Quantum Factor Graphs: Closing-the-Box Operation and Variational - - PowerPoint PPT Presentation

Quantum Factor Graphs: Closing-the-Box Operation and Variational Approaches

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

Department of Information Engineering, CUHK

August 29, 2016

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016

Overview

Marginal problems

• x
• a∈F

fa(x∂a) where fa is a non-negative function on

×

i∈∂a Xi.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 1

Overview

Marginal problems

• x
• a∈F

fa(x∂a) where fa is a non-negative function on

×

i∈∂a Xi.

Sum-Product Algorithm Closing-the-box Approach; Variational Approach (Bethe Approximation).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 1

Overview

Marginal problems

• x
• a∈F

fa(x∂a) where fa is a non-negative function on

×

i∈∂a Xi.

Sum-Product Algorithm Closing-the-box Approach; Variational Approach (Bethe Approximation). Marginal problems in Quantum Setup Tr

• exp
• a∈F

log ρa

• where ρa is a PSD operator on
• i∈∂a Hi.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 1

Overview

Marginal problems

• x
• a∈F

fa(x∂a) where fa is a non-negative function on

×

i∈∂a Xi.

Sum-Product Algorithm Closing-the-box Approach; Variational Approach (Bethe Approximation). Marginal problems in Quantum Setup Tr

• exp
• a∈F

log ρa

• where ρa is a PSD operator on
• i∈∂a Hi.

Quantum Sum-Product Algorithm

?

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 1

Factor Graphs/Preliminaries

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1 X2

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1 X2

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1 X2 X3

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1 X2 X3

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Example 1 Consider the factorization of g(x1, x2, x3, x4) given as g(x1, x2, x3, x4) = fA(x1, x3) · fB(x1, x2, x3) · fC(x3, x4). (1) fA fB fC

X1 X2 X3 X4

Factor Graph for (1).

fA fB fC X2 X1 X2 X3 X4

Normal Factor Graph for (1).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

F We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

F V We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

F V We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

F V We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs, or classical factor graph (CFG), describe factorizations. Definition 1 In general, we associate the factorization g(x)

• a∈F

fa(xa) to a factor graph with variable node set V, function node set F, and edge set E ⊆ V × F given by E = {(i, a) ∈ V × F : i ∈ ∂a} . In Example 1, fA fB fC

X1 X2 X3 X4

F V We call g the global function, f the local function/factors.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs are popular in describing (large-scale) probability systems. ✶

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs are popular in describing (large-scale) probability systems. Example 2 (LDPC Code described by CFG) + + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn ✶

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs are popular in describing (large-scale) probability systems. Example 2 (LDPC Code described by CFG) + + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs are popular in describing (large-scale) probability systems. Example 2 (LDPC Code described by CFG) + + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    . A problem of interest: Calculate the marginal distribution

• f Xi given fixed {yi}n

i=1.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

Factor Graphs/Preliminaries

Factor Graphs and Normal Factor Graphs

Factor graphs are popular in describing (large-scale) probability systems. Example 2 (LDPC Code described by CFG) + + . . . + x3 x2 x1 . . . xn pY1|X1 pY2|X2 pY3|X3 . . . pYn|Xn y1 y2 y3 . . . yn xi, yi ∈ F2 ∀i ∈ {1, · · · , n} ; f+ (x) ✶   

• i incoming

xi = 0    . A problem of interest: Calculate the marginal distribution

• f Xi given fixed {yi}n

i=1.

gy(x) =

• i

pYi=yi|Xi(xi) ·

• k

fk(x) ∝ pY=y|X(x)

• xj, j=i

gy(x) ∝ pY=y|Xi(xi) Symbol-wise ML decoding

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

Factor Graphs/Preliminaries

Partition Function/Sum

Definition 3 (Partition Function/Sum) In many applications, we are interested in calculating summations like (or similar to) Z

• x

g(x) =

• x
• a∈F

fa(x∂a), which is defined to be the partition function/sum of a CFG.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

Factor Graphs/Preliminaries

Partition Function/Sum

Definition 3 (Partition Function/Sum) In many applications, we are interested in calculating summations like (or similar to) Z

• x

g(x) =

• x
• a∈F

fa(x∂a), which is defined to be the partition function/sum of a CFG. Example 1 (continue) fA fB fC

X1 X2 X3 X4

g = fA(x1) · fB(x1, x2, x3) · fC(x3, x4). Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

Factor Graphs/Preliminaries

Partition Function/Sum

Definition 3 (Partition Function/Sum) In many applications, we are interested in calculating summations like (or similar to) Z

• x

g(x) =

• x
• a∈F

fa(x∂a), which is defined to be the partition function/sum of a CFG. Example 1 (continue) fA fB fC

X1 X2 X3 X4

g = fA(x1) · fB(x1, x2, x3) · fC(x3, x4). Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4). In general, calculation of Z is NP hard.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA fB fC

X1 X2 X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA fB fC

X1 X2 X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fB fC

X1 X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fB fC

X1 X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fAB fC

X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fAB fC

X3 X4

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• ·
• x4

fC(x3, x4)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fAB ˆ fC

X3

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• · ˆ

fC(x3)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA ˆ fAB ˆ fC

X3

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• · ˆ

fC(x3)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Example 1 Sum over local variables first... fA Z

X3

Z =

• x1,x2,x3,x4

fA(x1) · fB(x1, x2, x3) · fC(x3, x4) =

• x3
• x1
• fA(x1) ·
• x2

fB(x1, x2, x3)

• ·
• x4

fC(x3, x4) =

• x3

x1

• fA(x1) · ˆ

fB(x1, x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• ·
• x4

fC(x3, x4) =

• x3
• ˆ

fAB(x3)

• · ˆ

fC(x3) = Z

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Closing-the-box in general In general, “closing-the-box” means to replace the boxes with the result of the summing over the interior variable(s). x y f1 w1 z1 f2 w2 z2 f3 w3 z3

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 6

Factor Graphs/Preliminaries

Partition Sum and the Closing-the-box Operations

Closing-the-box in general In general, “closing-the-box” means to replace the boxes with the result of the summing over the interior variable(s). x y f1 w1 z1 f2 w2 z2 f3 w3 z3

• x,y

f1 · f2 · f3 w1 z1 w2 z2 w3 z3

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 6

Factor Graphs/Preliminaries

Sum-Product Algorithm on a tree

Example 1

fB fA fC

X1 X2 X3 X4

Closing the boxes from the inner ones to

• uter ones will yield the partition sum Z.

