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Toward automatic verification of quantum programs

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Formal Aspects of Computing

Abstract

This paper summarises the results obtained by the author and his collaborators in a program logic approach to the verification of quantum programs, including quantum Hoare logic, invariant generation and termination analysis for quantum programs. It also introduces the notion of proof outline and several auxiliary rules for more conveniently reasoning about quantum programs. Some problems for future research are proposed at the end of the paper.

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References

  1. Altenkirch T, Grattage J (2005) A functional quantum programming language. In: Proceedings of the 20th IEEE symposium on logic in computer science (LICS), pp 249–258

  2. Ambainis A, Bach E, Nayak A, Vishwanath A, Watrous J (2001) One-dimensional quantum walks. In: Proceedings of the 33rd ACM symposium on theory of computing (STOC), pp 37–49

  3. Apt, K.R., de Boer, F.S., Olderog, E.-R.: Verification of sequential and concurrent programs. Springer, London (2009)

    Book  MATH  Google Scholar 

  4. Ardeshir-Larijani E, Gay SJ, Nagarajan R (2014) Verification of concurrent quantum protocols by equivalence checking. In: Proceedings of the 20th international conference on tools and algorithms for the construction and analysis of systems (TACAS), pp 500–514

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Barthe G, Fournet C, Grégoire B, Strub P-Y, Swamy N, Zanella Béguelin S (2014) Probabilistic relational verification for cryptographic implementations. In: Proceedings of the 41st annual ACM symposium on principles of programming languages (POPL), pp 193–206

  7. Barthe G, Köpf B, Olmedo F, Zanella Béguelin Z (2013) Probabilistic relational reasoning for differential privacy. In: ACM transactions on programming languages and systems, vol 35, No. 9

  8. Brunet, O., Jorrand, P.: Dynamic quantum logic for quantum programs. Int J Quantum Inf 2, 45–54 (2004)

    Article  MATH  Google Scholar 

  9. Chadha, R., Mateus, P., Sernadas, A.: Reasoning about imperative quantum programs. Electron Notes Theor Comput Sci 158, 19–39 (2006)

    Article  MATH  Google Scholar 

  10. Chakarov A, Sankaranarayanan S (2013) Probabilistic program analysis with martingales. In: Proceedings of the 25th international conference on computer aided verification (CAV). Springer LNCS 8044, pp 511–526

  11. Chatterjee K, Fu HF, Novotný P, Hasheminezhad R (2016) Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs. In: Proceedings of the 43rd annual ACM symposium on principles of programming languages (POPL), pp 327–342

  12. Cleve, R., Buhrman, H.: Substituting quantum entanglement for communication. Phys Rev A 56, 1201–1204 (1997)

    Article  Google Scholar 

  13. Colón MA, Sankaranarayanan S, Sipma HB (2003) Linear invariant generation using non-linear constraint solving. In: Proceedings of the 15th international conference on computer aided verification (CAV). Springer LNCS, pp 420–433

  14. Dale, H., Jennings, D., Rudolph, T.: Provable quantum advantage in randomness processing. Nat Commun 6, 8203 (2015)

    Article  Google Scholar 

  15. D'Hondt, E., Panangaden, P.: Quantum weakest preconditions. Math Struct Comput Sci 16, 429–451 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feng, Y., Duan, R.Y., Ji, Z.F., Ying, M.S.: Proof rules for the correctness of quantum programs. Theor Comput Sci 386, 151–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng Y, Hahn EM, Turrini A, Zhang LJ (2015) QPMC: a model checker for quantum programs and protocols. In: Proceedings of the 20th international symposium on formal methods (FM). Springer LNCS 9109, pp 265–272

  18. Feng Y, Ying MS (2015) Toward automatic verification of quantum cryptographic protocols. In: Proceedings of the 26th international conference on concurrency theory (CONCUR), pp 441–455

