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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,

  1. 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 ) � f a ( x a ) a ∈F 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

  2. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 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

  3. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 E = { ( i , a ) ∈ V × F : i ∈ ∂ a } . F We call g the global function , f the local function/factors . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 2

  4. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 E = { ( i , a ) ∈ V × F : i ∈ ∂ a } . 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

  5. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 E = { ( i , a ) ∈ V × F : i ∈ ∂ a } . 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

  6. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 E = { ( i , a ) ∈ V × F : i ∈ ∂ a } . 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

  7. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs , or classical factor graph (CFG), describe factorizations. Definition 1 In Example 1, In general, we associate the factorization � g ( x ) � f a ( x a ) f A X 1 a ∈F f B X 2 to a factor graph with variable node set V , function node set F , and edge set E ⊆ V × F X 3 given by f C X 4 E = { ( i , a ) ∈ V × F : i ∈ ∂ a } . 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

  8. ✶ 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

  9. ✶ 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) y 1 x 1 p Y 1 | X 1 + y 2 x 2 p Y 2 | X 2 + y 3 x 3 p Y 3 | X 3 . . . . . . . . . . . . + y n x n p Y n | X n Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

  10. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs are popular in describing (large-scale) probability systems. x i , y i ∈ F 2 ∀ i ∈ { 1 , · · · , n } ; Example 2 (LDPC Code described by CFG)     � f + ( x ) � ✶ x i = 0  . y 1 x 1  i incoming p Y 1 | X 1 + y 2 x 2 p Y 2 | X 2 + y 3 x 3 p Y 3 | X 3 . . . . . . . . . . . . + y n x n p Y n | X n Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

  11. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs are popular in describing (large-scale) probability systems. x i , y i ∈ F 2 ∀ i ∈ { 1 , · · · , n } ; Example 2 (LDPC Code described by CFG)     � f + ( x ) � ✶ x i = 0  . y 1 x 1  i incoming p Y 1 | X 1 + A problem of interest: Calculate the marginal distribution y 2 x 2 p Y 2 | X 2 + of X i given fixed { y i } n i =1 . y 3 x 3 p Y 3 | X 3 . . . . . . . . . . . . + y n x n p Y n | X n Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

  12. Factor Graphs/Preliminaries Factor Graphs and Normal Factor Graphs Factor graphs are popular in describing (large-scale) probability systems. x i , y i ∈ F 2 ∀ i ∈ { 1 , · · · , n } ; Example 2 (LDPC Code described by CFG)     � f + ( x ) � ✶ x i = 0  . y 1 x 1  i incoming p Y 1 | X 1 + A problem of interest: Calculate the marginal distribution y 2 x 2 p Y 2 | X 2 + of X i given fixed { y i } n i =1 . � � g y ( x ) = p Y i = y i | X i ( x i ) · f k ( x ) y 3 x 3 p Y 3 | X 3 . . i k . . . . ∝ p Y = y | X ( x ) . . . . . . + � g y ( x ) ∝ p Y = y | X i ( x i ) x j , j � = i y n x n p Y n | X n Symbol-wise ML decoding Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 3

  13. 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 � g ( x ) = f a ( x ∂ a ) , x x a ∈F which is defined to be the partition function/sum of a CFG. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

  14. 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 � g ( x ) = f a ( x ∂ a ) , x x a ∈F which is defined to be the partition function/sum of a CFG. Example 1 (continue) f A X 1 g = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) . f B X 2 � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) . X 3 x 1 , x 2 , x 3 , x 4 f C X 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

  15. 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 � g ( x ) = f a ( x ∂ a ) , x x a ∈F which is defined to be the partition function/sum of a CFG. Example 1 (continue) f A X 1 g = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) . f B X 2 � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) . X 3 x 1 , x 2 , x 3 , x 4 f C X 4 In general, calculation of Z is NP hard. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 4

