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Automated Quantum Program Verification in Dynamic Quantum Logic

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Dynamic Logic. New Trends and Applications (DaLí 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14401))

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Abstract

Dynamic Quantum Logic (DQL) has been used as a logical framework for manually proving the correctness of quantum programs. This paper presents an automated approach to quantum program verification at the cost of simplifying DQL to Basic Dynamic Quantum Logic (BDQL). We first formalize quantum states, quantum gates, and projections in bra-ket notation and use a set of laws from quantum mechanics and matrix operations to reason on quantum computation. We then formalize the semantics of BQDL and specify the behavior and desired properties of quantum programs in the scope of BDQL. Formal verification of whether a quantum program satisfies desired properties is conducted automatically through an equational simplification process. We use Maude, a rewriting logic-based specification/programming language, to implement our approach. To demonstrate the effectiveness of our automated approach, we successfully verified the correctness of five quantum protocols: Superdense Coding, Quantum Teleportation, Quantum Secret Sharing, Entanglement Swapping, and Quantum Gate Teleportation, using our support tool.

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References

  1. Akatov, D.: The logic of quantum program verification. Master’s thesis, Oxford University (2005)

    Google Scholar 

  2. Baltag, A., Smets, S.: LQP: the dynamic logic of quantum information. Math. Struct. Comput. Sci. 16(3), 491–525 (2006)

    Article  MathSciNet  Google Scholar 

  3. Baltag, A., Bergfeld, J., Kishida, K., Sack, J., Smets, S., Zhong, S.: PLQP & company: decidable logics for quantum algorithms. Int. J. Theor. Phys. 53(10), 3628–3647 (2014)

    Article  MathSciNet  Google Scholar 

  4. Baltag, A., Smets, S.: Reasoning about quantum information: an overview of quantum dynamic logic. Appli. Sci. 12(9) (2022)

    Google Scholar 

  5. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

    Article  MathSciNet  Google Scholar 

  6. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)

    Article  MathSciNet  Google Scholar 

  7. Bergfeld, J.M., Sack, J.: Deriving the correctness of quantum protocols in the probabilistic logic for quantum programs. Soft. Comput. 21(6), 1421–1441 (2017)

    Article  Google Scholar 

  8. Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., Lloyd, S.: Quantum machine learning. Nature 549(7671), 195–202 (2017)

    Article  Google Scholar 

  9. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 57(4), 823–843 (1936)

    Article  MathSciNet  Google Scholar 

  10. Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.L. (eds.): All About Maude - A High-Performance Logical Framework, How to Specify, Program and Verify Systems in Rewriting Logic. LNCS, vol. 4350. Springer (2007). https://doi.org/10.1007/978-3-540-71999-1

  11. Do, C.M., Ogata, K.: Symbolic model checking quantum circuits in maude. In: The 35th International Conference on Software Engineering and Knowledge Engineering, SEKE 2023 (2023)

    Google Scholar 

  12. Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem. Science 292(5516), 472–475 (2001)

    Article  MathSciNet  Google Scholar 

  13. Gay, S., Nagarajan, R., Papanikolaou, N.: Probabilistic model-checking of quantum protocols. arXiv preprint quant-ph/0504007 (2005)

    Google Scholar 

  14. Gottesman, D., Chuang, I.L.: Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402(6760), 390–393 (1999)

    Article  Google Scholar 

  15. Hardegree, G.M.: The conditional in quantum logic. Synthese, 63–80 (1974)

    Google Scholar 

  16. Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press (2000)

    Google Scholar 

  17. Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009)

    Article  MathSciNet  Google Scholar 

  18. Herman, L., Marsden, E.L., Piziak, R.: Implication connectives in orthomodular lattices. Notre Dame J. Formal Logic 16(3), 305–328 (1975)

    Article  MathSciNet  Google Scholar 

  19. Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)

    Article  MathSciNet  Google Scholar 

  20. Meseguer, J.: Twenty years of rewriting logic. J. Log. Algebraic Methods Program 81(7–8), 721–781 (2012)

    Article  MathSciNet  Google Scholar 

  21. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press (2011)

    Google Scholar 

  22. Paykin, J., Rand, R., Zdancewic, S.: Qwire: a core language for quantum circuits. SIGPLAN Not. 52(1), 846–858 (2017)

    Article  Google Scholar 

  23. Rédei, M.: Quantum logic in algebraic approach. FTPH, vol. 91. Springer (1998). https://doi.org/10.1007/978-94-015-9026-6