  19. Feng, Y., Yu, N.K., Ying, M.S.: Model checking quantum Markov chains. J Comput Syst Sci 79, 1181–1198 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fioriti LMF, Hermanns H (2015) Probabilistic termination: soundness, completeness, and compositionality. In: Proceedings of the 42nd annual ACM symposium on principles of programming languages (POPL), pp 489–501

  21. Gay SJ, Papanikolaou N, Nagarajan R (2008) QMC: a model checker for quantum systems. In: Proceedings of the 20th international conference on computer aided verification (CAV). Springer LNCS 5123, pp 543–547

  22. Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J Math Mech 6, 885–893 (1957)

    MathSciNet  MATH  Google Scholar 

  23. Gorelick GA (1975) A complete axiomatic system for proving assertions about recursive and non-recursive programs. Technical Report, Department of Computer Science, University of Toronto

  24. Green A, Lumsdaine PL, Ross NJ, Selinger P, Valiron B (2013) Quipper: a scalable quantum programming language. In: Proceedings of the 34th ACM conference on programming language design and implementation (PLDI), pp 333–342

  25. Grover LK (1997) Quantum telecomputation. arXiv:quant-ph/9704012

  26. Harel D (1979) First-order dynamic logic, LNCS 68. Springer

  27. Den Hartog, J., de Vink, E.P.: Verifying probabilistic programs using a Hoare like logic. Int J Found Comput Sci 13, 315–340 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leymann, T.: Hoare logic and auxiliary variables. Formal Asp Comput 11, 541–566 (1999)

    Article  MATH  Google Scholar 

  29. Katoen J-P, McIver A, Meinicke L, Morgan CC (2010) Linear-invariant generation for probabilistic programs—automated support for proof-based methods. In: Proceedings 17th international symposium on static analysis (SAS). Springer LNCS 6337, pp 390–406

  30. Kubota, T., Kakutani, Y., Kato, G., Kawano, Y., Sakurada, H.: Semi-automated verification of security proofs of quantum cryptographic protocols. J Symb Comput 73, 192–220 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. JavadiAbhari, A., Patil, S., Kudrow, D., Heckey, J., Lvov, A., Chong, F.T., Martonosi, M.: ScaffCC: scalable compilation and analysis of quantum programs. Parallel Comput 45, 2–17 (2015)

    Article  Google Scholar 

  32. Kakutani Y (2009) A logic for formal verification of quantum programs. In: Proceedings of the 13th Asian computing science conference (ASIAN 2009). Springer LNCS 5913, pp 79–93

  33. Li YJ, Ying MS (2018) Algorithmic analysis of termination problems for quantum programs. In: Proceedings of the 45th annual ACM symposium on principles of programming languages (POPL), pp 35.1–35.29

  34. Li, Y.J., Yu, N.K., Ying, M.S.: Termination of nondeterministic quantum programs. Acta Inf 51, 1–24 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liu T, Li YJ, Wang SL, et al A theorem prover for quantum Hoare logic and its applications. arXiv:1601.03835

  36. Mateus, P., Sernadas, A.: Weakly complete axiomatization of exogenous quantum propositional logic. Inf Comput 204, 771–794 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. van Meter R, Munro WJ, Nemoto K, Itoh KM (2008) Arithmetic on a distributed-memory quantum multicomputer. ACM J Emerg Technol Comput Syst 3, Art. No. 17

  38. McIver, A., Morgan, C.: Abstraction, refinement and proof for probabilistic systems. Springer, Berlin (2005)

    MATH  Google Scholar 

  39. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  40. Paykin J, Rand R, Zdancewic S (2017) QWIRE: a core language for quantum circuits. In: Proceedings of the 44th annual ACM symposium on principles of programming languages (POPL), pp 846–858

  41. Rand R Verification logics for quantum programs. http://www.cis.upenn.edu/~rrand/wpe.pdf

  42. Sankaranarayanan S, Sipma HB, Manna Z (2004) Non-linear loop invariant generation using Gröbner bases. In: Proceedings of the 31st ACM symposium on principles of programming languages (POPL), pp 318–329