  16. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A X 1 � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 f B X 2 X 3 f C X 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  17. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A X 1 � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 f B X 2 X 3 f C X 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  18. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A X 1 � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f B � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 f C X 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  19. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� � � � f A X 1 = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f B � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 f C X 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  20. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f AB � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 f C X 4 � � � ˆ � = f AB ( x 3 ) · f C ( x 3 , x 4 ) x 3 x 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  21. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f AB � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 f C X 4 � � � ˆ � = f AB ( x 3 ) · f C ( x 3 , x 4 ) x 3 x 4 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  22. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f AB � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 ˆ f C � � � ˆ � = f AB ( x 3 ) · f C ( x 3 , x 4 ) x 3 x 4 � � � ˆ · ˆ = f AB ( x 3 ) f C ( x 3 ) x 3 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  23. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 ˆ f AB � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 ˆ f C � � � ˆ � = f AB ( x 3 ) · f C ( x 3 , x 4 ) x 3 x 4 � � � ˆ · ˆ = f AB ( x 3 ) f C ( x 3 ) x 3 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  24. Factor Graphs/Preliminaries Partition Sum and the Closing-the-box Operations Example 1 Sum over local variables first... � Z = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 1 , x 2 , x 3 , x 4 �� � �� f A � � � = f A ( x 1 ) · f B ( x 1 , x 2 , x 3 ) · f C ( x 3 , x 4 ) x 3 x 1 x 2 x 4 Z � � �� � � f A ( x 1 ) · ˆ � = f B ( x 1 , x 3 ) · f C ( x 3 , x 4 ) X 3 x 3 x 1 x 4 � � � ˆ � = f AB ( x 3 ) · f C ( x 3 , x 4 ) x 3 x 4 � � � ˆ · ˆ = f AB ( x 3 ) f C ( x 3 ) = Z x 3 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 5

  25. 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). z 1 w 1 f 1 w 3 x f 3 y z 3 f 2 z 2 w 2 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 6

  26. 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). z 1 z 1 w 1 w 1 f 1 w 3 w 3 � f 1 · f 2 · f 3 x f 3 x , y y z 3 z 3 f 2 z 2 z 2 w 2 w 2 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 6

  27. Factor Graphs/Preliminaries Sum-Product Algorithm on a tree Example 1 f A X 1 f B X 2 X 3 f C X 4 Closing the boxes from the inner ones to outer ones will yield the partition sum Z. Distributive Law on R Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 7

  28. Factor Graphs/Preliminaries Sum-Product Algorithm on a tree Example 1 Sum-Product Algorithm for Trees Require: Acyclic factor graph G = ( F , V , E ); root r ∈ V ; height of the tree h � 0. f A Ensure: Partition sum Z X 1 1: for d = h − 1 , · · · , 0 do for all i ∈ V d-step reachable a from r f B X 2 2: do X 3 Let f ( i ) be the parent factor b of i ; 3: f ( i ) ← � � a ∈ ∂ i f a ( x i ); f C X 4 4: x i end for 5: 6: end for Closing the boxes from the inner ones to 7: Z ← f ( r ) . outer ones will yield the partition sum Z. Distributive Law on R a i.e., there exists a path connecting r and i passing through d factors. b i.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

  29. Factor Graphs/Preliminaries Sum-Product Algorithm on a tree Example 1 Sum-Product Algorithm for Trees Require: Acyclic factor graph G = ( F , V , E ); root r ∈ V ; height of the tree h � 0. f A Ensure: Partition sum Z X 1 1: for d = h − 1 , · · · , 0 do for all i ∈ V d-step reachable a from r f B X 2 2: do X 3 Let f ( i ) be the parent factor b of i ; 3: f ( i ) ← � � a ∈ ∂ i f a ( x i ); f C X 4 4: x i end for 5: 6: end for Closing the boxes from the inner ones to 7: Z ← f ( r ) . outer ones will yield the partition sum Z. Distributive Law on R a i.e., there exists a path connecting r and i passing through d factors. Message-Passing Algorithm b i.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

  30. Factor Graphs/Preliminaries Sum-Product Algorithm on a tree Example 1 Sum-Product Algorithm for Trees Require: Acyclic factor graph G = ( F , V , E ); root r ∈ V ; height of the tree h � 0. f A Ensure: Partition sum Z X 1 ← 1: for d = h − 1 , · · · , 0 do for all i ∈ V d-step reachable a from r ← f B X 2 2: → do X 3 Let f ( i ) be the parent factor b of i ; 3: f ( i ) ← � ← � a ∈ ∂ i f a ( x i ); f C X 4 4: x i end for 5: 6: end for Closing the boxes from the inner ones to 7: Z ← f ( r ) . outer ones will yield the partition sum Z. Distributive Law on R a i.e., there exists a path connecting r and i passing through d factors. Message-Passing Algorithm b i.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