Distributive Law on R

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 7

Factor Graphs/Preliminaries

Sum-Product Algorithm on a tree

Example 1

fB fA fC

X1 X2 X3 X4

Closing the boxes from the inner ones to

• uter ones will yield the partition sum Z.

Distributive Law on R Sum-Product Algorithm for Trees Require: Acyclic factor graph G = (F, V, E); root r ∈ V; height of the tree h 0. Ensure: Partition sum Z

1: for d = h − 1, · · · , 0 do 2:

for all i ∈ V d-step reachablea from r do

3:

Let f (i) be the parent factorb of i;

4:

f (i) ←

xi

• a∈∂i fa(xi);

5:

end for

6: end for 7: Z ← f (r).

ai.e., there exists a path connecting r and i passing

through d factors.

bi.e. the unique factor node that is both on the path

from r to i and also adjacent to i.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 7

Factor Graphs/Preliminaries

Sum-Product Algorithm on a tree

Example 1

fB fA fC

X1 X2 X3 X4

Closing the boxes from the inner ones to

• uter ones will yield the partition sum Z.

Distributive Law on R Message-Passing Algorithm Sum-Product Algorithm for Trees Require: Acyclic factor graph G = (F, V, E); root r ∈ V; height of the tree h 0. Ensure: Partition sum Z

1: for d = h − 1, · · · , 0 do 2:

for all i ∈ V d-step reachablea from r do

3:

Let f (i) be the parent factorb of i;

4:

f (i) ←

xi

• a∈∂i fa(xi);

5:

end for

6: end for 7: Z ← f (r).

ai.e., there exists a path connecting r and i passing

through d factors.

bi.e. the unique factor node that is both on the path

from r to i and also adjacent to i.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 7

Factor Graphs/Preliminaries

Sum-Product Algorithm on a tree

Example 1

fB fA fC

X1 X2 X3 X4

← ← ← →

Closing the boxes from the inner ones to

• uter ones will yield the partition sum Z.

Distributive Law on R Message-Passing Algorithm Sum-Product Algorithm for Trees Require: Acyclic factor graph G = (F, V, E); root r ∈ V; height of the tree h 0. Ensure: Partition sum Z

1: for d = h − 1, · · · , 0 do 2:

for all i ∈ V d-step reachablea from r do

3:

Let f (i) be the parent factorb of i;

4:

f (i) ←

xi

• a∈∂i fa(xi);

5:

end for

6: end for 7: Z ← f (r).

ai.e., there exists a path connecting r and i passing

through d factors.

bi.e. the unique factor node that is both on the path

from r to i and also adjacent to i.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 7

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ←

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ←

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ←

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ←

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ∝

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ∝

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ∝

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ∝

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ∝

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ∝

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until convergence

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm as a Message Passing Algorithm

Sum-Product Algorithm Require: Factor graph G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← ✶;

3:

ma→i ← ✶;

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a(xi) ∝

c∈∂i\a ma→i(xi);

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i(xi) ∝

x∂a\i fa(x∂a) · j∈∂a\i mj→a(xj);

11:

end for

12: until convergence

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

Factor Graphs/Preliminaries

Sum-Product Algorithm and the Variational Approach

In acyclic case, it will always converge. bi(xi)

a∈∂i ma→i(xi) ∝ xV\i g(x);

ba(x∂a) fa(x∂a) ·

i∈∂a mi→a(xj) ∝ xV\∂a g(x).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 9

Factor Graphs/Preliminaries

Sum-Product Algorithm and the Variational Approach

In acyclic case, it will always converge. bi(xi)

a∈∂i ma→i(xi) ∝ xV\i g(x);

ba(x∂a) fa(x∂a) ·

i∈∂a mi→a(xj) ∝ xV\∂a g(x).

In general case, if it converges, then: [Yedidia et al., 2005] The above {bi}i∈V and {ba}a∈F correspond to the interior stationary points of the constrained Bethe approximation problem:

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 9

Factor Graphs/Preliminaries

Sum-Product Algorithm and the Variational Approach

In acyclic case, it will always converge. bi(xi)

a∈∂i ma→i(xi) ∝ xV\i g(x);

ba(x∂a) fa(x∂a) ·

i∈∂a mi→a(xj) ∝ xV\∂a g(x).

In general case, if it converges, then: [Yedidia et al., 2005] The above {bi}i∈V and {ba}a∈F correspond to the interior stationary points of the constrained Bethe approximation problem: min FBethe

• (ba)a∈F , (bi)i∈V
• −
• a∈F
• x∂a

ba(x∂a) log fa(x∂a) −

• a∈F

H(ba) +

• i∈V

(di − 1) · H(bi) s.t. ba probability on x∂a, bi probability on xi, ∀a ∈ F, ∀i ∈ V bi(xi) =

• x∂a\i

ba(x∂a) ∀xi, ∀ (i, a) ∈ E

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 9

Quantum Factor Graphs (QFGs)

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 10

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

Definition 4 ([Leifer and Poulin, 2008]) A quantum Factor graph (QFG) (V, F, E) with local factors {ρa} describes the “factorization” ρ

• a∈F

ρa = exp

• a∈F

log(ρa)

• ,

(1) where, for each a ∈ F, positive definite operator ρa is an operator on

i∈∂a Hi.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 10

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

Definition 4 ([Leifer and Poulin, 2008]) A quantum Factor graph (QFG) (V, F, E) with local factors {ρa} describes the “factorization” ρ

• a∈F

ρa = exp

• a∈F

log(ρa)

• ,

(1) where, for each a ∈ F, positive definite operator ρa is an operator on

i∈∂a Hi.

Example 5 ρA ρB ρC

H1 H2

A QFG describing ρ = ρA ⊙ ρB ⊙ ρC.

