Generalization of Factor Graphs and Belief Propagation for Quantum - - PowerPoint PPT Presentation

Generalization of Factor Graphs and Belief Propagation for Quantum Information Science

End-of-First-Year Oral Exam Michael X. CAO

Department of Information Engineering, CUHK

September 25, 2015

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015

Highlight

g (x) =

• a

fa (xa) where fa : Xa → R0 is a real nonnegetive valued function.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 1

Highlight

g (x) =

• a

fa (xa) where fa : Xa → R0 is a real nonnegetive valued function. g (x) =

• a

fa where fa ∈ L+

H (Xa) is a

PSD operator on Xa, i.e., [fa]x∂a,x′

∂a is a PSD matrix. Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 1

Highlight

?

g (x) =

• a

fa (xa) where fa : Xa → R0 is a real nonnegetive valued function. g (x) =

• a

fa where fa ∈ L+

H (Xa) is a

PSD operator on Xa, i.e., [fa]x∂a,x′

∂a is a PSD matrix. Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 1

Highlight

?

g (x) =

• a

fa (xa) where fa : Xa → R0 is a real nonnegetive valued function. g (x) =

• a

fa where fa ∈ L+

H (Xa) is a

PSD operator on Xa, i.e., [fa]x∂a,x′

∂a is a PSD matrix.

How about calculating the partition sum?

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 1

Classical Factor Graphs Modeling

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 2

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

+ + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn

An LDPC Code applied on a binary independent channel

✶

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 2

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

+ + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn

An LDPC Code applied on a binary independent channel

xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    .

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 2

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

+ + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn

An LDPC Code applied on a binary independent channel

xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    . A problem of interest: Calculate the marginal distribution of xi given fixed {yi}n

i=1.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 2

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

+ + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn

An LDPC Code applied on a binary independent channel

xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    . A problem of interest: Calculate the marginal distribution of xi given fixed {yi}n

i=1.

g =

• i

pYi|Xi ·

• k

fk ∝ pY|X

• xj, j=i

g(x, y) ∝ pY=y|Xi

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 2

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

x1,1 x2,1 x3,1 xn,1 x1,2 x2,2 x3,2 xn,2 x1,3 x2,3 x3,3 xn,3 x1,n x2,n x3,n xn,n

. . . . . . . . . . . . · · · · · · · · · · · · ...

f f f f f f f f f f f f f f f f f f f f f f f f f f f f h11 h12 h13 h15 h21 h22 h23 h25 h31 h32 h33 h35 h51 h52 h53 h55 A simplified n × n Ising Model

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 3

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

x1,1 x2,1 x3,1 xn,1 x1,2 x2,2 x3,2 xn,2 x1,3 x2,3 x3,3 xn,3 x1,n x2,n x3,n xn,n

. . . . . . . . . . . . · · · · · · · · · · · · ...

f f f f f f f f f f f f f f f f f f f f f f f f f f f f h11 h12 h13 h15 h21 h22 h23 h25 h31 h32 h33 h35 h51 h52 h53 h55 A simplified n × n Ising Model

xi,j ∈ {−1, 1} ∀i, j; hij (xi,j) = exp

• xi · ˜

hij T

• ∀i, j;

f (x1, x2) exp x1 · x2 T

• .

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 3

Classical Factor Graphs Modeling

Factor Graph representing Factorization

Classically, a factor graph describes a factorization of a global function, which is

• ften related to a probability model.

x1,1 x2,1 x3,1 xn,1 x1,2 x2,2 x3,2 xn,2 x1,3 x2,3 x3,3 xn,3 x1,n x2,n x3,n xn,n

. . . . . . . . . . . . · · · · · · · · · · · · ...

f f f f f f f f f f f f f f f f f f f f f f f f f f f f h11 h12 h13 h15 h21 h22 h23 h25 h31 h32 h33 h35 h51 h52 h53 h55 A simplified n × n Ising Model

xi,j ∈ {−1, 1} ∀i, j; hij (xi,j) = exp

• xi · ˜

hij T

• ∀i, j;

f (x1, x2) exp x1 · x2 T

• .

A problem of interest: Calculate the partition function Z (T)

• x
• a,b adjacent

f (xa, xb) ·

• i,j

hij (xi,j) .