  31. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ← � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ← � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  32. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ← � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ← � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  33. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ∝ � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ∝ � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  34. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ∝ � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ∝ � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  35. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ∝ � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ∝ � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until convergence Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  36. 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 m i → a ← ✶ ; 2: m a → i ← ✶ ; 3: 4: end for 5: repeat for all ( i , a ) ∈ E do 6: m i → a ( x i ) ∝ � c ∈ ∂ i \ a m a → i ( x i ); 7: end for 8: for all ( i , a ) ∈ E do 9: m a → i ( x i ) ∝ � x ∂ a \ i f a ( x ∂ a ) · � j ∈ ∂ a \ i m j → a ( x j ); 10: end for 11: 12: until convergence Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 8

  37. Factor Graphs/Preliminaries Sum-Product Algorithm and the Variational Approach In acyclic case, it will always converge. b i ( x i ) � � a ∈ ∂ i m a → i ( x i ) ∝ � x V\ i g ( x ); b a ( x ∂ a ) � f a ( x ∂ a ) · � i ∈ ∂ a m i → a ( x j ) ∝ � x V\ ∂ a g ( x ). Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 9

  38. Factor Graphs/Preliminaries Sum-Product Algorithm and the Variational Approach In acyclic case, it will always converge. b i ( x i ) � � a ∈ ∂ i m a → i ( x i ) ∝ � x V\ i g ( x ); b a ( x ∂ a ) � f a ( x ∂ a ) · � i ∈ ∂ a m i → a ( x j ) ∝ � x V\ ∂ a g ( x ). In general case, if it converges, then: [Yedidia et al., 2005] The above { b i } i ∈V and { b a } 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

  39. Factor Graphs/Preliminaries Sum-Product Algorithm and the Variational Approach In acyclic case, it will always converge. b i ( x i ) � � a ∈ ∂ i m a → i ( x i ) ∝ � x V\ i g ( x ); b a ( x ∂ a ) � f a ( x ∂ a ) · � i ∈ ∂ a m i → a ( x j ) ∝ � x V\ ∂ a g ( x ). In general case, if it converges, then: [Yedidia et al., 2005] The above { b i } i ∈V and { b a } a ∈F correspond to the interior stationary points of the constrained Bethe approximation problem : � � min F Bethe ( b a ) a ∈F , ( b i ) i ∈V � � � � � − b a ( x ∂ a ) log f a ( x ∂ a ) − H ( b a ) + ( d i − 1) · H ( b i ) x ∂ a a ∈F a ∈F i ∈V s . t . b a probability on x ∂ a , b i probability on x i , ∀ a ∈ F , ∀ i ∈ V � b i ( x i ) = b a ( x ∂ a ) ∀ x i , ∀ ( i , a ) ∈ E x ∂ a \ i Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 9

  40. Quantum Factor Graphs (QFGs) Outline Factor Graphs/Preliminaries 1 Quantum Factor Graphs (QFGs) 2 Closing-the-box Operations on QFGs 3 Variational Approach on QFGs 4 Numerical Result of QSPA 5 Conclusion & Outlook 6 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 10

  41. 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 { ρ a } describes the “factorization” �� � � ρ � ρ a = exp log( ρ a ) (1) , a ∈F a ∈F where, for each a ∈ F , positive definite operator ρ a is an operator on � i ∈ ∂ a H i . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 10

  42. 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 { ρ a } describes the “factorization” �� � � ρ � ρ a = exp log( ρ a ) (1) , a ∈F a ∈F where, for each a ∈ F , positive definite operator ρ a is an operator on � i ∈ ∂ a H i . Example 5 ρ A ∈ L + ( H 1 ) H 1 H 2 ρ A ρ B ρ C ρ B ∈ L + ( H 1 ⊗ H 2 ) ρ C ∈ L + ( H 2 ) A QFG describing ρ = ρ A ⊙ ρ B ⊙ ρ C . 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

  43. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (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

  44. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have � n 1 1 � ρ A ⊙ ρ B = lim ρ A ρ n n . (3) B n →∞ Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

  45. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have � n 1 1 � ρ A ⊙ ρ B = lim ρ A ρ n n . (3) B n →∞ Equation (3) can be used to generalize the ⊙ product to PSD operators. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