  43. Selinger, P.: Towards a quantum programming language. Math Struct Comput Sci 14, 527–586 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. Serafini A, Mancinians S, Bose S (2006) Distributed quantum computation via optical fibers. Phys Rev Lett 96, Art. No. 010503

  45. Svore K, Geller A, Troyer M, Azariah J, Granade C, Heim B, Kliuchnikov V, Mykhailova M, Paz A, Roetteler M (2018) Q#: enabling scalable quantum computing and development with a high-level DSL. In: Proceedings of the real world domain specific languages workshop 2018, Art. no. 8

  46. Tani S, Kobayashi H, Matsumoto K (2012) Exact quantum algorithms for the leader election problem. ACM Trans Comput Theory 4, Art. No. 1

  47. Temme, K., Osborne, T.J., Vollbrecht, K.G., Poulin, D., Verstraete, F.: Quantum metropolis sampling. Nature 471, 87–90 (2011)

    Article  Google Scholar 

  48. Unruh D Quantum relational Hoare logic. https://arxiv.org/pdf/1802.03188.pdf

  49. Wecker D, Svore KM (2014) LIQUi|>: a software design architecture and domain-specific language for quantum computing. arXiv:1402.4467

  50. Ying MS (2011) Floyd-Hoare logic for quantum programs. ACM Trans Program Lang Syst 39, Art. no. 19

  51. Ying MS (2016) Foundations of quantum programs. Morgan-Kaufmann

  52. Ying MS (2016) Toward automatic verification of quantum programs (extended abstract). In: Proceedings of the 2nd international symposium on dependable software engineering: theories, tools, and applications (SETTA)

  53. Ying, M.S., Chen, J.X., Feng, Y., Duan, R.Y.: Commutativity of quantum weakest preconditions. Inf Process Lett 104, 152–158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Ying, M.S., Feng, Y.: Quantum loop programs. Acta Inf 47, 221–250 (2010)

    Google Scholar 

  55. Ying, M.S., Feng, Y.: A flow-chart language for quantum programming. IEEE Trans Softw Eng 37, 466–485 (2011)

    Article  Google Scholar 

  56. Ying MS, Li YJ, Yu NK, Feng Y (2014) Model-checking linear-time properties of quantum systems. ACM Trans Comput Logic 15, Art. no. 22

  57. Ying MS, Ying SG, Wu XD (2017) Invariants of quantum programs: characterisations and generation. In: Proceedings of the 44th annual ACM symposium on principles of programming languages (POPL), pp 818–832

  58. Ying, M.S., Yu, N.K., Feng, Y., Duan, R.Y.: Verification of quantum programs. Sci Comput Program 78, 1679–1700 (2013)

    Article  Google Scholar 

  59. Ying SG, Feng Y, Yu NK, Ying MS (2013) Reachability analysis of quantum Markov chains. In: Proceedings of the 24th Int Conf Concurr Theory (CONCUR), pp 334–348

  60. Yu NK, Ying MS (2012) Reachability and termination analysis of concurrent quantum programs. In: Proceedings of the 23th international conference on concurrency theory (CONCUR), pp 69–83

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Acknowledgements

The author likes to thank Professors Martin Fränzle, DeepakKapur and NaijunZhan for invitingme to give a talk at SETTA’2016. This work was partly supported by the Australian Research Council (Grant No: DP160101652) and the Key Research Program of Frontier Sciences, Chinese Academy of Sciences.

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Authors and Affiliations

  1. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, NSW, 2007, Australia

    Mingsheng Ying

  2. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Mingsheng Ying

  3. Department of Computer Science and Technology, Tsinghua University, Beijing, China

    Mingsheng Ying

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  1. Mingsheng Ying
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Correspondence to Mingsheng Ying.

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Naijun Zhan, Martin Fraenzle, Deepak Kapur, and Heike Wehrheim

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Ying, M. Toward automatic verification of quantum programs. Form Asp Comp 31, 3–25 (2019). https://doi.org/10.1007/s00165-018-0465-3

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