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 3

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph p

x

uA uB bB bA

y

q

z z

z′ A normal factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph p = uA uB bB bA = q

z z

z′ x x1 x2 y1 y2 y A normal factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs Modeling

Factor Graph representing Factorization

In general, a factor graph for factorization g (x) =

• a∈F

fa (x∂a) is a bipartite graph G = (F, V, E) between F and V with edge set E = {(i, a) ∈ F × V : i ∈ ∂a} .

p uA uB bB bA qB z x z′ y A standard factor graph p = uA uB bB bA = q

x z z′ y

A normal factor graph

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 4

Classical Factor Graphs “Closing-the-box” Operation

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 5

Classical Factor Graphs “Closing-the-box” Operation

Normal Factor Graph and “Closing-the-box” Operation

Traditionally, we assume all the factors to be nonnegative. Thus, any marginal function is still a measure. p0 p1 p2 p3

x0 x1 x2 x3 y1 y2 y3

Normal Factor Graph for a hidden Markov model of length 3: p (y1, . . . , y3, x0, . . . , x3) = p0 (x0) 3

k=1 pk (yk, xk|xk−1) Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 5

Classical Factor Graphs “Closing-the-box” Operation

Normal Factor Graph and “Closing-the-box” Operation

Traditionally, we assume all the factors to be nonnegative. Thus, any marginal function is still a measure. p0 p1 p2 p3

x0 x1 x2 x3 y1 y2 y3

Normal Factor Graph for a hidden Markov model of length 3: p (y1, . . . , y3, x0, . . . , x3) = p0 (x0) 3

k=1 pk (yk, xk|xk−1)

Exterior Function of above dashed box: pY1,Y2,Y3|X0(y1, y2, y3|x0) =

• x1,x2,x3

p1(y1, x1|x0)p2(y2, x2|x1)p3(y3, x3|x2).

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 5

Classical Factor Graphs “Closing-the-box” Operation

Normal Factor Graph and “Closing-the-box” Operation

Traditionally, we assume all the factors to be nonnegative. Thus, any marginal function is still a measure. p0

x0 y1 y2 y3

pY1,Y2,Y3|X0(y1, y2, y3|x0)

Normal Factor Graph for a hidden Markov model of length 3: p (y1, . . . , y3, x0, . . . , x3) = p0 (x0) 3

k=1 pk (yk, xk|xk−1)

Exterior Function of above dashed box: pY1,Y2,Y3|X0(y1, y2, y3|x0) =

• x1,x2,x3

p1(y1, x1|x0)p2(y2, x2|x1)p3(y3, x3|x2). “Closing-the-box” Operation: Replacing the box with a factor corresponding to its exterior function

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 5

Quantum Factor Graphs A Motivating Example

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs A Motivating Example

Factor Graph representing Quantum Probabilities

Factor graphs can be used to represent quantum probabilities if more general factors are allowed [Loeliger and Vontobel, 2015, Loeliger and Vontobel, 2012]. p (x) = U UH BH B =

X Y

Factor graph for an elementary quantum system

The global function: g (x, y, ˜ x, ˜ x′) p (x) U (˜ x, x) UH (x, ˜ x′) BH (y, ˜ x) B (˜ x′, y) = p (x) U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y).

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs A Motivating Example

Factor Graph representing Quantum Probabilities

Factor graphs can be used to represent quantum probabilities if more general factors are allowed [Loeliger and Vontobel, 2015, Loeliger and Vontobel, 2012]. p (x) = U UH BH B =

X Y

pY |X (y|x) Factor graph for an elementary quantum system

The exterior function of the dashed box: pY |X(y|x) =

• ˜

x,˜ x′

U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) =

• ˜

x

U (˜ x, x) B (˜ x, y)

• 2

.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs A Motivating Example

Factor Graph representing Quantum Probabilities

Factor graphs can be used to represent quantum probabilities if more general factors are allowed [Loeliger and Vontobel, 2015, Loeliger and Vontobel, 2012]. p (x) = U UH BH B =

X Y

Factor graph for an elementary quantum system

p0 p1 p2 p3

x0 x1 x2 x3 y1 y2 y3

Normal Factor Graph for a hidden Markov model of length 3: p (y1, . . . , y3, x0, . . . , x3) = p0 (x0) 3

k=1 pk (yk, xk|xk−1) Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs A Motivating Example

Factor Graph representing Quantum Probabilities

Factor graphs can be used to represent quantum probabilities if more general factors are allowed [Loeliger and Vontobel, 2015, Loeliger and Vontobel, 2012]. p (x) = U UH BH B =

X Y

Factor graph for an elementary quantum system

p (x) = ˆ U BH B =

X Y

Redraw of above

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs A Motivating Example

Factor Graph representing Quantum Probabilities

Factor graphs can be used to represent quantum probabilities if more general factors are allowed [Loeliger and Vontobel, 2015, Loeliger and Vontobel, 2012]. p (x) = U UH BH B =

X Y

Factor graph for an elementary quantum system

p (x) = ˆ U ˆ B =

X Y

Redraw of above

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 6

Quantum Factor Graphs Quantum Factor Graph

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

In this case, we have ✶ ✶

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

In this case, we have Global function: g (x, x′, ˜ x, ˜ x′, ˜ y, ˜ y ′, y) = p(x)U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) ·✶ {x = x′} ✶ {˜ y = ˜ y ′ = y}

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y f1

Redraw of last example

In this case, we have Global function: g (x, x′, ˜ x, ˜ x′, ˜ y, ˜ y ′, y) = p(x)U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) ·✶ {x = x′} ✶ {˜ y = ˜ y ′ = y} Exterior function: f1

