Axiomatizing Probabilistic Logic of Quantum Programs Jort Bergfeld - PowerPoint PPT Presentation

Axiomatizing Probabilistic Logic of Quantum Programs Jort Bergfeld and Joshua Sack Amsterdam, 2014 April 1 1/32

Motivation and Background Quantum Algorithms and Protocols: use logic for a better understanding. Probabilistic Logic of Quantum Programs We involve a logic that expresses probabilities of outcomes of quantum tests effects of quantum tests and unitary operations separation operations that characterize subsystems. and is decidable: PLQP & company by Baltag et al 2014 In this talk, we provide a sound proof system use it to prove that “leader election protocol” is correct 2/32

Structures for reasoning about quantum systems Hilbert spaces are commonly used: Quantum states are one-dimensional subspaces. Probabilities of outcomes of tests characterized by inner product of vector representatives of the states |� x , y �| 2 | x || y | . Composite systems are constructed from the tensor product of Hilbert spaces (subsystems) In this talk, we Involve finite dimensional Hilbert spaces Build each structure from the set of states for a basis of the Hilbert space Involve agents, each corresponding to a basis of a subsystem 3/32

Bases and states Let H be a Hilbert space with an orthonormal basis � B = { � b 1 , . . . , � b n } . For every state (one dimensional space) s , there is a unit vector � s in state s , such that 1 there exists an i ∈ N such that s ,� � � b j � = 0 for all j < i , and 1 s , � � � b i � ∈ (0 , 1], 2 b k �| 2 ∈ [0 , 1], and s ,� 2 |� � b k �| 2 = 1. s ,� 3 � k ∈ N |� � s , ·� : � The function � � B → C characterizes s . s , ·�| 2 : � The function |� � B → [0 , 1] is a probability mass function. 4/32

Complex probability mass function B = { b i | i ∈ N } a finite totally ordered set (called basis states). f : B → C is called a complex probability mass function on B if 1 there exists an i ∈ N such that f ( b j ) = 0 for all j < i , and 1 f ( b i ) ∈ (0 , 1], 2 2 | f ( b k ) | 2 ∈ [0 , 1], and k ∈ N | f ( b k ) | 2 = 1. 3 � F B is the set of all complex probability mass functions on B . 5/32

Complex probability mass functions are states Proposition Given a set of basis states B = { b 1 , . . . , b n } there is a Hilbert space H with orthonormal basis { � b 1 , . . . ,� b n } such that for any complex probability mass function s : B → C , s ,� there is a vector � s, such that for each i, s ( b i ) = � � b i � . As the complex probability mass function s uniquely identifies the state (one dimensional subspace) generated by � s , we identify s with that state. 6/32

Generating structure from F B Define S := F B (all complex probability mass functions) inner product µ ( s , t ) := � i ∈ n s ( b i ) t ( b i ) for all s , t ∈ S , z is the complex conjugate of z ∈ C nonorthogonality relation R = { ( s , t ) ∈ S × S | µ ( s , t ) � = 0 } , orthocomplement ∼ X := { s ∈ S | ( s , x ) / ∈ R for all x ∈ X } , for any set X ⊆ S , testable properties T := { P ⊆ S | P = ∼∼ P } , P -test relation R P := { ( s , t ) ∈ R | t ∈ P and | µ ( s , u ) | 2 ≤ | µ ( s , t ) | 2 for all u ∈ P } , unitary operators U := { U : S → S | U is a permutation and µ ( s , t ) = µ ( Us , Ut ) for all s , t ∈ S } , unitary relation R U := { ( s , t ) ∈ S × S | t = Us } . 7/32

Tensor product Definition (Tensor product) The tensor product of state bases B = { b 1 , . . . , b n } and C = { c 1 , . . . , c m } is B ⊗ C = { b i c j | b i ∈ B , c j ∈ C } The elements of B ⊗ C are totally ordered by the dictionary order. 8/32

Separability Definition (Separable and entangled states) A complex probability mass function f ∈ F B ⊗ C is separable (into B and C ) if there exist s ∈ F B and t ∈ F C , such that f ( bc ) = s ( b ) t ( c ) for all b ∈ B and c ∈ C . We write f = s ⊗ t . If f is not separable we call f entangled. Definition (Separable Unitaries) A unitary operator U if F B ⊗ C is separable if there exists unitaries U B and U C , such that for all s ∈ F B and t ∈ F C , U ( s ⊗ t ) = U B ( s ) ⊗ U C ( t ). We then write U = U B ⊗ U C . 9/32

Multi-agent models Let A = { 0 , 1 , . . . , N − 1 } be a finite set of agents. Let Prop be a set of atomic propositions. Definition (multi-agent probabilistic quantum model (PQM)) An A -PQF (probabilistic quantum frame) is a pair F = ( B , { B i } i ∈A ), where B is a basis of states and B i is a two-state basis for each i ∈ A , such that B = � i ∈A B i . Then an A -PQM (probabilistic quantum model) is a pair ( F , V ), such that F = ( B , { B i } i ∈A ) is an A -PQF and V : Prop → P ( F B ) is a valuation. Given a subset I ⊆ A of agents, let B I = � i ∈ I B i S I = F B I If s is separable over B I and B A\ I , let s I denote the complex probability mass function such that there exists s A\ I such that s = s I ⊗ s A\ I . 10/32