  46. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have � n 1 1 � ρ A ⊙ ρ B = lim ρ A ρ n n . (3) B n →∞ Equation (3) can be used to generalize the ⊙ product to PSD operators. Properties of ⊙ Associativity: ( ρ A ⊙ ρ B ) ⊙ ρ C = ρ A ⊙ ( ρ B ⊙ ρ C ); Commutativity: ρ A ⊙ ρ B = ρ B ⊙ ρ A ; Closeness: ρ A ⊙ ρ B is positive (semi) definite if ρ A , ρ B are positive (semi) definite. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

  47. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have � n 1 1 � ρ A ⊙ ρ B = lim ρ A ρ n n . (3) B n →∞ Equation (3) can be used to generalize the ⊙ product to PSD operators. Properties of ⊙ Associativity: ( ρ A ⊙ ρ B ) ⊙ ρ C = ρ A ⊙ ( ρ B ⊙ ρ C ); Commutativity: ρ A ⊙ ρ B = ρ B ⊙ ρ A ; Closeness: ρ A ⊙ ρ B is positive (semi) definite if ρ A , ρ B 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

  48. Quantum Factor Graphs (QFGs) Quantum Factor Graphs (QFGs) For ρ A , ρ B ∈ L ++ ( H ), define [Warmuth, 2005] ρ A ⊙ ρ B � exp � � log( ρ A ) + log( ρ B ) , (2) where exp and log denote the operator exponential and the operator natural logarithm, respectively. By the Lie Product formula, we have � n 1 1 � ρ A ⊙ ρ B = lim ρ A ρ n n . (3) B n →∞ Equation (3) can be used to generalize the ⊙ product to PSD operators. ρ A ∈ L + ( H 1 ) Properties of ⊙ � H 1 � = H 2 ρ B ∈ L + ( H 2 ) Associativity: ( ρ A ⊙ ρ B ) ⊙ ρ C = ρ A ⊙ ( ρ B ⊙ ρ C ); Commutativity: ρ A ⊙ ρ B = ρ B ⊙ ρ A ; H 1 → H 3 ← H 2 Closeness: ρ A ⊙ ρ B is positive (semi) definite if ρ A , ρ B are positive (semi) definite. ρ A ⊙ ρ B � ˜ ρ A ⊙ ˜ ρ B . �L + ( H ) , ⊙� (or �L ++ ( H ) , ⊙� ) is an Abelian group. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 11

  49. Quantum Factor Graphs (QFGs) Quantum Factor Graphs Example 5: continue H 1 = C 2 H 2 = C 2 ρ A ρ B ρ C A QFG describing ρ = ρ A ⊙ ρ B ⊙ ρ C . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

  50. Quantum Factor Graphs (QFGs) Quantum Factor Graphs Example 5: continue H 1 = C 2 H 2 = C 2 ρ A ρ B ρ C A QFG describing ρ = ρ A ⊙ ρ B ⊙ ρ C . Suppose H 1 = H 2 = C 2 , and   0 � � � � +3 − 1 1 +3 − 1   ρ A = , ρ B =  , ρ C = .   − 1 +3 1 − 1 +3  0 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

  51. Quantum Factor Graphs (QFGs) Quantum Factor Graphs Example 5: continue H 1 = C 2 H 2 = C 2 ρ A ρ B ρ C A QFG describing ρ = ρ A ⊙ ρ B ⊙ ρ C . Suppose H 1 = H 2 = C 2 , and   0 � � � � +3 − 1 1 +3 − 1   ρ A = , ρ B =  , ρ C = .   − 1 +3 1 − 1 +3  0 We have,   9 − 3 − 3 1 − 3 9 1 − 3   ρ = ρ A ⊙ ρ B ⊙ ρ C =  .   − 3 1 9 − 3  1 − 3 − 3 9 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 12

  52. 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 ρ a = Tr exp log( ρ a ) , a ∈F a ∈F which is defined to be the partition function/sum of a QFG. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 13

  53. 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 ρ a = Tr exp log( ρ a ) , a ∈F a ∈F 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