• (x, ˜

y) , (x′, ˜ y ′)

• =
• ˜

x,˜ x′

ˆ U (˜ x, x; ˜ x′, x′) · ˆ B (˜ x, ˜ y; ˜ x′, ˜ y ′)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y f1

Redraw of last example

In this case, we have Global function: g (x, x′, ˜ x, ˜ x′, ˜ y, ˜ y ′, y) = p(x)U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) ·✶ {x = x′} ✶ {˜ y = ˜ y ′ = y} Exterior function: f1

• (x, ˜

y) , (x′, ˜ y ′)

• =
• ˆ

U, ˆ B

• LH( ˜

X)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y f1 f2

Redraw of last example

In this case, we have Global function: g (x, x′, ˜ x, ˜ x′, ˜ y, ˜ y ′, y) = p(x)U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) ·✶ {x = x′} ✶ {˜ y = ˜ y ′ = y} Exterior function: f1

• (x, ˜

y) , (x′, ˜ y ′)

• =
• ˆ

U, ˆ B

• LH( ˜

X)

Exterior function: f2 (y) =

• x

p(x)f1

• (x, y) , (x, y)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y f1 f2

Redraw of last example

In this case, we have Global function: g (x, x′, ˜ x, ˜ x′, ˜ y, ˜ y ′, y) = p(x)U (˜ x, x) B (˜ x′, y) U (˜ x′, x) B (˜ x, y) ·✶ {x = x′} ✶ {˜ y = ˜ y ′ = y} Exterior function: f1

• (x, ˜

y) , (x′, ˜ y ′)

• =
• ˆ

U, ˆ B

• LH( ˜

X)

Exterior function: f2 (y) =

• ˆ

P ⊗ Iy, f1

• LH(X⊗ ˜

Y)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

Definition 1 (Quantum Normal Factor Graph) A quantum normal factor graph or QNFG is a normal factor graph where each variable edge may stands for one or a pair of variables. For each factor (indexed by a ∈ F) fa (x∂a, x′

∂a; yδa)

is a PSD operator over X∂a, given yδa fixed arbitrarily.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

Definition 1 (Quantum Normal Factor Graph) A quantum normal factor graph or QNFG is a normal factor graph where each variable edge may stands for one or a pair of variables. For each factor (indexed by a ∈ F) fa (x∂a, x′

∂a; yδa)

is a PSD operator over X∂a, given yδa fixed arbitrarily.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

Definition 1 (Quantum Normal Factor Graph) A quantum normal factor graph or QNFG is a normal factor graph where each variable edge may stands for one or a pair of variables. For each factor (indexed by a ∈ F) fa (x∂a, x′

∂a; yδa)

is a PSD operator over X∂a, given yδa fixed arbitrarily.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

Definition 1 (Quantum Normal Factor Graph) A quantum normal factor graph or QNFG is a normal factor graph where each variable edge may stands for one or a pair of variables. For each factor (indexed by a ∈ F) fa (x∂a, x′

∂a; yδa)

is a PSD operator over X∂a, given yδa fixed arbitrarily.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Quantum Factor Graph

Quantum Normal Factor Graph (QNFG) as a simplified model

diag (p(x))

ˆ P ˆ U ˆ B

Iy

X X ′ ˜ X ˜ X ′ ˜ Y ˜ Y ′ Y

Redraw of last example

Definition 1 (Quantum Normal Factor Graph) A quantum normal factor graph or QNFG is a normal factor graph where each variable edge may stands for one or a pair of variables. For each factor (indexed by a ∈ F) fa (x∂a, x′

∂a; yδa)

is a PSD operator over X∂a, given yδa fixed arbitrarily. We can defined quantum factor graph (QFG) similarly, allowing some variable nodes to have degree higher than 2.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 7

Quantum Factor Graphs Construction of a QNFG

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 8

Quantum Factor Graphs Construction of a QNFG

Conversion into QNFG: Squeeze

Classical factor graph U UH Quantum Normal Factor Graph ˆ U ˆ U ((˜ x, x), (˜ x′, x′)) U (˜ x, x) · U (˜ x′, x′) = vec (U) vec (U)H

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 8

Quantum Factor Graphs Construction of a QNFG

Conversion into QNFG: Equality

Classical factor graph = X X ′ Quantum Normal Factor Graph I X X ′ I is the identity matrix, i.e., I (x, x′) = δ (x, x′).