ρA ∈ L+ (H1) ρB ∈ L+ (H1 ⊗ H2) ρC ∈ L+ (H2)

Here, L+ (H) stands for the set of all positive semi-definite operators on the Hilbert space H.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 10

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have ρA ⊙ ρB = lim

n→∞

• ρ

1 n

Aρ

1 n

B

n . (3)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have ρA ⊙ ρB = lim

n→∞

• ρ

1 n

Aρ

1 n

B

n . (3) Equation (3) can be used to generalize the ⊙ product to PSD operators.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have ρA ⊙ ρB = lim

n→∞

• ρ

1 n

Aρ

1 n

B

n . (3) Equation (3) can be used to generalize the ⊙ product to PSD operators. Properties of ⊙ Associativity: (ρA⊙ρB)⊙ρC = ρA⊙(ρB⊙ρC); Commutativity: ρA ⊙ ρB = ρB ⊙ ρA; Closeness: ρA ⊙ ρB is positive (semi) definite if ρA, ρB are positive (semi) definite.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have ρA ⊙ ρB = lim

n→∞

• ρ

1 n

Aρ

1 n

B

n . (3) Equation (3) can be used to generalize the ⊙ product to PSD operators. Properties of ⊙ Associativity: (ρA⊙ρB)⊙ρC = ρA⊙(ρB⊙ρC); Commutativity: ρA ⊙ ρB = ρB ⊙ ρA; Closeness: ρA ⊙ ρB is positive (semi) definite if ρA, ρB are positive (semi) definite.

L+ (H) , ⊙ (or L++ (H) , ⊙) is an Abelian group.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs (QFGs)

For ρA, ρB ∈ L++ (H), define [Warmuth, 2005] ρA ⊙ ρB exp

• log(ρA) + log(ρB)
• ,

(2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have ρA ⊙ ρB = lim

n→∞

• ρ

1 n

Aρ

1 n

B

n . (3) Equation (3) can be used to generalize the ⊙ product to PSD operators. Properties of ⊙ Associativity: (ρA⊙ρB)⊙ρC = ρA⊙(ρB⊙ρC); Commutativity: ρA ⊙ ρB = ρB ⊙ ρA; Closeness: ρA ⊙ ρB is positive (semi) definite if ρA, ρB are positive (semi) definite.

L+ (H) , ⊙ (or L++ (H) , ⊙) is an Abelian group.

ρA ∈ L+ (H1) ρB ∈ L+ (H2)

• H1 = H2

H1 → H3 ← H2 ρA ⊙ ρB ˜ ρA ⊙ ˜ ρB.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs

Example 5: continue

ρA ρB ρC

H1 = C2 H2 = C2

A QFG describing ρ = ρA ⊙ ρB ⊙ ρC.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs

Example 5: continue

ρA ρB ρC

H1 = C2 H2 = C2

A QFG describing ρ = ρA ⊙ ρB ⊙ ρC.

Suppose H1 = H2 = C2, and ρA =

• +3

−1 −1 +3

• , ρB =

    1 1     , ρC =

• +3

−1 −1 +3

• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

Quantum Factor Graphs (QFGs)

Quantum Factor Graphs

Example 5: continue

ρA ρB ρC

H1 = C2 H2 = C2

A QFG describing ρ = ρA ⊙ ρB ⊙ ρC.

Suppose H1 = H2 = C2, and ρA =

• +3

−1 −1 +3

• , ρB =

    1 1     , ρC =

• +3

−1 −1 +3

• .

We have, ρ = ρA ⊙ ρB ⊙ ρC =     9 −3 −3 1 −3 9 1 −3 −3 1 9 −3 1 −3 −3 9     .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

Quantum Factor Graphs (QFGs)

Quantum Partition-Sum Problem

Definition 6 (Partition Function/Sum) In a number of applications, we are interested in calculating Z Tr (ρ) = Tr

• a∈F

ρa

• = Tr
• exp
• a∈F

log(ρa)

• ,

which is defined to be the partition function/sum of a QFG.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 13

Quantum Factor Graphs (QFGs)

Quantum Partition-Sum Problem

Definition 6 (Partition Function/Sum) In a number of applications, we are interested in calculating Z Tr (ρ) = Tr

• a∈F

ρa

• = Tr
• exp
• a∈F

log(ρa)

• ,

which is defined to be the partition function/sum of a QFG. In general calculation of the partition function/sum of a QFG is NP hard.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 13

Closing-the-box Operations on QFGs

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in CFG fA fB fC

X1 X2

Z =

x1,x2 fA(x1)fB(x1, x2)fC(x2) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in CFG fA fB fC

X1 X2

Z =

x1,x2 fA(x1)fB(x1, x2)fC(x2) = x1 fA(x1) x2 fB(x1, x2)fC(x2) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in CFG = ⇒ Applying distributive law fA fB fC

X1 X2

Z =

x1,x2 fA(x1)fB(x1, x2)fC(x2) = x1 fA(x1) x2 fB(x1, x2)fC(x2) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in QFG ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in QFG ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC)

?

= Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in QFG = ⇒ Distributive law over (partial) trace ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC)

?

= Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in QFG = ⇒ Distributive law over (partial) trace ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC)

?

= Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• However, in general,

Tr(ρA ⊙ ρB ⊙ ρC) = Tr1

• Tr2(ρA ⊙ ρB ⊙ ρC)
• = Tr1
• ρA ⊙ Tr2(ρB ⊙ ρC)
• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Closing-the-box operations and partial trace functions

Closing the box in QFG = ⇒ Distributive law over (partial) trace ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC)

?

= Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• However, in general,

Tr(ρA ⊙ ρB ⊙ ρC) = Tr1

• Tr2(ρA ⊙ ρB ⊙ ρC)
• = Tr1
• ρA ⊙ Tr2(ρB ⊙ ρC)
• .

Example 7 Let H1 = H2 C2. Suppose ρA 1

2 ·

+1 −1

−1 +1

• , ρB ⊙ ρC diag (0, 1, 1, 0) . In

this case, Tr(ρA ⊙ ρB ⊙ ρC) = 0 and Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• = 1.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

Closing-the-box Operations on QFGs

Cases when factors are close to identity matrix

Oftentimes, we can still have an approximate closing-the-box rule. Lemma 8 We have bounds S

• κ(ρA)

−1 Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• S
• κ(ρA)
• .

