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

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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, …

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Introduction | Formal verification Formal verification of quantum protocols Equivalence between processes

• (Weak) bisimulation
• Barbed congruence

Model Spec.

≈

Equivalence

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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]

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Introduction | Motivation Not bisimilar but intuitively equivalent processes Example: [KKKKS‘12]

• Measures a qubit + + and …
• Applies ℰ to a qubit + + and …

» ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1

+ + 0 0 1 1 + +

1 2 0 0 + 1 2 1 1

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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]

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Quantum process calculus qCCS

Quantum process calculus qCCS | Syntax Quantum processes (classical constructs)

Receive classical data Send classical data Nondeterministic choice Parallel composition

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Quantum process calculus qCCS | Syntax Quantum processes (quantum constructs)

Receive qubit Send qubit Applying super-operator Measurement

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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)

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Quantum process calculus qCCS | Semantics Example:

+ c 𝑟 𝑠

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Quantum process calculus qCCS | Semantics Example:

Φ c 𝑟 𝑠

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Φ

Quantum process calculus qCCS | Semantics Example:

c 𝑟 𝑠

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Quantum process calculus qCCS | Semantics Example: Probabilistic transition

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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

» tr𝑟𝑤 𝑄 𝜍 = tr𝑟𝑤 𝑅 𝜏

• For any super-operator ℰ acting on the environment,

whenever there is some 𝜉 s.t.

• (Symmetric condition)

≈𝑝: largest open bisimulation

Adding/removing 𝜐 transitions

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Open bisimulation on qCCS | Example Intuitively equivalent processes [KKKKS‘12]

• Measures a qubit + + and …
• Applies ℰ to a qubit + + and …

» ℰ 𝜍 = 0 0 𝜍 0 0 + 1 1 𝜍 1 1

+ + 0 0 1 1 + +

1 2 0 0 + 1 2 1 1

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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

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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

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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

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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

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

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Our equivalence relation | Solving nondeterminism Schedulers solve nondeterminism

Scheduler 𝐺: configuration → next transition 𝐺 𝐺 𝐺

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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

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Observational equivalence | Definition 𝑄, 𝜍 , 𝑅, 𝜏 are observationally equivalent ( 𝑄, 𝜍 ≈𝑝𝑓 𝑅, 𝜏 ) if

• 𝑄 and 𝑅 hold the same quantum variables
• Their environment are the same
• 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)

Attacker

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Observational equivalence | Sketch Run parallel with any process 𝑆

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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

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Observational equivalence | Example

No schedulers

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

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Observational equivalence | Strategy Strategies: schedulers with this limitation

Strategy 𝐺: configuration → next transition Not allowed

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Observational equivalence | Strategy 𝑄, 𝜍 , 𝑅, 𝜏 are

• bservationally equivalent with strategies

( 𝑄, 𝜍 ≈𝑝𝑓

𝒕𝒖

𝑅, 𝜏 ) if

• 𝑄 and 𝑅 hold the same quantum 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)

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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

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Observational equivalence | Comparing with others Relation among ≈𝑝, ≈𝑝𝑓, ≈𝑝𝑓

𝑡𝑢 ?

≈𝑝⊆≈𝑝𝑓⊆≈𝑝𝑓

𝑡𝑢 ?

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Observational equivalence | Comparing with others ≈𝑝, ≈𝑝𝑓, ≈𝑝𝑓

𝑡𝑢 are incomparable

≈𝑝 ≈𝑝𝑓 ≈𝑝𝑓

𝑡𝑢

𝐷 ≈𝑝 𝐸 but 𝐷 ≉𝑝𝑓 𝐸, 𝐷 ≉𝑝𝑓

𝑡𝑢 𝐸

𝐷 ≈𝑝𝑓 𝐸 but 𝐷 ≉𝑝 𝐸, 𝐷 ≉𝑝𝑓

𝑡𝑢 𝐸

𝐷 ≈𝑝𝑓

𝑡𝑢 𝐸 but

𝐷 ≉𝑝 𝐸, 𝐷 ≉𝑝𝑓 𝐸

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Conclusion

Conclusion | Summary

• Introduce qCCS and open bisimulation ≈𝑝
• Define observational equivalence

» With schedulers: ≈𝑝𝑓 » With strategies: ≈𝑝𝑓

𝑡𝑢

• Show motivating examples are ≈𝑝𝑓

𝑡𝑢

• Show ≈𝑝, ≈𝑝𝑓, ≈𝑝𝑓

𝑡𝑢 are incomparable

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Conclusion | Future work

• Formalize our “intuition”

» Is observational equivalence really “intuitive”?

≈𝑝 ≈𝑝𝑓 ≈𝑝𝑓

𝑡𝑢

Motivating example Artificial?

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Conclusion | Future work

• Check congruence property

» Congruence for parallel compositions holds: 𝑄 ≈𝑝𝑓

𝑡𝑢 𝑅 ⟹ 𝑄| 𝑆 ≈𝑝𝑓 𝑡𝑢 𝑅 |𝑆

» Does congruence for other constructs hold?

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Conclusion » Summary

• Define observational equivalence
• With schedulers ≈𝑝𝑓
• With strategies ≈𝑝𝑓

𝑡𝑢

• Show motivating examples are ≈𝑝𝑓

𝑡𝑢

• Show ≈𝑝, ≈𝑝𝑓, ≈𝑝𝑓

𝑡𝑢 are incomparable

» Future work

• Formalize our “intuition”
• Check congruence property

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