Model checking quantum Markov chains Yuan Feng, Nengkun Yu, and - PowerPoint PPT Presentation

Model checking quantum Markov chains Yuan Feng, Nengkun Yu, and Mingsheng Ying University of Technology Sydney, Australia, Tsinghua University, China Model checking quantum Markov chains. Journal of Computer and System Sciences 79, 1181-1198, (2013) Reachability of recursive quantum Markov chains. Proceedings of the 38th Int. Symp. on Mathematical Foundations of Computer Science (MFCS’13) 385-396.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Outline Motivation 1 Basic notions from quantum information theory 2 Quantum Markov chain 3 Quantum computation tree logic 4 Algorithm 5 Summary 6

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Outline Motivation 1 Basic notions from quantum information theory 2 Quantum Markov chain 3 Quantum computation tree logic 4 Algorithm 5 Summary 6

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Motivation Quantum mechanics is highly counterintuitive; flaws and errors creep in during the design of quantum programs and quantum protocols. So, it is indispensable to develop techniques of verifying and debugging quantum systems.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Model checking Model-checking is one of the dominant techniques for verification of classical hardware as well as software systems. It has proved mature as witnessed by a large number of successful industrial applications. Quantum model checking???

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Outline Motivation 1 Basic notions from quantum information theory 2 Quantum Markov chain 3 Quantum computation tree logic 4 Algorithm 5 Summary 6

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Probability Theory v.s. Quantum Information Theory Quantum bit: Binary Random Varable X: Unit vector in a 2D Hilbert space | φ � = a 0 | 0 � + a 1 | 1 � , X = 0 or X = 1 a i ∈ C , | a 0 | 2 + | a 1 | 2 = 1 1 | 1 � | φ � 0 | 0 �

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Probability Theory v.s. Quantum Information Theory Evolution: Unitary Matrices Evolution: Stochastic Matrices Preserve l 2 -norm Preserve l 1 -norm p ′ = S · p | φ ′ � = U · | φ � � � � a 0 � � � 1 � � p 0 � � 1 � � 1 1 1 2 ( a 0 + a 1 ) 1 √ √ √ 2 2 2 2 2 = = 1 1 1 1 − 1 1 2 ( a 0 − a 1 ) p 1 √ √ a 1 √ 2 2 2 2 2

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Probability Theory v.s. Quantum Information Theory Measurement: A measurement of | φ � according to a Hermitian operator M = ∑ i λ i | b i �� b i | is a projection onto the orthonormal vectors | b i � , and Observation: Pr [ outcome is λ i ] = |� φ | b i �| 2 . Pr ( X = b ) = p b , | φ � | 1 � p b ∈ [ 0, 1 ] | φ � ⊥ | 0 �

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Density operators Mixed state: Classical distribution over (pure) quantum states.  | φ 1 � ,  with probability p 1  . . . . ρ = . .   | φ k � , with probability p k { p i : | φ i �} . Ensemble: ρ = ∑ k i = 1 p i | φ i �� φ i | (hermitian, Density operator: trace 1, positive) Contains all information about the state. Different ensembles can have the same density operator.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Density operators Different ensembles can have the same density operator. � 1 1 2 ( | 0 � − | 1 � ) , √ w.p. 2 = 1 | 0 � , w.p. 2  √ 2 | 0 � − 1 3 1  2 | 1 � , √ w.p. � �  3 − 1 3 3 1 4 4 | 0 � , 4 ( 1 − 3 ) = w.p. √ − 1 1   4 4 1 1 | 1 � , 4 ( 1 − 3 ) w.p. √

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Super-operators and Kraus theorem Super-operators: (special) mapping from density operators to density operators. A map E is a Kraus representation theorem: super-operator if and only if d E i ρ E † ∑ E ( ρ ) = i i = 1 for some set of matrices { E i , i = 1, . . . , d } with ∑ i E † i E i ≤ I . Special case: ρ → U ρ U † Unitary transformation: Measurement with outcome i : ρ → | b i �� b i | ρ | b i �� b i | Measurement with reading outcome: ρ → ∑ i | b i �� b i | ρ | b i �� b i |

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Matrix representation of super-operators Let E = { E i : i ∈ I } be a super-operator. The matrix representation of E is defined as M E = ∑ E i ⊗ E ∗ i . i ∈ I Here the complex conjugate is taken according to the orthonormal basis {| k � : k ∈ K } . It is easy to check that M E is independent of the choice of orthonormal basis and the Kraus operators E i .

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Outline Motivation 1 Basic notions from quantum information theory 2 Quantum Markov chain 3 Quantum computation tree logic 4 Algorithm 5 Summary 6

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Markov chains A Markov chain (MC) is a tuple ( S , P ) where S is a countable set of states; P : S × S → [ 0, 1 ] such that for each s ∈ S , ∑ P ( s , t ) = 1, t ∈ S or equivalently, P ( s , · ) is a probabilistic distribution over S .

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Quantum Markov chains ( S , P ) ⇒ ( H , E ) ⇒ Hilbert space H Set S ⇒ Prob. distributions Density operators ⇒ P : Dist ( S ) → Dist ( S ) E : D ( H ) → D ( H )

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Obstacles for model checking quantum system The set of all possible quantum states, H , is a continuum, even when it is finite dimensional. The techniques of classical model checking, which normally work for finite state spaces, cannot be applied directly.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary In this talk, we propose... A super-operator weighted Markov chain model which aims at providing finite models for general quantum programs and quantum communication protocols. A quantum extension QCTL of the logic PCTL to descibe properties we are interested in for QMCs. An algorithm to model check logic formulas in QCTL against a QMC model.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Some more notations Let SO ( H ) be the set of super-operators on H , ranged over by E , F , · · · . Definition Let E , F ∈ SO ( H ) . 1 E ⊑ F if for any ρ ∈ D ( H ) , F ( ρ ) − E ( ρ ) is positive semi-definite; 2 E � F if for any ρ ∈ D ( H ) , tr ( E ( ρ )) ≤ tr ( F ( ρ )) . Let � be � ∩ � ; it is obviously an equivalence relation.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Some notations Let SI ( H ) = {E ∈ SO ( H ) : E � I H } be the ‘quantum’ correspondence of the unit interval [ 0, 1 ] for real numbers.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Quantum Markov chains A super-operator weighted Markov chain, or quantum Markov chain (QMC), over H is a tuple ( S , Q , AP , L ) , where S is a countable set of states; Q : S × S → SI ( H ) such that for each s ∈ S , ∑ t ∈ S Q ( s , t ) � I H , AP is a finite set of atomic propositions; L is a mapping from S to 2 AP . A classical Markov chain may be viewed as a degenerate quantum Markov chain in which all super-operators appear in the transition matrix have the form p I H for some 0 ≤ p ≤ 1.

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Example: quantum loop A simple quantum loop program goes as follows: q : = F ( q ) l 0 : l 1 while M [ q ] do : q : = E ( q ) l 2 : l 3 : od where M = λ 0 | 0 �� 0 | + λ 1 | 1 �� 1 | .

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Example: quantum loop l 0 F q l 1 E q E 0 q E 1 q l 2 I l 3 Here E 0 q = {| 0 � q � 0 |} and E 1 q = {| 1 � q � 1 |} .

Motivation Basic notions from QIP Quantum Markov chain Quantum computation tree logic Algorithm Summary Outline Motivation 1 Basic notions from quantum information theory 2 Quantum Markov chain 3 Quantum computation tree logic 4 Algorithm 5 Summary 6