Given ρA ∈ L++ (H1); ρB ∈ L++ (H1 ⊗ H2); ρC ∈ L++ (H2); κ(·) 1 is the condition number function; S(·) is the Specht ratio function defined as S(r) (r − 1) · r

1 r−1

e · log r .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

Closing-the-box Operations on QFGs

Cases when factors are close to identity matrix

Oftentimes, we can still have an approximate closing-the-box rule. Lemma 8 We have bounds S

• κ(ρA)

−1 Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• S
• κ(ρA)
• .

Given ρA ∈ L++ (H1); ρB ∈ L++ (H1 ⊗ H2); ρC ∈ L++ (H2); κ(·) 1 is the condition number function; S(·) is the Specht ratio function defined as S(r) (r − 1) · r

1 r−1

e · log r . The proof utilizes the Golden–Thompson inequality [Bourin and Seo, 2007].

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

Closing-the-box Operations on QFGs

Cases when factors are close to identity matrix

Oftentimes, we can still have an approximate closing-the-box rule. Lemma 8 We have bounds S

• κ(ρA)

−1 Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• S
• κ(ρA)
• .

Given

20 40 60 80 100 2 4 6 8 condition number r Specht Ratio Function

The proof utilizes the Golden–Thompson inequality [Bourin and Seo, 2007]. Considering ρA ≈ I,

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

Closing-the-box Operations on QFGs

Cases when factors are close to identity matrix

Oftentimes, we can still have an approximate closing-the-box rule. Lemma 8 We have bounds S

• κ(ρA)

−1 Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• S
• κ(ρA)
• .

Given

20 40 60 80 100 2 4 6 8 condition number r Specht Ratio Function

The proof utilizes the Golden–Thompson inequality [Bourin and Seo, 2007]. Considering ρA ≈ I, we expect to have Tr(ρA ⊙ ρB ⊙ ρC) ≈ Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρA is “close” to identity matrix I

ρA ρB ρC

H1 H2

ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρA is “close” to identity matrix I

ρA ρB ρC

H1 H2

≈

ρA ρB ρC

H1 H2

Z = Tr(ρA ⊙ ρB ⊙ ρC) Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Lemma (Type-1 Approximation)

Given X ∈ LH (H1), and Y ∈ LH (H1 ⊗ H2), for t close to 0, we have Tr2

• (I + tX) ⊙ (I + tY )
• = (I + tX) ⊙ Tr2(I + tY ) + O(t3).

(4) Theorem (Type-1 Approximation) Following the same setup, we have Tr

• (I + tX) ⊙ (I + tY )
• = Tr1
• (I + tX) ⊙ Tr2(I + tY )
• + O(t4).

(5)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Linear” close to I: i.e., ρ1 = I + tX and ρ1,2 = I + tY .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Linear” close to I: i.e., ρ1 = I + tX and ρ1,2 = I + tY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• =

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Linear” close to I: i.e., ρ1 = I + tX and ρ1,2 = I + tY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = 1 + t · Tr( ˜ X + Y ) + t2 · Tr( ˜ XY + Y ˜ X) 2 + t3 · 0 + t4 · Tr

• ˜

XY ˜ XY − ˜ X 2Y 2 12 + · · · Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• = 1 + t·Tr1(X + Tr2(Y )) + t2· Tr1
• XTr2(Y ) + Tr2(Y )X
• 2

+ t4· Tr1

• XTr2(Y )XTr2(Y ) − X 2Tr2(Y )2

12 + · · ·

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-1 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Linear” close to I: i.e., ρ1 = I + tX and ρ1,2 = I + tY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = 1 + t · Tr( ˜ X + Y ) + t2 · Tr( ˜ XY + Y ˜ X) 2 + t3 · 0 + t4 · Tr

• ˜

XY ˜ XY − ˜ X 2Y 2 12 + · · · Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• = 1 + t·Tr1(X + Tr2(Y )) + t2· Tr1
• XTr2(Y ) + Tr2(Y )X
• 2

+ t4· Tr1

• XTr2(Y )XTr2(Y ) − X 2Tr2(Y )2

12 + · · ·

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Michael X. CAO (IE@CUHK)

Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• Lemma (Type-2 Approximation)

Given X ∈ LH (H1), and Y ∈ LH (H1 ⊗ H2), for t close to 0, we have Tr2

• etX ⊙ etY

= etX ⊙ Tr2(etY ) + O(t3). (4) Theorem (Type-2 Approximation) Following the same setup, we have Tr

• etX ⊙ etY

= Tr1

• etX ⊙ Tr2(etY )
• + O(t4).

(5)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Exponential” close to I: i.e., ρ1 = etX and ρ1,2 = etY .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Exponential” close to I: i.e., ρ1 = etX and ρ1,2 = etY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• =

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Exponential” close to I: i.e., ρ1 = etX and ρ1,2 = etY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = 1 + t · Tr( ˜ X + Y ) + t2 2! · Tr( ˜ X + Y )2 + t3 3! · Tr( ˜ X + Y )3 + t4 4! · Tr( ˜ X + Y )4 + · · · Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• = 1 + t · Tr1
• X + Tr2(Y )
• + t2

2 · Tr1

• X 2 + XTr2(Y ) + Tr2(Y )X + Tr2(Y 2)
• + t3

6 · Tr1

• X 3 + 3 · X 2 · Tr2(Y ) + 3 · X · Tr2(Y 2) + Tr2(Y 3)
• + · · ·

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

Type-2 Approximation

when ρ1 or ρ1,2 is “close” to identity matrix I

ρ1 ρ1,2

H1 H2

≈

ρ1 ρ1,2

H1 H2

Z = Tr(ρ1 ⊙ ρ1,2) Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• “Exponential” close to I: i.e., ρ1 = etX and ρ1,2 = etY .