  24. Shor, P.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994)

    Google Scholar 

  25. Ying, M.: Floyd-hoare logic for quantum programs. ACM Trans. Program. Lang. Syst. (TOPLAS) 33(6), 1–49 (2012)

    Article  Google Scholar 

  26. Ying, M., Feng, Y.: Quantum loop programs. Acta Informatica 47(4), 221–250 (2010)

    Article  MathSciNet  Google Scholar 

  27. Ying, M., Feng, Y.: Model checking quantum systems – a survey (2018)

    Google Scholar 

  28. Żukowski, M., Zeilinger, A., Horne, M.A., Ekert, A.K.: “Event-ready-detectors” Bell experiment via entanglement swapping. Phys. Rev. Lett. 71, 4287–4290 (1993)

    Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable feedback. The research was supported by JAIST Research Grant for Fundamental Research. The research of the first author was supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP22KJ1483. The research of the second and the third authors was supported by JST SICORP Grant Number JPMJSC20C2 and JSPS KAKENHI Grant Number JP24H03370.

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Authors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Tsubasa Takagi, Canh Minh Do & Kazuhiro Ogata

Authors
  1. Tsubasa Takagi
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  2. Canh Minh Do
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  3. Kazuhiro Ogata
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Corresponding author

Correspondence to Tsubasa Takagi .

Editor information

Editors and Affiliations

  1. Technical University of Denmark, Lyngby, Denmark

    Nina Gierasimczuk

  2. University of Bergen, Bergen, Norway

    Fernando R. Velázquez-Quesada

Appendix

Appendix

1.1 Proof of Theorem 1

Before embarking on the proof of Theorem 1, we show a lemma.

Lemma 1

The following holds for any M:

  1. 1.

    \([\![[\textbf{skip}]A]\!]^M=[\![A]\!]^M\);

  2. 2.

    \([\![[\textbf{abort}]A]\!]^M=\mathcal {H}\);

  3. 3.

    \([\![[\textbf{abort};b]A]\!]^M=[\![[\textbf{abort}]A]\!]^M\);

  4. 4.

    \([\![[\textbf{skip};b]A]\!]^M=[\![[b]A]\!]^M\);

  5. 5.

    \([\![[(a\mathrel {;}b)\mathrel {;}c]A]\!]^M=[\![[a\mathrel {;}(b\mathrel {;}c)]A]\!]^M\);

  6. 6.

    \([\![[(a\cup b)\mathrel {;}c]A]\!]^M=[\![(a\mathrel {;}c)\cup (b\mathrel {;}c)]A]\!]^M\);

  7. 7.

    \([\![[a\mathrel {;}b]A]\!]^M=[\![[a][b]A]\!]^M\);

  8. 8.

    \([\![[a\cup b]A]\!]^M=[\![[a]A]\!]^M\cap [\![[b]A]\!]^M\);

  9. 9.

    \([\![[B?]A]\!]^M=[\![B\rightarrow A]\!]^M\in \mathcal {C}(\mathcal {H})\), where \(B\rightarrow A\) denotes the Sasaki hook [18] defined as \(\lnot (A\wedge \lnot (A\wedge B))\).

Proof

1 to 8 are easy to show. For 9, some knowledge of Hilbert space theory is required. Observe that \([\![[B?]A]\!]^M\) is the inverse image \(P_{[\![B]\!]^M}^{-1}([\![A]\!]^M)\) of \([\![A]\!]^M\) under \(P_{[\![B]\!]^M}\). That is,

$$\begin{aligned}{}[\![[B?]A]\!]^M&=P_{[\![B]\!]^M}^{-1}([\![A]\!]^M)=\{s\in \mathcal {H}:P_{[\![B]\!]^M}(s)\in [\![A]\!]^M\} \\ &=\{s\in \mathcal {H}:P_{[\![A]\!]^M}P_{[\![B]\!]^M}(s)=P_{[\![B]\!]^M}(s)\}. \end{aligned}$$

Therefore, \([\![[B?]A]\!]^M=[\![B\rightarrow A]\!]^M\in \mathcal {C}(\mathcal {H})\) (see [15]).

We use Lemma 1 to prove Theorem 1 without mentioning it.

Proof

We prove by simultaneous structural induction on formulas in BDQL and star-free regular programs. The case \(A=p\in L_0\) is immediate. The cases \(A=\lnot B\) and \(A=B\wedge C\) follow from the basic fact in Hilbert space theory. Thus, we only discuss the case \(A=[a]B\).

  • Case 1 \(a=\textbf{skip}\). We have \([\![[a]B]\!]^M=[\![B]\!]^M\in \mathcal {C}(\mathcal {H})\) by the induction hypothesis \([\![B]\!]^M\in \mathcal {C}(\mathcal {H})\).