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 8

Quantum Factor Graphs Construction of a QNFG

Conversion into QNFG: Merge

Classical factor graph p (x) = X X ′ Quantum Normal Factor Graph

diag (p(x))

X X ′ diag (p) (x, x′) =

• p (x)

if x = x′

• therwise

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 8

Quantum Factor Graphs Construction of a QNFG

Conversion into QNFG: Parametrize

Classical factor graph = ˜ Y ˜ Y ′ Y Quantum Normal Factor Graph Iy ˜ Y ˜ Y ′ Y Iy (˜ y, ˜ y ′) =

• 1

if ˜ y = ˜ y ′ = y

• therwise

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 8

Quantum Factor Graphs Several Examples

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

Unitary Evolution over time in n steps followed by a single projective measure

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1 ρ2

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

ρ2 (x2, x′

2) =

• x1,x′

1

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ρ (x1, x′

1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1 ρ2

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

ρ2 (x2, x′

2) =

• x1,x′

1

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ρ (x1, x′

1) =

• ˆ

U2, ρ1

• LH(X1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1 ρ2 ρn

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

ρ2 (x2, x′

2) =

• x1,x′

1

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ρ (x1, x′

1) =

• ˆ

U2, ρ1

• LH(X1)

ρn (xn, x′

n) =

• xn−1,x′

n−1

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ρ

• xn−1, x′

n−1

• Michael X. CAO (IE@CUHK)

Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1 ρ2 ρn

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

ρ2 (x2, x′

2) =

• x1,x′

1

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ρ (x1, x′

1) =

• ˆ

U2, ρ1

• LH(X1)

ρn (xn, x′

n) =

• xn−1,x′

n−1

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ρ

• xn−1, x′

n−1

• =
• ˆ

Un, ρn−1

• LH(Xn−1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ρ1 ρ2 ρn

Unitary Evolution over time in n steps followed by a single projective measure

ρ1 (x1, x′

1) =

• x,x′

ˆ U1

• (x1, x) , (x′

1, x′)

• p (x, x′) δ (x, x′) =
• ˆ

U1, diag (p)

• LH(X)

ρ2 (x2, x′

2) =

• x1,x′

1

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ρ (x1, x′

1) =

• ˆ

U2, ρ1

• LH(X1)

ρn (xn, x′

n) =

• xn−1,x′

n−1

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ρ

• xn−1, x′

n−1

• =
• ˆ

Un, ρn−1

• LH(Xn−1)

Schr¨

• dinger representation.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• Michael X. CAO (IE@CUHK)

Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1 ϕ1

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

ϕ1 =

• x2,x′

2

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ϕ2 (x2, x′

2)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1 ϕ1

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

ϕ1 =

• x2,x′

2

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ϕ2 (x2, x′

2) =

• ˆ

U2, ϕ2

• LH( ˜

X2)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1 ϕ1 ϕ0

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

ϕ1 =

• x2,x′

2

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ϕ2 (x2, x′

2) =

• ˆ

U2, ϕ2

• LH( ˜

X2)

ϕ0 =

• x1,x′

1

ˆ U1

• (x1, x) , (x′

1, x′)

• ϕ1 (x1, x′

1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1 ϕ1 ϕ0

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

ϕ1 =

• x2,x′

2

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ϕ2 (x2, x′

2) =

• ˆ

U2, ϕ2

• LH( ˜

X2)

ϕ0 =

• x1,x′

1

ˆ U1

• (x1, x) , (x′

1, x′)

• ϕ1 (x1, x′

1) =

• ˆ

U1, ϕ1

• LH( ˜

X1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 1

diag (p(x))

ˆ U1 ˆ U2 · · · ˆ Un ˆ B

Iy

X X ′ X1 X ′

1

Xn X ′

n

˜ Y ˜ Y ′ Y

ϕn ϕn−1 ϕ1 ϕ0

Unitary Evolution over time in n steps followed by a single projective measure

ϕn (xn, x′

n) =

• y

ˆ B

• (xn, y) , (x′

n, y)

• =
• ˆ

B, Iy

• LH( ˜

Y)

ϕn−1

• xn−1, x′

n−1

• =
• xn,x′

n

ˆ Un−1

• (xn, xn−1) ,
• x′

n, x′ n−1

ϕ (xn, x′

n) =

• ˆ

Un, ϕn

• LH( ˜

Xn)

ϕ1 =

• x2,x′

2

ˆ U2

• (x2, x1) , (x′

2, x′ 1)

• ϕ2 (x2, x′

2) =

• ˆ

U2, ϕ2

• LH( ˜

X2)

ϕ0 =

• x1,x′

1

ˆ U1

• (x1, x) , (x′

1, x′)

• ϕ1 (x1, x′

1) =

• ˆ

U1, ϕ1

• LH( ˜

X1)

Heisenberg representation.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 9

Quantum Factor Graphs Several Examples

Example 2

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

Y1 Y2

A Two-Measurement Quantum System

Here, we assume

• yk
• xk

ˆ Ak

• (˜

xk, xk) , (˜ xk, x′

k)

• = δ (xk, x′

k)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 10

Quantum Factor Graphs Several Examples

Example 2

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

Y1 Y2

A Two-Measurement Quantum System

Here, we assume

• yk
• xk

ˆ Ak

• (˜

xk, xk) , (˜ xk, x′

k)

• = δ (xk, x′

k)