Taylor Series Expansion:

Tr(ρ1 ⊙ ρ1,2) = 1 + t · Tr( ˜ X + Y ) + t2 2! · Tr( ˜ X + Y )2 + t3 3! · Tr( ˜ X + Y )3 + t4 4! · Tr( ˜ X + Y )4 + · · · Tr1

• ρ1 ⊙ Tr2(ρ1,2)
• = 1 + t · Tr1
• X + Tr2(Y )
• + t2

2 · Tr1

• X 2 + XTr2(Y ) + Tr2(Y )X + Tr2(Y 2)
• + t3

6 · Tr1

• X 3 + 3 · X 2 · Tr2(Y ) + 3 · X · Tr2(Y 2) + Tr2(Y 3)
• + · · ·

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

Closing-the-box Operations on QFGs

t-close Approximation

Definition (t-close to I) A set of operators {ρk}k are said to be t-close to I, if ρk = I + tχk or ρk = etχk ∀k, for some Hermitian operators {χk}k, and t close to 0.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 18

Closing-the-box Operations on QFGs

t-close Approximation

Definition (t-close to I) A set of operators {ρk}k are said to be t-close to I, if ρk = I + tχk or ρk = etχk ∀k, for some Hermitian operators {χk}k, and t close to 0. Lemma 9 (t-close Approximation) Given ρ1 ∈ L+ (H1), and ρ1,2 ∈ LH (H1 ⊗ H2), t close to I, we have Tr2

• ρ1 ⊙ ρ1,2
• = ρ1 ⊙ Tr2(ρ1,2) + O(t3).

(4) Theorem 10 (t-close Approximation) Following the same setup, we have Tr

• ρ1 ⊙ ρ1,2
• = Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• + O(t4).

(5)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 18

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere. Diagonal matrix Λ = diag(λ1, · · · , λn) with {λk}k to be i.i.d.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

We are interested in a numerical comparison between Tr1

• Tr2(ρ1 ⊙ ρ1,2)
• and Tr1
• ρ1 ⊙ Tr2(ρ1,2)
• .

for random ρ1 ∈ L+ C2 and ρ1,2 ∈ L+ C4 . ρ = UHΛU Unitary matrix U = [u1, · · · , un] with u1 ⊥ · · · ⊥ un uniformly distributed on Cn unit sphere. Diagonal matrix Λ = diag(λ1, · · · , λn) with {λk}k to be i.i.d. Consider the statistics of the relative error: η

• Tr1
• ρA ⊙ Tr2(ρB)
• − Tr1
• Tr2(ρA ⊙ ρB)
• Tr1
• Tr2(ρA ⊙ ρB)
• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 19

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1 0.12 0.14 0.16 0.18 0.2 100 200 300 400 500 Relative Error η Frequency Density =

Frequency Interval Length

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 0.25)

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 0.5)

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 1) Uniformly distributed Eigenvalues (a = 0, b = 1)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 20

Closing-the-box Operations on QFGs

Numerical Result of Closing-the-Box Approximation

2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1 0.12 0.14 0.16 0.18 0.2 10−4 10−3 10−2 10−1 100 101 102 103 Relative Error η Frequency Density =

Frequency Interval Length

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 0.25)

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 0.5)

• N
• µ, σ2

distributed Eigenvalues (µ = 1, σ = 1) Uniformly distributed Eigenvalues (a = 0, b = 1)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 20

Closing-the-box Operations on QFGs

Closing-the-box on a Chain QFG

ρ1 ρ2 ρ3 ρN−1 ρN · · ·

H1 H2 H3 HN−2 HN−1

A Chain QFG. (N 3)

Corollary 11 (Closing-the-box on a Chain QFG) Consider the chain QFG above where {ρk}k are t-close to I. Then, Tr {ρ1 ⊙ ρ2 ⊙ · · · ⊙ ρN−1 ⊙ ρN} (6)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 21

Closing-the-box Operations on QFGs

Closing-the-box on a Chain QFG

ρ1 ρ2 ρ3 ρN−1 ρN · · ·

H1 H2 H3 HN−2 HN−1

A Chain QFG. (N 3)

Corollary 11 (Closing-the-box on a Chain QFG) Consider the chain QFG above where {ρk}k are t-close to I. Then, Tr {ρ1 ⊙ ρ2 ⊙ · · · ⊙ ρN−1 ⊙ ρN} = Tr2 {Tr1(ρ1 ⊙ ρ2) ⊙ · · · ⊙ ρN−1 ⊙ ρN} + O(t4) (6)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 21

Closing-the-box Operations on QFGs

Closing-the-box on a Chain QFG

ρ1 ρ2 ρ3 ρN−1 ρN · · ·

H1 H2 H3 HN−2 HN−1

A Chain QFG. (N 3)

Corollary 11 (Closing-the-box on a Chain QFG) Consider the chain QFG above where {ρk}k are t-close to I. Then, Tr {ρ1 ⊙ ρ2 ⊙ · · · ⊙ ρN−1 ⊙ ρN} = Tr2 {Tr1(ρ1 ⊙ ρ2) ⊙ · · · ⊙ ρN−1 ⊙ ρN} + O(t4) = Tr3 {Tr2[Tr1(ρ1 ⊙ ρ2) ⊙ ρ3] ⊙ · · · ⊙ ρN−1 ⊙ ρN] + O(t4) (6)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 21

Closing-the-box Operations on QFGs

Closing-the-box on a Chain QFG

ρ1 ρ2 ρ3 ρN−1 ρN · · ·

H1 H2 H3 HN−2 HN−1

A Chain QFG. (N 3)

Corollary 11 (Closing-the-box on a Chain QFG) Consider the chain QFG above where {ρk}k are t-close to I. Then, Tr {ρ1 ⊙ ρ2 ⊙ · · · ⊙ ρN−1 ⊙ ρN} = Tr2 {Tr1(ρ1 ⊙ ρ2) ⊙ · · · ⊙ ρN−1 ⊙ ρN} + O(t4) = Tr3 {Tr2[Tr1(ρ1 ⊙ ρ2) ⊙ ρ3] ⊙ · · · ⊙ ρN−1 ⊙ ρN] + O(t4) = TrN−1 {TrN−2 [TrN−3 (· · · Tr1 (ρ1 ⊙ ρ2) · · · ) ⊙ ρN−1] ⊙ ρN} + O(t4). (6)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 21

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Quantum Sum-Product Algorithm for Trees Require: Acyclic QFG G = (F, V, E); root r ∈ V; height of the tree h 0; local

• perators {ρa}a t-close to I.