Language Let A = { 0 , 1 , . . . , N − 1 } be a finite set of agents. Let Prop be a (countable) set of atomic propositions. φ ::= ⊤ I | p | t ≥ ρ | ¬ φ | φ ∧ φ | � φ | [ α ] φ α ::= ⊤ I | φ ? | U | U † | α ∪ α | α ; α t ::= ρ Pr( φ ) | t + t where p ∈ Prop, U ∈ U , I ⊆ A and ρ ∈ R . Language “ L ” is defined to the the set of all such φ Set “Terms” is defined to be the set of all terms t ⊤ I means “ I -separable” [ ⊤ I ] ranges over the “ I -subsystem” (is equivalent to K A\ I ) 11/32

Semantics Let (( B , { B i } i ∈A ) , V ) be an A -PQM, and let S = F B . We define an extended valuation � · � : L → P S , and for each s ∈ S , � · � s : Terms → R : � ⊤ I � := { s ∈ S | s = s I ⊗ s A\ I for some s I ∈ S I and s A\ I ∈ S A\ I } , � p � := V ( p ) , � t ≥ ρ � := { s ∈ S | � t � s ≥ ρ } � ¬ φ � := S \ � φ � , � φ ∧ ψ � := � φ � ∩ � ψ � , � � φ � := { s ∈ S | R ( s ) ⊆ � φ � } ( R is non-orthogonality relation) � [ α ] φ � := { s ∈ S | R α ( s ) ⊆ � φ � } ( R α is defined on next slide) . � | µ ( s , t ) | 2 , where P = ∼∼ � φ � , � ρ Pr( φ ) � s := ρ t ∈ R P ( s ) � t 1 + t 2 � s := � t 1 � s + � t 2 � s 12/32

More semantics Here R α can be inductively defined by R ⊤ I := { ( s , t ) | t = ( U I ⊗ Id A\ I )( s ) for some U I ∈ U I } , R φ ? := R P , where P = ∼∼ � φ � , R U := R U , R U † := R c U , R α ∪ β := R α ∪ R β , and R α ; β := R α ; R β . 13/32

Probabilistic abbreviations � n k =1 a k Pr( φ k ) := a 1 Pr( φ 1 ) + · · · + a n Pr( φ n ) ρ � n � n k =1 a k Pr( φ k ) := k =1 ρ a k Pr( φ k ) t < ρ := ¬ t ≥ ρ t ≤ ρ := − t ≥ − ρ t = ρ := t ≥ ρ ∧ t ≤ ρ t 1 ≥ t 2 := t 1 − t 2 ≥ 0 14/32

Abbreviations ∼ φ := � ¬ φ (orthocomplement) φ ∨ ψ := ¬ ( ¬ φ ∧ ¬ ψ ) (disjunction) φ ⊔ ψ := ∼ ( ∼ φ ∧ ∼ ψ ) (quantum join) A φ := �� φ (global universal) E φ := ♦♦ φ (global existential) ( φ ≤ ψ ) := A( φ → ψ ) ( φ = ψ ) := A( φ ↔ ψ ) φ ⊥ ψ := φ ≤ ∼ ψ (orthogonal) T ( φ ) := ∼∼ φ = φ (testable) φ I := ⊤ I ∧ �⊤ N \ I � φ ( I -component) φ = I ψ := ( φ ≤ ⊤ I ) ∧ ( ψ ≤ ⊤ I ) ∧ ( φ I = ψ I ) ( I -equivalent) I ( φ ) := ( φ = φ I ) ( I -local) � φ ? � = ρ ψ := Pr( φ ) = ρ ∧ � φ ? � ψ � φ ? � >ρ ψ := Pr( φ ) > ρ ∧ � φ ? � ψ 15/32

Example: Leader Election Protocol Example Setting: There are N agents. Goal: Each should have an equal (1 / N ) chance of being chosen to be the leader. Strategy: Prepare a quantum state that has equal probability of collapsing into any of N basis elements when measured. Solution: This state is the W -state in a 2 N -dimensional Hilbert space (a subsystem for each agent). The basis for the 2 N -dimensional space is the product of the bases { 0 k , 1 k } , for each of the N agents. The k -th agent is associated with the basis element b k = (0 0 ⊗ · · · ⊗ 0 k − 1 ⊗ 1 k ⊗ 0 k +1 ⊗ · · · ⊗ 0 N − 1 ). The W -state is an equally weighted superposition of the b k . E. D’Hondt and P. Panangaden, The Computational Power of the W and GHZ States, Quantum Information and Computation 6 (2006), 173–83. 16/32

Expressing the Leader Election Protocol Let A = { 0 , 1 , . . . , N − 1 } be a finite set of agents. � � Basis( B ) := ( b i = ⊤ ) ∧ ( b i ⊥ b j ) . i � = j i ∈ 2 N � � Separable( B ) := ( b i ≤ ⊤ a ) . a ∈A i ∈ 2 N Let W = { W i | i ∈ { 0 , . . . , N }} ⊆ B . Think of W N as (0 0 ⊗ · · · ⊗ 0 N − 1 ).   � �  . QLE( W ) :=  (( W a ) a � = a ( W N ) a ) ∧ (( W a ) b = b ( W N ) b ) a ∈A b ∈A\ a The correctness of the quantum leader election is expressed by (Pr( W a ) = 1 � Basis( B ) ∧ Separable( B ) ∧ QLE( W ) → E N ) . a ∈A 17/32

Duality result Theorem Each quantum dynamic frame is dual to a Piron lattice. A quantum dynamic frame is a special Kripke frame that satisfies Atomicity, Intersection, Orthocomplement, Adequacy, Repeatability, Partial functionality, Self-adjointness, Proper superposition, Cover law. 18/32

Duality result Theorem Each quantum dynamic frame is dual to a Piron lattice. A quantum dynamic frame is a special Kripke frame that satisfies Atomicity, Intersection, Orthocomplement, Adequacy, Repeatability, Partial functionality, Self-adjointness, Proper superposition, Cover law. 18/32