• r, equivalently
• yk
• ˆ

Ayk

k , δ ˆ Xk, ˆ X ′

k

• LH( ˆ

X) = δXk,X ′

k Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 10

Quantum Factor Graphs Several Examples

Example 2

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

Y1 Y2

A Two-Measurement Quantum System

Here, we assume

• yk
• xk

ˆ Ak

• (˜

xk, xk) , (˜ xk, x′

k)

• = δ (xk, x′

k)

• r, equivalently
• yk
• ˆ

Ayk

k , δ ˆ Xk, ˆ X ′

k

• LH( ˆ

X) = δXk,X ′

k

A special example:

ˆ Bk Iyk ˆ BH

k

Yk

Projective Measurement with 1-dim Eigenspaces

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 10

Quantum Factor Graphs Several Examples

Example 3

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 ˆ U2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

X3 X ′

3

W1 W ′

1

W2 W ′

2

Y1 Y2

A Quantum System with partial measurement

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 11

Quantum Factor Graphs Several Examples

Example 3

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 ˆ U2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

X3 X ′

3

W1 W ′

1

W2 W ′

2

Y1 Y2

A Quantum System with partial measurement

X0 = X1 × W1

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 11

Quantum Factor Graphs Several Examples

Example 3

diag (p(x0))

ˆ U0 ˆ A1 ˆ U1 ˆ A2 ˆ U2 I

X0 X ′ X1 X ′

1

˜ X1 ˜ X ′

1

X2 X ′

2

˜ X2 ˜ X ′

2

X3 X ′

3

W1 W ′

1

W2 W ′

2

Y1 Y2

A Quantum System with partial measurement

X0 = X1 × W1 This QFG contains cycles.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 11

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) a b d c e f

x1 x2 x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) 1 a b d c e f

x1 x2 x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) a bd c e f

x2 x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) 2 a bd c e f

x2 x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) a bd ce f

x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) 3 a bd ce f

x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) a bd cef

x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) 4 a bd cef

x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3) fabd (x5) =

x4

fa (x4, x5) fbd (x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) abd cef

x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3) fabd (x5) =

x4

fa (x4, x5) fbd (x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) 5 abd cef

x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3) fabd (x5) =

x4

fa (x4, x5) fbd (x4) Z = fabcdef =

x5

fabd (x5) fcef (x5)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) abcdef

Sum-Product Algorithm on a normal factor graph with no cycles

fbd (x4) =

x1

fb (x1, x4) fd (x1) fce (x3, x5) =

x2

fc (x2, x3, x5) fe (x2) fcef (x5) =

x3

fce (x3, x5) ff (x3) fabd (x5) =

x4

fa (x4, x5) fbd (x4) Z = fabcdef =

x5

fabd (x5) fcef (x5)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x

g (x) a b d c e f

x1 x2 x3 x4 x5

Sum-Product Algorithm on a normal factor graph with no cycles

fbd = fb, fdx1 fce = fc, fex2 fcef = fce, ff x3 fabd = fa, fbdx4 Z = fabcdef = fabd, fcef x5

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm for Acyclic Factor Graphs

Target: Calculate Z (G)

x,x′ g (x, x′)

a b d c e f

x1 x′

1

x2 x′

2

x3 x′

3

x4 x′

4

x5 x′

5

Sum-Product Algorithm on a quantum normal factor graph with no cycles

fbd = fb, fdLH(X1) fce = fc, feLH(X2) fcef = fce, ff LH(X3) fabd = fa, fbdLH(X4) Z = fabcdef = fabd, fcef LH(X5)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 12

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm: General Rules

More generally, we are applying following two rules: a i a1 a2

mi→a PSD ma1→i PSD ma2→i PSD

mi→a ←

• b∈∂i\{a}

mb→i (xi, x′

i )

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 13

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm: General Rules

More generally, we are applying following two rules: a i a1 a2

mi→a PSD ma1→i PSD ma2→i PSD

mi→a ←

• b∈∂i\{a}

mb→i (xi, x′

i )

i a i1 i2

ma→i PSD mi1→a PSD mi2→a PSD

ma→i ←

• j∈∂a\{i}

mj→a, fa

• ∂a\{i}

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 13

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Sum-Product Algorithm: General Rules

More generally, we are applying following two rules: a i a1 a2

mi→a PSD ma1→i PSD ma2→i PSD

mi→a ←

• b∈∂i\{a}

mb→i (xi, x′

i )

i a i1 i2

ma→i PSD mi1→a PSD mi2→a PSD

ma→i ←

• j∈∂a\{i}

mj→a, fa

• ∂a\{i}

with initialization at the leaf factors ma→i (xi, x′

i ) = fa (xi, x′ i )

where {i} = ∂a

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 13

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

For general QFGs with cycles

Definition 2 (General Sum-Product / Belief Propagation Algorithm for QFG) For a general QFG G = {F, V, E} with global funciton g (x, x′) =

• a∈F

fa (x∂a, x′

∂a)

• i∈V

hi (xi, x′

i ) .

