GENERALIZED SATISFIABILITY PROBLEMS VIA OPERATOR ASSIGNMENTS - PowerPoint PPT Presentation

GENERALIZED SATISFIABILITY PROBLEMS VIA OPERATOR ASSIGNMENTS Albert Atserias, UPC Barcelona Phokion Kolaitis, UCSC and IBM Almaden Simone Severini, UCL and Shangai

Talk plan I. Background and motivation II. Problem statement III. Results 2

Part I BACKGROUND AND MOTIVATION 3

Systems of Polynomial Equations over Rings Variables : X 1 , . . . , X n Systems of polynomial equations (arithmetic in a ring) : X 2 X 3 − 1 = 0 X 1 X 2 X 3 − 2 X 1 X 3 + X 2 = 0 X 3 + X 4 − 2 X 1 + 1 = 0 X 1 + X 2 + X 3 − 2 = 0 4

An important example: Mermin-Peres Magic Square Nine variables, fifteen equations : X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 X 31 X 32 X 33 = +1 X 11 X 21 X 31 = +1 X 12 X 22 X 32 = +1 X 13 X 23 X 33 = − 1 X 2 11 = X 2 12 = · · · = X 2 33 = +1 5

An important example: Mermin-Peres Magic Square Nine variables, fifteen equations : X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 X 31 X 32 X 33 = +1 = +1 = +1 = − 1 X 2 11 = X 2 12 = · · · = X 2 33 = +1 6

Proof of unsatisfiability over commutative rings (e.g., C ) X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 = +1 X 11 X 21 X 31 X 12 X 22 X 32 X 13 X 23 X 33 = − 1 7

Proof of unsatisfiability over commutative rings (e.g., C ) X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 = +1 X 11 X 21 X 31 X 12 X 22 X 32 X 13 X 23 X 33 = − 1 Remarks : ◮ Do not even need X 2 ij = +1 . ◮ Relies heavily on the fact that product commutes. 7

Indeed ... There is a solution in 4x4 complex matrices I ⊗ Z Z ⊗ I Z ⊗ Z = + I X ⊗ I I ⊗ X X ⊗ X = + I X ⊗ Z Z ⊗ X Y ⊗ Y = + I = = = + I + I − I where X, Y, Z are the Pauli matrices: � 0 � � 0 � � 1 � 1 − i 0 X = Y = Z = . 1 0 0 0 − 1 i 8

Indeed ... There is a solution in 4x4 complex matrices I ⊗ Z Z ⊗ I Z ⊗ Z = + I X ⊗ I I ⊗ X X ⊗ X = + I X ⊗ Z Z ⊗ X Y ⊗ Y = + I = = = + I + I − I where X, Y, Z are the Pauli matrices: � 0 � � 0 � � 1 � 1 − i 0 X = Y = Z = . 1 0 0 0 − 1 i Note: ( I ⊗ Z ) 2 = ( X ⊗ I ) 2 = · · · ( Y ⊗ Y ) 2 = + I 8

Where does this come from? Quantum entanglement [Einstein-Podolsky-Rosen 1935], [Bell 1964], [Mermin 1990] p ij,ab := “empirical probability that ij lights as ab ” ˆ 9

Where does this come from? Quantum entanglement [Einstein-Podolsky-Rosen 1935], [Bell 1964], [Mermin 1990] p ij,ab := “empirical probability that ij lights as ab ” ˆ 9

Where does this come from? Quantum entanglement [Einstein-Podolsky-Rosen 1935], [Bell 1964], [Mermin 1990] p ij,ab := “empirical probability that ij lights as ab ” ˆ 9

Where does this come from? Quantum entanglement [Einstein-Podolsky-Rosen 1935], [Bell 1964], [Mermin 1990] p ij,ab := “empirical probability that ij lights as ab ” ˆ ◮ Cannot be explained by a classical probability distribution ; ◮ Can be explained by quantum entanglement ; 9

Back to equations [Cleve and Mittal, ICALP 2014] Variables ranging over Hermitian matrices X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 10

Back to equations [Cleve and Mittal, ICALP 2014] Variables ranging over Hermitian matrices Equations enforcing unitaries X 11 X 12 X 13 X 21 X 22 X 23 X 31 X 32 X 33 X 2 ij = +1 for all i and j 10

Back to equations [Cleve and Mittal, ICALP 2014] Variables ranging over Hermitian matrices Equations enforcing unitaries Equations enforcing constraints X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 X 31 X 32 X 33 = +1 = − 1 = +1 = +1 X 2 ij = +1 for all i and j 10

