Observational equivalence using scheduler for quantum processes K. - PowerPoint PPT Presentation

Observational equivalence using scheduler for quantum processes K. Yasuda / T. Kubota / Y. Kakutani Dept. of Computer Science University of Tokyo

Outline 1. Introduction 2. Quantum process calculus qCCS 3. Open bisimulation on qCCS 4. Our equivalence relation » Observational equivalence » Scheduler / Strategy 5. Conclusion 2

Introduction

Introduction | Quantum process calculi Quantum communication protocols • Quantum key distribution: BB84, B92, … • Quantum bit commitment • Quantum oblivious transfer Quantum process calculi » To analyze/verify quantum processes formally » QPAlg, CQP , qCCS, … 4

Introduction | Formal verification Formal verification of quantum protocols ≈ Model Spec. Equivalence Equivalence between processes • (Weak) bisimulation • Barbed congruence 5

Introduction | Motivation Not bisimilar but intuitively equivalent processes Example: • Sends 0 or 1 with the same prob. • Sends + or − with the same prob. » The same density matrix expresses these qubits: 1 2 0 0 + 1 2 1 1 = 1 2 + + + 1 2 − − » Used in Shor & Preskill’s security proof of BB84 [SP’00] 6

Introduction | Motivation Not bisimilar but intuitively equivalent processes Example: [KKKKS‘12 ] • Measures a qubit + + and … 0 0 + + 1 1 • Applies ℰ to a qubit + + and … » ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1 + + 2 1 1 1 2 0 0 + 1 7

Introduction | Motivation To define more intuitive equivalence qCCS [FDY’12] Existing notions of equivalence: • (Weak) bisimulation [FDY’12] • (Weak) open bisimulation [DF’12] • Reduction barbed congruence [DF’12] 8

Quantum process calculus qCCS

Quantum process calculus qCCS | Syntax Quantum processes (classical constructs) Receive classical data Send classical data Nondeterministic choice Parallel composition 10

Quantum process calculus qCCS | Syntax Quantum processes (quantum constructs) Receive qubit Send qubit Applying super-operator Measurement 11

Quantum process calculus qCCS | Semantics State of a process: configuration 𝐷 = 𝑄, 𝜍 » 𝑄 : quantum process » 𝜍 : quantum state (density operator) Operational semantics: labeled transition system Labels: • 𝑑? 𝑤 / 𝑑! 𝑤 : receive/send data 𝑤 using 𝑑 • c ? 𝑟 / c ! 𝑟 : receive/send qubit 𝑟 using c • 𝜐 : internal transition (cannot be observed) 12

Quantum process calculus qCCS | Semantics Example: 𝑟 𝑠 c + 0 13

Quantum process calculus qCCS | Semantics Example: 𝑟 𝑠 c Φ 14

Quantum process calculus qCCS | Semantics Example: 𝑟 𝑠 c Φ 15

Quantum process calculus qCCS | Semantics Example: Probabilistic transition 16

Open bisimulation on qCCS

Open bisimulation on qCCS | Definition ℛ is a (weak) open bisimulation if 𝑄, 𝜍 ℛ 𝑅, 𝜏 ⟹ • 𝑄 and 𝑅 hold the same quantum variables » 𝑟𝑤 𝑄 = 𝑟𝑤 𝑅 • Their environment (states associated with the qubits that 𝑄 and 𝑅 do not hold) are the same » tr 𝑟𝑤 𝑄 𝜍 = tr 𝑟𝑤 𝑅 𝜏 • For any super-operator ℰ acting on the environment, whenever there is some 𝜉 s.t. • (Symmetric condition) Adding/removing 𝜐 transitions ≈ 𝑝 : largest open bisimulation 18

Open bisimulation on qCCS | Example Intuitively equivalent processes [KKKKS‘12] • Measures a qubit + + and … 0 0 + + 1 1 • Applies ℰ to a qubit + + and … » ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1 + + 2 1 1 1 2 0 0 + 1 19

Open bisimulation on qCCS | Example Intuitively equivalent processes » 𝑁 : projective measurement 0 , 1 » ℰ : super-operator ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1 Not open bisimilar 20

Our equivalence relation

Our equivalence relation | Informal definition When are two processes equivalent ? They are observed the same by any attackers » Observable actions = Receiving/sending data » Attackers = Processes They use the same channels with the same prob. whenever they run parallel with any other process 22

Our equivalence relation | Related notions • Barbed congruence » Defined in qCCS [ DF’12] » Coincides with ≈ 𝑝 [DF’12] • Testing equivalence » Not defined in quantum process calculi 23

Our equivalence relation | Informal definition When are two processes equivalent ? They are observed the same by any attackers » Observable actions = Receiving/sending data » Attackers = Processes They use the same channels with the same prob. whenever they run parallel with any other process 24

Our equivalence relation | Solving nondeterminism Processes have nondeterministic transitions Probabilities of using each channel? 25

Our equivalence relation | Solving nondeterminism Schedulers solve nondeterminism Scheduler 𝐺 : configuration → next transition 𝐺 𝐺 𝐺 26