(1) Update rules for belief propagation (BP) algorithm: m(t+1)

a→i

∝

• j∈∂a\{i}

m(t)

j→a, fa

• Lh(X∂a\{i})

(2) m(t+1)

i→a ∝ hi ·

• b∈∂i\{a}

m(t)

b→i

(3) The messages are said to be fixed-point messages when above equations holds without time-stamp superscripts.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 14

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

For general QFGs with cycles

Definition 2 (General Sum-Product / Belief Propagation Algorithm for QFG) For a general QFG G = {F, V, E} with global funciton g (x, x′) =

• a∈F

fa (x∂a, x′

∂a)

• i∈V

hi (xi, x′

i ) .

(1) Update rules for belief propagation (BP) algorithm: ma→i ∝

• j∈∂a\{i}

mj→a, fa

• Lh(X∂a\{i})

(2) mi→a ∝ hi ·

• b∈∂i\{a}

mb→i (3) The messages are said to be fixed-point messages when above equations holds without time-stamp superscripts.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 14

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Loop Calculus for SP / BP Algorithms

Theorem 3 (Loop Calculus [Chertkov and Chernyak, 2006, Mori, 2015a, Mori, 2015b]) At BP-fixed point, we have Z

• a∈F

fa,

• i∈V

hi

• L(XV)

= ZBethe

• 1 +
• E⊂E′

K (E)

• where the extended loop set is defined as

E′ {E ⊂ E\ {φ} : di (E) = 1 ∀i ∈ V, da (E) = 1 ∀i ∈ F} where K (E) is some function depending on E, and K (φ) = 1, and ZBethe

• a∈F

Za

• i∈V

Zi

• (i,a)∈E

Zi,a =

• a∈F

i∈∂a

m(t)

i→a, fa i∈V

• hi,

a∈∂i

m(t)

a→i

• (i,a)∈E

ma→i, mi→a .

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 15

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Loop Calculus for BP Algorithms

Interpretation Bethe Approximation is exact for acyclic QFG;

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 15

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm

Loop Calculus for BP Algorithms

Interpretation Bethe Approximation is exact for acyclic QFG; Bethe Approximation is close to the exact value for QFGs with small number

• f cycles.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 15

Problem of Calculating the Partition Sum Exploration on Variational Approach

Outline

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

End Matters

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G);

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G); II: Minimize FGibbs (b) over all possible global probability function b (x); min

b is a probability function FGibbs (b) −

• a∈F
• x

b (x) ln fa (x∂a) −

• i∈V
• x

b (x) ln hi (xi) +

• x

b (x) ln b (x) = FH + D(b p) FH.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G); II: Minimize FGibbs (b) over all possible global probability function b (x); III: Minimize FBethe

• {ba}a∈F , {bi}i∈V
• ver all valid marginal probability

functions {ba}a∈F , {bi}i∈V; FBethe

• (ba)a∈F , (bi)i∈V
• −
• a∈F
• x∂a

ba (x∂a) ln fa (x∂a) −

• i∈V
• xi

bi (xi) ln hi (xi) +

• a∈F
• x∂a

ba (x∂a) ln ba (x∂a) −

• i∈V

(di − 1)

• xi

bi (xi) ln bi (xi) =FGibbs for acyclic factor graphs.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G); II: Minimize FGibbs (b) over all possible global probability function b (x); III: Minimize FBethe

• {ba}a∈F , {bi}i∈V
• ver all valid marginal probability

functions {ba}a∈F , {bi}i∈V; IV: Study the Stationary Condition of above optimization problem, which turned

• ut to be equivalent to

min

ba,bi probability functionsFBethe

• (ba)a∈F , (bi)i∈V
• s.t.
• x∂a\{i}

ba (xa) = bi (xi) ∀ (i, a) ∈ E, ∀xi ∈ Xi

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G); II: Minimize FGibbs (b) over all possible global probability function b (x); III: Minimize FBethe

• {ba}a∈F , {bi}i∈V
• ver all valid marginal probability

functions {ba}a∈F , {bi}i∈V; IV: Study the Stationary Condition of above optimization problem, which turned

• ut to be equivalent to

ba ∝ fa ·

• i∈∂a

mi→a bi ∝ h1 ·

• a∈∂i

ma→i

• x∂a\{i}

ba (xa) = bi (xi) ∀ (i, a) ∈ E, ∀xi ∈ Xi

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Variational Approach for Classic Factor Graphs

Target: Calculate Z (G)

x

g (x), where g (x) =

a∈F

fa (xa)

i∈V

hi (xi)

I: Calculate FH − ln Z (G); II: Minimize FGibbs (b) over all possible global probability function b (x); III: Minimize FBethe

• {ba}a∈F , {bi}i∈V
• ver all valid marginal probability

functions {ba}a∈F , {bi}i∈V; IV: Study the Stationary Condition of above optimization problem, which turned