Ensure: Approximate Partition sum Z

1: for d = h − 1, · · · , 0 do 2:

for all i ∈ V d-step reachable from r do

3:

Let ρ(i) be the parent factor of i;

4:

ρ(i) ← Tri

• a∈∂i fa(xi)
• ;

5:

end for

6: end for 7: ˜

Z ← ρ(r).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Closing-the-box on a Tree QFG

Applying the same trick on a tree QFG

Quantum Sum-Product Algorithm for Trees Require: Acyclic QFG G = (F, V, E); root r ∈ V; height of the tree h 0; local

• perators {ρa}a t-close to I.

Ensure: Approximate Partition sum Z

1: for d = h − 1, · · · , 0 do 2:

for all i ∈ V d-step reachable from r do

3:

Let ρ(i) be the parent factor of i;

4:

ρ(i) ← Tri

• a∈∂i fa(xi)
• ;

5:

end for

6: end for 7: ˜

Z ← ρ(r).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 22

Closing-the-box Operations on QFGs

Quantum Sum-Product Algorithm (QSPA)

Quantum Sum-Product Algorithm [Leifer and Poulin, 2008] Require: QFG G = (F, V, E); Ensure: ???

1: for all (i, a) ∈ E do 2:

mi→a ← I ∈ L (Hi);

3:

ma→i ← I ∈ L (Hi);

4: end for 5: repeat 6:

for all (i, a) ∈ E do

7:

mi→a ∝

c∈∂i\a ma→i;

8:

end for

9:

for all (i, a) ∈ E do

10:

ma→i ∝ Tr∂a\i

• ρa ⊙

j∈∂a\i mj→a

• ;

11:

end for

12: until converge

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 23

Closing-the-box Operations on QFGs

Decomposition of density operators on acyclic QFGs

Lemma 12 Consider a QFG with no cycles.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 24

Closing-the-box Operations on QFGs

Decomposition of density operators on acyclic QFGs

Lemma 12 Consider a QFG with no cycles. Given t-close density operators

• σa ∈ L+

1 (Ha)

• a∈F ,
• σi ∈ L+

1 (Hi)

• i∈V

satisfying the local marginal constrains σi = Tr∂a\i (σa) ∀ (i, a) ∈ E.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 24

Closing-the-box Operations on QFGs

Decomposition of density operators on acyclic QFGs

Lemma 12 Consider a QFG with no cycles. Given t-close density operators

• σa ∈ L+

1 (Ha)

• a∈F ,
• σi ∈ L+

1 (Hi)

• i∈V

satisfying the local marginal constrains σi = Tr∂a\i (σa) ∀ (i, a) ∈ E. There exists a global density operator ˜ σ σa ≈ TrV\∂a (˜ σ) ∀a ∈ F, σi ≈ TrV\i (˜ σ) ∀i ∈ V.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 24

Closing-the-box Operations on QFGs

Decomposition of density operators on acyclic QFGs

Lemma 12 Consider a QFG with no cycles. Given t-close density operators

• σa ∈ L+

1 (Ha)

• a∈F ,
• σi ∈ L+

1 (Hi)

• i∈V

satisfying the local marginal constrains σi = Tr∂a\i (σa) ∀ (i, a) ∈ E. There exists a global density operator ˜ σ σa ≈ TrV\∂a (˜ σ) ∀a ∈ F, σi ≈ TrV\i (˜ σ) ∀i ∈ V. Proof [Idea] Just consider the closing-the-box operations on ˜ σ = exp

a∈F

log(σa) −

i∈V

(di − 1) log(σi)

• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 24

Closing-the-box Operations on QFGs

Decomposition of density operators on acyclic QFGs

Lemma 12 Consider a QFG with no cycles. Given t-close density operators

• σa ∈ L+

1 (Ha)

• a∈F ,
• σi ∈ L+

1 (Hi)

• i∈V

satisfying the local marginal constrains σi = Tr∂a\i (σa) ∀ (i, a) ∈ E. There exists a global density operator ˜ σ σa ≈ TrV\∂a (˜ σ) ∀a ∈ F, σi ≈ TrV\i (˜ σ) ∀i ∈ V. Proof [Idea] Just consider the closing-the-box operations on ˜ σ = exp

a∈F

log(σa) −

i∈V

(di − 1) log(σi)

• .

[Remark] If σ is the “true” global density operator, then ˜ σ ≈ σ.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 24

Variational Approach on QFGs

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Helmholtz Energy and Gibbs Free Energy

Definition 13 In analogy to CFGs, we define the quantum Helmholtz energy and quantum Gibbs free energy as FH − log(Z), FGibbs(σ) −

• a∈F

σ, log ρa − S(σ) = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ) .

where S (·) stands for the von Neumann entropy function.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Helmholtz Energy and Gibbs Free Energy

Definition 13 In analogy to CFGs, we define the quantum Helmholtz energy and quantum Gibbs free energy as FH − log(Z), FGibbs(σ) −

• a∈F

σ, log ρa − S(σ) = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ) .

where S (·) stands for the von Neumann entropy function. Direct calculation yields FGibbs(σ) = FH + S(σ ˜ ρ) FH.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Helmholtz Energy and Gibbs Free Energy

Definition 13 In analogy to CFGs, we define the quantum Helmholtz energy and quantum Gibbs free energy as FH − log(Z), FGibbs(σ) −

• a∈F

σ, log ρa − S(σ) = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ) .

where S (·) stands for the von Neumann entropy function. Direct calculation yields FGibbs(σ) = FH + S(σ ˜ ρ) FH. In other words, FH = min

σ∈L+

1 (H) FGibbs(σ). Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Helmholtz Energy and Gibbs Free Energy

Definition 13 In analogy to CFGs, we define the quantum Helmholtz energy and quantum Gibbs free energy as FH − log(Z), FGibbs(σ) −

• a∈F

σ, log ρa − S(σ) = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ) .

where S (·) stands for the von Neumann entropy function. Direct calculation yields FGibbs(σ) = FH + S(σ ˜ ρ) FH. In other words, FH = min

σ∈L+

1 (H) FGibbs(σ).