  54. Closing-the-box Operations on QFGs Outline Factor Graphs/Preliminaries 1 Quantum Factor Graphs (QFGs) 2 Closing-the-box Operations on QFGs 3 Variational Approach on QFGs 4 Numerical Result of QSPA 5 Conclusion & Outlook 6 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  55. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in CFG X 1 X 2 f A f B f C Z = � x 1 , x 2 f A ( x 1 ) f B ( x 1 , x 2 ) f C ( x 2 ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  56. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in CFG X 1 X 2 f A f B f C Z = � x 1 , x 2 f A ( x 1 ) f B ( x 1 , x 2 ) f C ( x 2 ) = � x 1 f A ( x 1 ) � x 2 f B ( x 1 , x 2 ) f C ( x 2 ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  57. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in CFG = ⇒ Applying distributive law X 1 X 2 f A f B f C Z = � x 1 , x 2 f A ( x 1 ) f B ( x 1 , x 2 ) f C ( x 2 ) = � x 1 f A ( x 1 ) � x 2 f B ( x 1 , x 2 ) f C ( x 2 ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  58. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in QFG H 1 H 2 ρ A ρ B ρ C Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  59. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in QFG H 1 H 2 ρ A ρ B ρ C ? � � Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  60. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in QFG = ⇒ Distributive law over (partial) trace H 1 H 2 ρ A ρ B ρ C ? � � Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  61. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in QFG = ⇒ Distributive law over (partial) trace H 1 H 2 ρ A ρ B ρ C ? � � Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) However, in general, � � � � Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 Tr 2 ( ρ A ⊙ ρ B ⊙ ρ C ) � = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  62. Closing-the-box Operations on QFGs Closing-the-box operations and partial trace functions Closing the box in QFG = ⇒ Distributive law over (partial) trace H 1 H 2 ρ A ρ B ρ C ? � � Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) However, in general, � � � � Tr( ρ A ⊙ ρ B ⊙ ρ C ) = Tr 1 Tr 2 ( ρ A ⊙ ρ B ⊙ ρ C ) � = Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) . Example 7 � +1 − 1 Let H 1 = H 2 � C 2 . Suppose ρ A � 1 � , ρ B ⊙ ρ C � diag (0 , 1 , 1 , 0) . In 2 · − 1 +1 � � this case, Tr( ρ A ⊙ ρ B ⊙ ρ C ) = 0 and Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) = 1. Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 14

  63. 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 Given ρ A ∈ L ++ ( H 1 ); � − 1 � � S κ ( ρ A ) ρ B ∈ L ++ ( H 1 ⊗ H 2 ); Tr( ρ A ⊙ ρ B ⊙ ρ C ) ρ C ∈ L ++ ( H 2 ); � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) κ ( · ) � 1 is the condition number function; � S � � κ ( ρ A ) . S ( · ) is the Specht ratio function defined as 1 S ( r ) � ( r − 1) · r r − 1 . e · log r Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

  64. 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 Given ρ A ∈ L ++ ( H 1 ); � − 1 � � S κ ( ρ A ) ρ B ∈ L ++ ( H 1 ⊗ H 2 ); Tr( ρ A ⊙ ρ B ⊙ ρ C ) ρ C ∈ L ++ ( H 2 ); � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) κ ( · ) � 1 is the condition number function; � S � � κ ( ρ A ) . S ( · ) is the Specht ratio function defined as 1 S ( r ) � ( r − 1) · r 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

  65. 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 Given Specht Ratio Function � − 1 � 8 � S κ ( ρ A ) 6 Tr( ρ A ⊙ ρ B ⊙ ρ C ) � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) 4 � S � � κ ( ρ A ) . 2 0 0 20 40 60 80 100 condition number r The proof utilizes the Golden–Thompson inequality [Bourin and Seo, 2007]. Considering ρ A ≈ I , Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

  66. 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 Given Specht Ratio Function � − 1 � 8 � S κ ( ρ A ) 6 Tr( ρ A ⊙ ρ B ⊙ ρ C ) � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) 4 � S � � κ ( ρ A ) . 2 0 0 20 40 60 80 100 condition number r The proof utilizes the Golden–Thompson inequality [Bourin and Seo, 2007]. Considering ρ A ≈ I , we expect to have � � Tr( ρ A ⊙ ρ B ⊙ ρ C ) ≈ Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 15

