i
iall entries of U(π/4) havethe same absolute value
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
U(π/4) = 1√2
11i
iall entries of U(π/4) havethe same absolute value
We can consider the average behaviour over time:
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
U(π/4) = 1√2
11i
iall entries of U(π/4) havethe same absolute value
We can consider the average behaviour over time:
1T
∫ T
0U(t) ◦ U(t)dt
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/2) = i0
01
1
U(π/4) = 1√2
11i
iall entries of U(π/4) havethe same absolute value
We can consider the average behaviour over time:
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)
Special behaviour of K2
There are some times when U(t) has a special form:
U(π/4) = 1√2
11i
i
perfect state transfer between u and v
exists τ , U(τ) = γ0
01
1 ||γ|| = 1
all entries of U(π/4) havethe same absolute value
We can consider the average behaviour over time:
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)
Special behaviour of K2
There are some times when U(t) has a special form:
perfect state transfer between u and v
exists τ , U(τ) = γ0
01
1 ||γ|| = 1
uniform mixing at time τ
all entries of U(τ) havethe same absolute value
We can consider the average behaviour over time:
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)
Special behaviour of K2
There are some times when U(t) has a special form:
perfect state transfer between u and v
exists τ , U(τ) = γ0
01
1 ||γ|| = 1
uniform mixing at time τ
all entries of U(τ) havethe same absolute value
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)average mixing matrix
M :=
Special behaviour of K2
There are some times when U(t) has a special form:
perfect state transfer between u and v
exists τ , U(τ) = γ0
01
1 ||γ|| = 1
uniform mixing at time τ
all entries of U(τ) havethe same absolute value
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)average mixing matrix
M :=
Special behaviour of K2
There are some times when U(t) has a special form:
perfect state transfer between u and v
exists τ , U(τ) = γ0
01
1 ||γ|| = 1
uniform mixing at time τ
all entries of U(τ) havethe same absolute value
1T
∫ T
0U(t) ◦ U(t)dt =
(12
12
12
12
)average mixing matrix
M :=
”Algebraically” symmetric
There is an automorphism of G swapping uand v
there exists a permutation matrix P such that
”Algebraically” symmetric
There is an automorphism of G swapping uand v
(a) PA = AP ;
there exists a permutation matrix P such that
”Algebraically” symmetric
There is an automorphism of G swapping uand v
(a) PA = AP ;
(b) Peu = ev;
there exists a permutation matrix P such that
”Algebraically” symmetric
There is an automorphism of G swapping uand v
(a) PA = AP ;
(b) Peu = ev;
(c) P 2 = I.
there exists a permutation matrix P such that
”Algebraically” symmetric
There is an automorphism of G swapping uand v
(a) PA = AP ;
(b) Peu = ev;
(c) P 2 = I.
there exists a permutation matrix P such that
We can take a spectral relaxation of this property.
”Algebraically” symmetric
We can take a spectral relaxation of this property.
Vertices u and v are cospectral
there exists a orthogonal matrix Q such that
(a) QA = AQ;
(b) Qeu = ev;
(c) Q2 = I.
”Algebraically” symmetric
Vertices u and v are cospectral
there exists a orthogonal matrix Q such that
(a) QA = AQ;
(b) Qeu = ev;
(c) Q2 = I.
To study quantum walks, we need another concept:
”Algebraically” symmetric
To study quantum walks, we need another concept:
Vertices u and v are strongly cospectral
there exists a orthogonal matrix Q such that
(a) Q is a polynomial in A with rational entries;
(b) Qeu = ev;
(c) Q2 = I.
Strong Cospectrality in Quantum Walks
perfect state transfer between u and v
average mixing matrix
Strong Cospectrality in Quantum Walks
perfect state transfer between u and v
average mixing matrix
u and v are strongly cospectral
Strong Cospectrality in Quantum Walks
perfect state transfer between u and v
average mixing matrix
u and v are strongly cospectral
(Godsil 2018) Two columns of M are equal ifand only if the corresponding vertices arestrongly cospectral.
Perfect state transfer on distance-regular graphsand association schemes. G. Coutinho, C. Godsil,K. Guo, F. Vanhove. Linear Algebra and itsApplications 449 (2015) P108-130.
State transfer in strongly regular graphs with anedge perturbation. C. Godsil, K. Guo, M.Kempton and G. Lippner.
A new perspective on the average mixing matrix.G. Coutinho, C. Godsil, K. Guo and H. Zhan.