Thus, we transfer the calculation of Z into the optimization problem above.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Helmholtz Energy and Gibbs Free Energy

Definition 13 In analogy to CFGs, we define the quantum Helmholtz energy and quantum Gibbs free energy as FH − log(Z), FGibbs(σ) −

• a∈F

σ, log ρa − S(σ) = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ) .

where S (·) stands for the von Neumann entropy function. Direct calculation yields FGibbs(σ) = FH + S(σ ˜ ρ) FH. In other words, FH = min

σ∈L+

1 (H) FGibbs(σ).

Thus, we transfer the calculation of Z into the optimization problem above. However, this optimization problem is in general not tractable.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 25

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively,

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively, FGibbs = −

• a∈F
• TrV\∂a(σ), log ρa
• − S (σ)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively, FGibbs = −

• a∈F

σa, log ρa − S (σ)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively, FGibbs = −

• a∈F

σa, log ρa − S (σ) ≈ −

• a∈F
• TrV\∂a(σ), log ρa
• − S (˜

σ)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively, FGibbs = −

• a∈F

σa, log ρa − S (σ) ≈ −

• a∈F
• TrV\∂a(σ), log ρa
• − S (˜

σ) ≈ −

• a∈F
• TrV\∂a(σ), log ρa
• −
• a∈F

S(σa) +

• i∈V

(di − 1) · S(σi)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

In acyclic cases, by Lemma 12, we can approximate σ by ˜ σ. Thus, intuitively, FGibbs = −

• a∈F

σa, log ρa − S (σ) ≈ −

• a∈F
• TrV\∂a(σ), log ρa
• − S (˜

σ) ≈ −

• a∈F
• TrV\∂a(σ), log ρa
• −
• a∈F

S(σa) +

• i∈V

(di − 1) · S(σi) Definition 14 (Quantum Bethe free energy) We define the quantum Bethe free energy function of a QFG to be FBethe

• (σa)a∈F, (σi)i∈V)
• −
• a∈F

σa, log ρa −

• a∈F

S(σa) +

• i∈V

(di − 1) · S(σi), (7) where (σa)a∈F and (σi)i∈V are density operators.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Quantum Bethe Free Energy

Theorem (FBethe approximate FGibbs) Consider a QFG with no cycles. Suppose the global density operator σ and its marginals density operator σa TrV\∂a(σ) for all a ∈ F, and σi = TrV\i(σ) for all i ∈ V are all t-close to identity. Then, FGibbs (σ) = FBethe

• (σa)a∈F, (σi)i∈V
• + O(t3).

Definition 14 (Quantum Bethe free energy) We define the quantum Bethe free energy function of a QFG to be FBethe

• (σa)a∈F, (σi)i∈V)
• −
• a∈F

σa, log ρa −

• a∈F

S(σa) +

• i∈V

(di − 1) · S(σi), (7) where (σa)a∈F and (σi)i∈V are density operators.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 26

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Definition 15 ( ) min FGibbs (σ) s.t. σ ∈ L+

1 (H)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 27

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Definition 15 ( ) min FBethe

• (σa)a∈F , (σi)i∈V
• s.t.

σa ∈ L+

1 (Ha)

∀a ∈ F σi ∈ L+

1 (Hi)

∀i ∈ V σi = Tr∂a\i(σa) ∀ (i, a) ∈ E

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 27

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Definition 15 (Constrained Quantum Bethe Approximation Problem) min FBethe

• (σa)a∈F , (σi)i∈V
• s.t.

σa ∈ L+

1 (Ha)

∀a ∈ F σi ∈ L+

1 (Hi)

∀i ∈ V σi = Tr∂a\i(σa) ∀ (i, a) ∈ E

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 27

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Definition 15 (Constrained Quantum Bethe Approximation Problem) min FBethe

• (σa)a∈F , (σi)i∈V
• s.t.

σa ∈ L+

1 (Ha)

∀a ∈ F σi ∈ L+

1 (Hi)

∀i ∈ V σi = Tr∂a\i(σa) ∀ (i, a) ∈ E The Lagrangian is given by L FBethe +

• a∈F

γa · (Tr (σa) − 1) +

• i∈V

γi · (Tr (σi) − 1) +

• (i,a)∈E

Tr

• λa,i ·
• σi − Tr∂a\i (σa)
• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 27

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Stationary condition ∂L ∂γa = ∂L ∂γi = 0 d dt L

• λ∗

a,i + tC

• t=0

= 0 d dt L (σ∗

a + tC)

• t=0

= 0 d dt L (σ∗

i + tC)

• t=0

= 0

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Stationary condition ∂L ∂γa = ∂L ∂γi = 0 d dt L

• λ∗

a,i + tC

• t=0

= 0 d dt L (σ∗

a + tC)

• t=0

= 0 d dt L (σ∗

i + tC)

• t=0

= 0 We have ∀a ∈ F, ∀i ∈ V σ∗

a = exp

• log ρa +
• i∈∂a

λ∗

a,i − (1 + γ∗ a ) I

• σ∗

i = exp

• 1

di − 1 ·

• (1 + γ∗

i ) I +

• a∈∂i

λ∗

a,i

• .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Stationary condition ∂L ∂γa = ∂L ∂γi = 0 d dt L

• λ∗

a,i + tC

• t=0

= 0 d dt L (σ∗

a + tC)

• t=0

= 0 d dt L (σ∗

i + tC)

• t=0

= 0 We have ∀a ∈ F, ∀i ∈ V σ∗

a = exp

• log ρa +
• i∈∂a

λ∗

a,i − (1 + γ∗ a ) I

• σ∗

i = exp

• 1

di − 1 ·

• (1 + γ∗

i ) I +

• a∈∂i

λ∗

a,i

• .

Now, suppose {mi→a}(i,a)∈E and {ma→i}(i,a)∈E are given s.t. λ∗

a,i = log mi→a

λ∗

a,i =

• c∈∂i\a

log mc→i ∀ (i, a) ∈ E.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Stationary condition ∂L ∂γa = ∂L ∂γi = 0 d dt L

• λ∗

a,i + tC

• t=0

= 0 d dt L (σ∗

a + tC)

• t=0

= 0 d dt L (σ∗

i + tC)

• t=0

= 0 We have ∀a ∈ F, ∀i ∈ V σ∗

a ∝ exp

• log(ρa) +
• i∈∂a

log(mi→a)

• σ∗

i ∝ exp

• a∈∂i

log(ma→i)

• Now, suppose {mi→a}(i,a)∈E and {ma→i}(i,a)∈E are given s.t.