Our equivalence relation | Informal definition When are two processes equivalent ? They are observed the same by any attackers » Observable actions = Receiving/sending data » Attackers = Processes They use the same channels with the same prob. whenever they run parallel with any other process 27

Observational equivalence | Definition 𝑄, 𝜍 , 𝑅, 𝜏 are observationally equivalent ( 𝑄, 𝜍 ≈ 𝑝𝑓 𝑅, 𝜏 ) if • 𝑄 and 𝑅 hold the same qu antum variables • Their environment are the same Attacker • For any process 𝑆 and scheduler 𝐺 , there exists a scheduler 𝐺 ′ s.t. for any channel 𝑑 , if ⟨𝑄| 𝑆, 𝜍 uses 𝑑 w.p. 𝑞 according to 𝐺 , then ⟨𝑅| 𝑆, 𝜏 also uses 𝑑 w.p. 𝑞 according to 𝐺′ • (Symmetric condition) 28

Observational equivalence | Sketch Run parallel with any process 𝑆 29

Observational equivalence | Example Not bisimilar but intuitively equivalent processes » 𝑁 : projective measurement 0 , 1 » ℰ : super-operator ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1 Not observationally equivalent 30

Observational equivalence | Example No schedulers 31

Observational equivalence | Example Schedulers can choose different transitions after measurement Processes are the same ⟹ Schedulers should choose the same transitions 32

Observational equivalence | Strategy Strategies : schedulers with this limitation Strategy 𝐺 : configuration → next transition Not allowed 33

Observational equivalence | Strategy 𝑄, 𝜍 , 𝑅, 𝜏 are observationally equivalent with strategies ( 𝑄, 𝜍 ≈ 𝑝𝑓 𝑅, 𝜏 ) if 𝒕𝒖 • 𝑄 and 𝑅 hold the same qu antum variables • Their environment are the same • For any process 𝑆 and strategy 𝐺 , there exists a strategy 𝐺 ′ s.t. for any channel 𝑑 , if ⟨𝑄| 𝑆, 𝜍 uses 𝑑 w.p. 𝑞 according to 𝐺 , then ⟨𝑅| 𝑆, 𝜏 also uses 𝑑 w.p. 𝑞 according to 𝐺′ • (Symmetric condition) 34

Observational equivalence | Example Not bisimilar but intuitively equivalent processes » 𝑁 : projective measurement 0 , 1 » ℰ : super-operator ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1 Not observationally equivalent Observationally equivalent with strategies 35

Observational equivalence | Comparing with others 𝑡𝑢 ? Relation among ≈ 𝑝 , ≈ 𝑝𝑓 , ≈ 𝑝𝑓 𝑡𝑢 ? ≈ 𝑝 ⊆≈ 𝑝𝑓 ⊆≈ 𝑝𝑓 36

Observational equivalence | Comparing with others 𝑡𝑢 are incomparable ≈ 𝑝 , ≈ 𝑝𝑓 , ≈ 𝑝𝑓 𝑡𝑢 𝐸 𝐷 ≈ 𝑝 𝐸 but 𝐷 ≉ 𝑝𝑓 𝐸, 𝐷 ≉ 𝑝𝑓 ≈ 𝑝 𝑡𝑢 𝐸 but 𝐷 ≈ 𝑝𝑓 𝐸 but 𝐷 ≈ 𝑝𝑓 𝑡𝑢 𝐸 𝐷 ≉ 𝑝 𝐸, 𝐷 ≉ 𝑝𝑓 𝐸 𝐷 ≉ 𝑝 𝐸, 𝐷 ≉ 𝑝𝑓 𝑡𝑢 ≈ 𝑝𝑓 ≈ 𝑝𝑓 37

Conclusion

Conclusion | Summary • Introduce qCCS and open bisimulation ≈ 𝑝 • Define observational equivalence » With schedulers: ≈ 𝑝𝑓 𝑡𝑢 » With strategies: ≈ 𝑝𝑓 𝑡𝑢 • Show motivating examples are ≈ 𝑝𝑓 𝑡𝑢 are incomparable • Show ≈ 𝑝 , ≈ 𝑝𝑓 , ≈ 𝑝𝑓 39

Conclusion | Future work • Formalize our “intuition” » Is observational equivalence really “intuitive”? Artificial? ≈ 𝑝 Motivating example 𝑡𝑢 ≈ 𝑝𝑓 ≈ 𝑝𝑓 40

Conclusion | Future work • Check congruence property » Congruence for parallel compositions holds: 𝑡𝑢 𝑅 ⟹ 𝑄| 𝑆 ≈ 𝑝𝑓 𝑡𝑢 𝑅 |𝑆 𝑄 ≈ 𝑝𝑓 » Does congruence for other constructs hold? 41

Conclusion » Summary • Define observational equivalence • With schedulers ≈ 𝑝𝑓 𝑡𝑢 • With strategies ≈ 𝑝𝑓 𝑡𝑢 • Show motivating examples are ≈ 𝑝𝑓 𝑡𝑢 are incomparable • Show ≈ 𝑝 , ≈ 𝑝𝑓 , ≈ 𝑝𝑓 » Future work • Formalize our “intuition” • Check congruence property 42