  • Case 2 \(a=\textbf{abort}\). We have \([\![[a]B]\!]^M=\mathcal {H}\in \mathcal {C}(\mathcal {H})\).

  • Case 3 \(a=\pi \in \varPi _0\). Observe that \([\![[a]B]\!]^M\) is the inverse image of \([\![B]\!]^M\) under v(a). In other words, \([\![[a]B]\!]^M\) is the image \((v(a)^\dagger )([\![B]\!]^M)\) of \([\![B]\!]^M\) under the adjoint operator \(v(a)^\dagger \) of v(a). Let \(X^\perp \) be the orthogonal complement of a subspace X of \(\mathcal {H}\), and write \(X^{\perp \perp }\) for \((X^\perp )^\perp \). Recall that \(X\in \mathcal {C}(\mathcal {H})\) if and only if \(X^{\perp \perp }=X\). By the induction hypothesis \([\![B]\!]^M\in \mathcal {C}(\mathcal {H})\),

    $$\begin{aligned} ([\![[a]B]\!]^M)^{\perp \perp }&=((v(a)^\dagger )([\![B]\!]^M))^{\perp \perp }=((v(a)^\dagger )(([\![B]\!]^M)^\perp ))^\perp \\ &=(v(a)^\dagger )(([\![B]\!]^M)^{\perp \perp })=(v(a)^\dagger )([\![B]\!]^M)=[\![[a]B]\!]^M. \end{aligned}$$

    Consequently, \([\![[a]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

  • Case 4 \(a=b\mathrel {;}c\). We further split the case with respect to b.

    • Case 4.1 \(b=\textbf{skip}\). \([\![[a]B]\!]^M=[\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\) by the induction hypothesis \([\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

    • Case 4.2 \(b=\textbf{abort}\). \([\![[a]B]\!]^M=[\![[\textbf{abort}]B]\!]^M=\mathcal {H}\in \mathcal {C}(\mathcal {H})\).

    • Case 4.3 \(b=\pi \). \([\![[a]B]\!]^M=[\![[\pi ][c]B]\!]^M\). By the induction hypothesis, \([\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\). Thus, it follows from the similar argument of case 3 above that \([\![[a]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

    • Case 4.4 \(b=b_1\mathrel {;}b_2\).

      $$ [\![[a]B]\!]^M=[\![[b_1\mathrel {;}(b_2\mathrel {;}c)]B]\!]^M=[\![[b_1][b_2\mathrel {;}c]B]\!]^M\in \mathcal {C}(\mathcal {H}) $$

      by the induction hypothesis \([\![[b_1][b_2\mathrel {;}c]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

    • Case 4.5 \(b=b_1\cup b_2\).

      $$ [\![[a]B]\!]^M=[\![[(b_1\mathrel {;}c)\cup (b_2\mathrel {;}c)]B]\!]^M=[\![[b_1\mathrel {;}c]B]\!]^M\cap [\![[b_2\mathrel {;}c]B]\!]^M\in \mathcal {C}(\mathcal {H}) $$

      by the induction hypothesis \([\![[b_1\mathrel {;}c]B]\!]^M,[\![[b_2\mathrel {;}c]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

    • Case 4.6 \(b=C?\).

      $$ [\![[a]B]\!]^M=[\![[C?][c]B]\!]^M=[\![C\rightarrow [c]B]\!]^M\in \mathcal {C}(\mathcal {H}) $$

      by the induction hypothesis \([\![C]\!]^M,[\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

  • Case 5 \(a=b\cup c\). We have \([\![[a]B]\!]^M=[\![[b]B]\!]^M\cap [\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\) by the induction hypothesis \([\![[b]B]\!]^M,[\![[c]B]\!]^M\in \mathcal {C}(\mathcal {H})\).

  • Case 6 \(a=C?\). We have \([\![[C?]B]\!]^M=[\![C\rightarrow B]\!]^M\in \mathcal {C}(\mathcal {H})\) by the induction hypothesis \([\![B]\!]^M,[\![C]\!]^M\in \mathcal {C}(\mathcal {H})\).

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Takagi, T., Do, C.M., Ogata, K. (2024). Automated Quantum Program Verification in Dynamic Quantum Logic. In: Gierasimczuk, N., Velázquez-Quesada, F.R. (eds) Dynamic Logic. New Trends and Applications. DaLí 2023. Lecture Notes in Computer Science, vol 14401. Springer, Cham. https://doi.org/10.1007/978-3-031-51777-8_5

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  • DOI: https://doi.org/10.1007/978-3-031-51777-8_5

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