λ∗

a,i = log mi→a

λ∗

a,i =

• c∈∂i\a

log mc→i ∀ (i, a) ∈ E.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Stationary condition ∂L ∂γa = ∂L ∂γi = 0 d dt L

• λ∗

a,i + tC

• t=0

= 0 d dt L (σ∗

a + tC)

• t=0

= 0 d dt L (σ∗

i + tC)

• t=0

= 0 We have ∀a ∈ F, ∀i ∈ V σ∗

a ∝ exp

• log(ρa) +
• i∈∂a

log(mi→a)

• σ∗

i ∝ exp

• a∈∂i

log(ma→i)

• mi→a ∝ exp

 

c∈∂i\a

log(mc→i)   , ma→i ∝ Tr∂a\i   exp  log(ρa) +

• j∈∂a\i

log(mj→a)      .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Theorem 16 (Interior Stationary Points)

• (σ∗

a)a∈F , (σ∗ i )i∈V

• is an internal stationary point of the constrained quantum

Bethe approximation problem if and only if, for all a ∈ F, i ∈ V, σ∗

a ∝ exp

• log(ρa) +
• i∈∂a

log(mi→a)

• σ∗

i ∝ exp

• a∈∂i

log(ma→i)

• with {mi→a, ma→i}(i,a)∈E defined before.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Variational Approach on QFGs

Constrained Quantum Bethe Approximation Problem

Theorem 16 (Interior Stationary Points)

• (σ∗

a)a∈F , (σ∗ i )i∈V

• is an internal stationary point of the constrained quantum

Bethe approximation problem if and only if, for all a ∈ F, i ∈ V, σ∗

a ∝ exp

• log(ρa) +
• i∈∂a

log(mi→a)

• σ∗

i ∝ exp

• a∈∂i

log(ma→i)

• with {mi→a, ma→i}(i,a)∈E defined before.

If QSPA converges, then it converges to an internal stationary point

• f the constrained quantum Bethe approximation problem.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 28

Numerical Result of QSPA

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 29

Numerical Result of QSPA

Numerical Result of QSPA

For the QFG below, we generate the factors {ρk}6

k=1 (independently) in the same

fashion as in the last numerical example.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 29

Numerical Result of QSPA

Numerical Result of QSPA

For the QFG below, we generate the factors {ρk}6

k=1 (independently) in the same

fashion as in the last numerical example.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6

We apply the QSPA to the QFG on the LHS, and estimate Z by Z QSPA = − log(F ∗

Bethe).

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 29

Numerical Result of QSPA

Numerical Result of QSPA

For the QFG below, we generate the factors {ρk}6

k=1 (independently) in the same

fashion as in the last numerical example.

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6

We apply the QSPA to the QFG on the LHS, and estimate Z by Z QSPA = − log(F ∗

Bethe).

Relative error: η |Z QSPA − Z| Z .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 29

Numerical Result of QSPA

Numerical Result

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 Relative Error η Frequency Density

|N (µ, σ)| distributed Eigenvalues (µ = 1, σ = 1) Uniformly distributed Eigenvalues (a = 0, b = 1)

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 30

Conclusion & Outlook

Outline

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 31

Conclusion & Outlook

Conclusion

1

The closing-the-box operations on QFGs holds approximately, namely, Tr(ρA ⊙ ρB ⊙ ρC) ≈ Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• for ρA, ρB, ρC close to identity matrix.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 31

Conclusion & Outlook

Conclusion

1

The closing-the-box operations on QFGs holds approximately, namely, Tr(ρA ⊙ ρB ⊙ ρC) ≈ Tr1

• ρA ⊙ Tr2(ρB ⊙ ρC)
• for ρA, ρB, ρC close to identity matrix.

2

The fixed-points of QSPA correspond to the interior stationary points of quantum Bethe free energy minimization problem.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 31

Conclusion & Outlook

Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013];

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013];

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013];

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013];

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013];

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013]; Near Future Implications of the theory on QSPA for practical problems; Convergence condition of QSPA (nontrivial special cases QSPA will always converge); .

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

Outlook

On-going Migration of other methods to QFGs, e.g., of loop calculus [Chernyak and Chertkov, 2007], graph cover [Vontobel, 2013]; Near Future Implications of the theory on QSPA for practical problems; Convergence condition of QSPA (nontrivial special cases QSPA will always converge); Future Minimum mathematical requirements s.t. the Sum-Product algorithm works (approximately). Sufficient condition: commutative ring FR→R, +, ·.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 32

Conclusion & Outlook

References I

Bourin, J.-C. and Seo, Y. (2007). Reverse inequality to Golden–Thompson type inequalities: Comparison of eA+B and eAeB. Linear Algebra and its Applications, 426(2-3):312–316. Chernyak, V. Y. and Chertkov, M. (2007). Loop calculus and belief propagation for q-ary alphabet: Loop tower. In Proc. IEEE Int. Symp. Inf. Theory, pages 316–320. IEEE. Leifer, M. and Poulin, D. (2008). Quantum graphical models and belief propagation. Annals of Physics, 323(8):1899–1946. Vontobel, P. O. (2013). Counting in graph covers: A combinatorial characterization of the bethe entropy function. IEEE Transactions on Information Theory, 59(9):6018–6048.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 33

Conclusion & Outlook

References II

Warmuth, M. K. (2005). A Bayes rule for density matrices. In Advances in Neural Information Processing Systems, pages 1457–1464. Yedidia, J. S., Freeman, W. T., and Weiss, Y. (2005). Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inf. Theory, 51(7):2282–2312.

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 34

Conclusion & Outlook

Q&A

1

Factor Graphs/Preliminaries

2

Quantum Factor Graphs (QFGs)

3

Closing-the-box Operations on QFGs

4

Variational Approach on QFGs

5

Numerical Result of QSPA

6

Conclusion & Outlook

Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 35