• ut to be equivalent to

ba ∝ fa ·

• i∈∂a

mi→a bi ∝ h1 ·

• a∈∂i

ma→i mi→a ←

• b∈∂i\{a}

mb→i (xi, x′

i ) ;

ma→i ←

• j∈∂a\{i}

mj→a, fa

• ∂a\{i}

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 16

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration of Variational Approach for QFGs

Definition 4 (Helmholtz free energy and Gibbs free energy for QFGs) FH − ln Z (G) FGibbs (b(x, x′)) −

• a∈F
• ˜

ba, Ln (fa)

• −
• i∈V
• ˜

bi, Ln (hi)

• + b, Ln (b)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 17

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration of Variational Approach for QFGs

Definition 4 (Helmholtz free energy and Gibbs free energy for QFGs) FH − ln Z (G) FGibbs (b(x, x′)) −

• a∈F
• ˜

ba, Ln (fa)

• −
• i∈V
• ˜

bi, Ln (hi)

• + b, Ln (b)

where the induced

• ˜

ba

• a∈F and
• ˜

bi

• a∈V are defined as

˜ ba 1, bLh(XV\{∂a}) , ˜ bi 1, bLh(XV\{i}) . (4)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 17

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration of Variational Approach for QFGs

Definition 4 (Helmholtz free energy and Gibbs free energy for QFGs) FH − ln Z (G) FGibbs (b(x, x′)) −

• a∈F
• ˜

ba, Ln (fa)

• −
• i∈V
• ˜

bi, Ln (hi)

• + b, Ln (b)

where the induced

• ˜

ba

• a∈F and
• ˜

bi

• a∈V are defined as

˜ ba 1, bLh(XV\{∂a}) , ˜ bi 1, bLh(XV\{i}) . (4) Here, Ln (·) is performed on matrix level, i.e., Ln

• UΛUH

Udiag

• {ln Λk,k}k
• UH

for any PSD matrix UΛUH, where U is unitary and Λ is a non-negative diagonal matrix.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 17

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration of Variational Approach for QFGs

Definition 4 (Helmholtz free energy and Gibbs free energy for QFGs) FH − ln Z (G) FGibbs (b(x, x′)) −

• a∈F
• ˜

ba, Ln (fa)

• −
• i∈V
• ˜

bi, Ln (hi)

• + b, Ln (b)

Von Neumann Entropy

where the induced

• ˜

ba

• a∈F and
• ˜

bi

• a∈V are defined as

˜ ba 1, bLh(XV\{∂a}) , ˜ bi 1, bLh(XV\{i}) . (4) Here, Ln (·) is performed on matrix level, i.e., Ln

• UΛUH

Udiag

• {ln Λk,k}k
• UH

for any PSD matrix UΛUH, where U is unitary and Λ is a non-negative diagonal matrix.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 17

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Theorem 5 We have following relationship between Gibbs free energy and Helmholtz free energy FGibbs(b(x, x′)) = FH + D(b p) (5) where D(b p) of two normalized PSD operator is defined as D(b p) b, Ln (b)Lh − b, Ln (p)Lh (6) and the quantum probability p is the normalized global function, i.e., p (x, x′) =

1 Z(G)g (x, x′).

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 18

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Theorem 5 We have following relationship between Gibbs free energy and Helmholtz free energy FGibbs(b(x, x′)) = FH + D(b p) (5) where D(b p) of two normalized PSD operator is defined as D(b p) b, Ln (b)Lh − b, Ln (p)Lh (6) and the quantum probability p is the normalized global function, i.e., p (x, x′) =

1 Z(G)g (x, x′).

Lemma 6 (Non-negativity of Von Neumann Divergence) D(b p) 0 “ = ” ⇐ ⇒ b = q (7)

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 18

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Definition 7 (Approximate FGibbs) For above QGF, the Bethe free energy w.r.t. marginal beliefs (ba)a∈F , (bi)i∈V is defined as FBethe

• (ba)a∈F , (bi)i∈V
• −
• a∈F

ba, Ln (fa) −

• i∈V

bi, Ln (hi) +

• a∈F

ba, Ln (ba) −

• i∈V

(di − 1) bi, Ln (bi) . Here, {ba}a∈F and {bi}i∈V are some given normalized PSD operators on {X∂a}a∈F and {Xi}i∈V, respectively.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 19

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Definition 7 (Approximate FGibbs) For above QGF, the Bethe free energy w.r.t. marginal beliefs (ba)a∈F , (bi)i∈V is defined as FBethe

• (ba)a∈F , (bi)i∈V
• −
• a∈F

ba, Ln (fa) −

• i∈V

bi, Ln (hi) +

• a∈F

ba, Ln (ba) −

• i∈V

(di − 1) bi, Ln (bi) . Here, {ba}a∈F and {bi}i∈V are some given normalized PSD operators on {X∂a}a∈F and {Xi}i∈V, respectively. Question: Does this “approximation” make sense?