Back to equations [Cleve and Mittal, ICALP 2014] Variables ranging over Hermitian matrices Equations enforcing unitaries Equations enforcing constraints Equations enforcing joint measurability X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 X 31 X 32 X 33 = +1 = − 1 = +1 = +1 X 2 ij = +1 for all i and j X ij X kl = X kl X ij if i = k or j = l 10

Part II PROBLEM STATEMENT 11

Informal statement Characterize the types of constraints that allow such gedanken experiments that are quantum realizable but not classically realizable. 12

Informal statement Characterize the types of constraints that allow such gedanken experiments that are quantum realizable but not classically realizable. E.g.: parity constraints of arity ≥ 3 do, but parity constraints of arity ≤ 2 don’t. 12

Schaefer’s framework for generalized satisfiability 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; Constraint language: a set A of relations R ⊆ {± 1 } r 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; Constraint language: a set A of relations R ⊆ {± 1 } r relations ↔ predicates ↔ polynomial equations 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; Constraint language: a set A of relations R ⊆ {± 1 } r relations ↔ predicates ↔ polynomial equations characteristic function R : {± 1 } r → {± 1 } 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; Constraint language: a set A of relations R ⊆ {± 1 } r relations ↔ predicates ↔ polynomial equations characteristic function Fourier-Welsh transform R : {± 1 } r → {± 1 } R ( X 1 , . . . , X r ) = − 1 13

Schaefer’s framework for generalized satisfiability Boolean domain: {± 1 } with +1 = false and and − 1 = true ; Constraint language: a set A of relations R ⊆ {± 1 } r relations ↔ predicates ↔ polynomial equations characteristic function Fourier-Welsh transform R : {± 1 } r → {± 1 } R ( X 1 , . . . , X r ) = − 1 Examples : OR disjunctions of literals LIN linear equations over Z 2 1-IN-3 triples with one − 1 and two +1 components NAE triples with not-all-equal components 13

Generalized Satisfiability Problems: SAT( A ) ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) 14

Generalized Satisfiability Problems: SAT( A ) ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n range over {± 1 } 14

Generalized Satisfiability Problems: SAT( A ) ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over {± 1 } of the form R ( Y 1 , . . . , Y r ) = 1 14

Generalized Satisfiability Problems: SAT( A ) ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over {± 1 } of the form R ( Y 1 , . . . , Y r ) = 1 in A X i ’s or ± 1 14

Generalized Satisfiability Problems: SAT( A ) ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over {± 1 } of the form R ( Y 1 , . . . , Y r ) = 1 in A X i ’s or ± 1 Examples : 3-SAT 1-IN-3-SAT HORN-SAT NAE-SAT LIN-SAT ... [Schaefer 1978] 14

... via Operator Assignments ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) 15

... via Operator Assignments ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n range over B ( H ) , the self-adjoint linear operators of a Hilbert space H 15

... via Operator Assignments ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over B ( H ) , the of the form R ( Y 1 , . . . , Y r ) = I self-adjoint linear operators Y i Y j = Y j Y i for all i, j ∈ [ r ] of a Hilbert space H 15

... via Operator Assignments ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over B ( H ) , the of the form R ( Y 1 , . . . , Y r ) = I self-adjoint linear operators Y i Y j = Y j Y i for all i, j ∈ [ r ] of a Hilbert space H and X 2 i = I for all i ∈ [ n ] 15

... via Operator Assignments ∃ X 1 · · · X n ( C 1 ∧ · · · ∧ C m ) variables X 1 , . . . , X n constraints C 1 , . . . , C m each range over B ( H ) , the of the form R ( Y 1 , . . . , Y r ) = I self-adjoint linear operators Y i Y j = Y j Y i for all i, j ∈ [ r ] of a Hilbert space H and X 2 i = I for all i ∈ [ n ] SAT( A ) satisfiability over Boolean domain SAT ∗ ( A ) satisfiability over some finite-dimensional H SAT ∗∗ ( A ) satisfiability over some arbitrary H 15

Gap Instances Mermin-Peres Magic Square X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 Unsatisfiable SAT-instance of LIN X 31 X 32 X 33 = − 1 Satisfiable SAT ∗ -instance of LIN = +1 = +1 = +1 a SAT-vs-SAT ∗ gap for LIN 16

Gap Instances Mermin-Peres Magic Square X 11 X 12 X 13 = +1 X 21 X 22 X 23 = +1 Unsatisfiable SAT-instance of LIN X 31 X 32 X 33 = − 1 Satisfiable SAT ∗ -instance of LIN = +1 = +1 = +1 a SAT-vs-SAT ∗ gap for LIN SAT-vs-SAT ∗ gap of the first kind SAT-vs-SAT ∗∗ gap of the second kind SAT ∗ -vs-SAT ∗∗ gap of the third kind 16