  67. Closing-the-box Operations on QFGs Type-1 Approximation when ρ A is “close” to identity matrix I H 1 H 2 H 1 H 2 ρ A ρ B ρ C ρ A ρ B ρ C Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  68. Closing-the-box Operations on QFGs Type-1 Approximation when ρ A is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ A ρ B ρ C ρ A ρ B ρ C Z = Tr( ρ A ⊙ ρ B ⊙ ρ C ) � � Tr 1 ρ A ⊙ Tr 2 ( ρ B ⊙ ρ C ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  69. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 ρ 1 , 2 ρ 1 ρ 1 , 2 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) � � Tr 1 ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  70. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  71. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  72. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � Lemma (Type-1 Approximation) Given X ∈ L H ( H 1 ), and Y ∈ L H ( H 1 ⊗ H 2 ), for t close to 0, we have = ( I + tX ) ⊙ Tr 2 ( I + tY ) + O ( t 3 ) . � � Tr 2 ( I + tX ) ⊙ ( I + tY ) (4) Theorem (Type-1 Approximation) Following the same setup, we have + O ( t 4 ) . � � � � Tr ( I + tX ) ⊙ ( I + tY ) = Tr 1 ( I + tX ) ⊙ Tr 2 ( I + tY ) (5) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  73. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � “Linear” close to I : i.e., ρ 1 = I + tX and ρ 1 , 2 = I + tY . Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  74. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � “Linear” close to I : i.e., ρ 1 = I + tX and ρ 1 , 2 = I + tY . Taylor Series Expansion: Tr( ρ 1 ⊙ ρ 1 , 2 ) = � � Tr 1 ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) = Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  75. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � “Linear” close to I : i.e., ρ 1 = I + tX and ρ 1 , 2 = I + tY . Taylor Series Expansion: X + Y ) + t 2 · Tr( ˜ XY + Y ˜ X ) + t 3 · 0 Tr( ρ 1 ⊙ ρ 1 , 2 ) = 1 + t · Tr( ˜ 2 � X 2 Y 2 � XY ˜ ˜ XY − ˜ Tr + t 4 · + · · · 12 = 1 + t · Tr 1 ( X + Tr 2 ( Y )) + t 2 · Tr 1 � X Tr 2 ( Y ) + Tr 2 ( Y ) X � � � Tr 1 ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) 2 X Tr 2 ( Y ) X Tr 2 ( Y ) − X 2 Tr 2 ( Y ) 2 � + t 4 · Tr 1 � + · · · 12 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  76. Closing-the-box Operations on QFGs Type-1 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � “Linear” close to I : i.e., ρ 1 = I + tX and ρ 1 , 2 = I + tY . Taylor Series Expansion: X + Y ) + t 2 · Tr( ˜ XY + Y ˜ X ) + t 3 · 0 Tr( ρ 1 ⊙ ρ 1 , 2 ) = 1 + t · Tr( ˜ 2 � X 2 Y 2 � XY ˜ ˜ XY − ˜ Tr + t 4 · + · · · 12 = 1 + t · Tr 1 ( X + Tr 2 ( Y )) + t 2 · Tr 1 � X Tr 2 ( Y ) + Tr 2 ( Y ) X � � � Tr 1 ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) 2 X Tr 2 ( Y ) X Tr 2 ( Y ) − X 2 Tr 2 ( Y ) 2 � + t 4 · Tr 1 � + · · · 12 Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 16

  77. Closing-the-box Operations on QFGs Type-2 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

  78. Closing-the-box Operations on QFGs Type-2 Approximation when ρ 1 or ρ 1 , 2 is “close” to identity matrix I ≈ H 1 H 2 H 1 H 2 ρ 1 , 2 ρ 1 , 2 ρ 1 ρ 1 Z = Tr( ρ 1 ⊙ ρ 1 , 2 ) Tr 1 � ρ 1 ⊙ Tr 2 ( ρ 1 , 2 ) � Lemma (Type-2 Approximation) Given X ∈ L H ( H 1 ), and Y ∈ L H ( H 1 ⊗ H 2 ), for t close to 0, we have e tX ⊙ e tY � = e tX ⊙ Tr 2 ( e tY ) + O ( t 3 ) . � Tr 2 (4) Theorem (Type-2 Approximation) Following the same setup, we have e tX ⊙ e tY � e tX ⊙ Tr 2 ( e tY ) + O ( t 4 ) . � � � Tr = Tr 1 (5) Michael X. CAO (IE@CUHK) Quantum Factor Graphs August 29, 2016 17

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