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 19

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Definition 7 (Approximate FGibbs) For above QGF, the Bethe free energy w.r.t. marginal beliefs (ba)a∈F , (bi)i∈V is defined as FBethe

• (ba)a∈F , (bi)i∈V
• −
• a∈F

ba, Ln (fa) −

• i∈V

bi, Ln (hi) +

• a∈F

ba, Ln (ba) −

• i∈V

(di − 1) bi, Ln (bi) . Here, {ba}a∈F and {bi}i∈V are some given normalized PSD operators on {X∂a}a∈F and {Xi}i∈V, respectively. Question: Does this “approximation” make sense? Given arbitrary compatible (ba)a∈F , (bi)i∈V, can we always find a corresponding global quantum belief matrix b?

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 19

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Lemma 8 Given an acyclic QFG, for any marginal quantum belief {ba}a∈F and {bi}i∈V with compatibility constrain bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E (8) ∃b ∈ LH (X), normalized with ba 1, bLh(XV\{∂a}) , bi 1, bLh(XV\{i}) .

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 20

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Lemma 8 Given an acyclic QFG, for any marginal quantum belief {ba}a∈F and {bi}i∈V with compatibility constrain bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E (8) ∃b ∈ LH (X), normalized with ba 1, bLh(XV\{∂a}) , bi 1, bLh(XV\{i}) . Conjecture 9 The operator b above is positive semi-definite.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 20

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Lemma 8 Given an acyclic QFG, for any marginal quantum belief {ba}a∈F and {bi}i∈V with compatibility constrain bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E (8) ∃b ∈ LH (X), normalized with ba 1, bLh(XV\{∂a}) , bi 1, bLh(XV\{i}) . Conjecture 9 The operator b above is positive semi-definite. In such case, min

b FGibbs(b) =

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 20

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Lemma 8 Given an acyclic QFG, for any marginal quantum belief {ba}a∈F and {bi}i∈V with compatibility constrain bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E (8) ∃b ∈ LH (X), normalized with ba 1, bLh(XV\{∂a}) , bi 1, bLh(XV\{i}) . Conjecture 9 The operator b above is positive semi-definite. In such case, min

b FGibbs(b) =

min

{ba}a∈F,{bi}i∈V

FBethe

• (ba)a∈F , (bi)i∈V
• s.t. Equation (8) holds.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 20

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Conjecture 10 The stationary condition of optimization problem min

{ba}a∈F,{bi}i∈V

FBethe

• (ba)a∈F , (bi)i∈V
• s.t.bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 21

Problem of Calculating the Partition Sum Exploration on Variational Approach

Exploration on Variational Approach for QFGs

Conjecture 10 The stationary condition of optimization problem min

{ba}a∈F,{bi}i∈V

FBethe

• (ba)a∈F , (bi)i∈V
• s.t.bi (xi, x′

i ) =

• x∂a\{i},x′

∂a\{i}

ba (x∂a, x′

∂a)

∀ (i, a) ∈ E is equivalent to ba (x∂a, x′

∂a) 1

Za fa (x∂a, x′

∂a)

• i∈∂a

mi→a (xi, x′

i ) ,

bi (xi, x′

i ) 1

Zi hi (xi, x′

i )

• a∈∂i

ma→i (xi, x′

i ) .

for some fixed-point messages.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 21

End Matters

Outlooks

Continue exploration on Variational Approach;

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 22

End Matters

Outlooks

Continue exploration on Variational Approach; Details in Loop Calculus: Any bound for K (E)?

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 22

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Outlooks

Continue exploration on Variational Approach; Details in Loop Calculus: Any bound for K (E)? Looking for practical class of factorizations where this model can be applied.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 22

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References

Chertkov, M. and Chernyak, V. Y. (2006). Loop calculus in statistical physics and information science. Physical Review E, 73(6):065102. Loeliger, H.-A. and Vontobel, P. O. (2012). A factor-graph representation of probabilities in quantum mechanics. In Proc. IEEE Int. Symp. Inf. Theory, pages 656–660. Loeliger, H.-A. and Vontobel, P. O. (2015). Factor graphs for quantum probabilities. Mori, R. (2015a). Holographic transformation, belief propagation and loop calculus for quantum information science. In Proc. IEEE Int. Symp. Inf. Theory. Mori, R. (2015b). Loop calculus for nonbinary alphabets using concepts from information geometry. IEEE Trans. Inf. Theory, 61(4):1887–1904.

Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 23

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Q&A

1

Classical Factor Graphs Modeling “Closing-the-box” Operation

2

Quantum Factor Graphs A Motivating Example Quantum Factor Graph Construction of a QNFG Several Examples

3

Problem of Calculating the Partition Sum Sum-Product / Belief Propagation Algorithm Exploration on Variational Approach

4

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Michael X. CAO (IE@CUHK) Quantum Factor Graph September 25, 2015 24