research-article

Quantum Hoare logic with ghost variables

Author:

Dominique UnruhAuthors Info & Claims

LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science

Article No.: 47, Pages 1 - 13

Published: 08 June 2021 Publication History

Abstract

Quantum Hoare logic allows us to reason about quantum programs. We present an extension of quantum Hoare logic that introduces "ghost variables" to extend the expressive power of pre-/postconditions. Ghost variables are variables that do not actually occur in the program and are allowed to have arbitrary quantum states (in a sense, they are existentially quantified), and be entangled with program variables. Ghost variables allow us to express properties such as the distribution of a program variable or the fact that a variable has classical content. And as a case study, we show how quantum Hoare logic with ghost variables can be used to prove the security of the quantum one-time pad.

References

[1]

Samson Abramsky and Bob Coecke. "A categorical semantics of quantum protocols". In: LICS '04. IEEE, 2004, pp. 415--425.

Digital Library

[2]

Gilles Barthe, Benjamin Grégoire, and Santiago Zanella Béguelin. "Formal Certification of Code-Based Cryptographic Proofs". In: POPL 2009. ACM, 2009, pp. 90--101.

Digital Library

[3]

Nick Benton. "Simple Relational Correctness Proofs for Static Analyses and Program Transformations". In: POPL '04. ACM, 2004, pp. 14--25. isbn: 1-58113-729-X.

Digital Library

[4]

Garrett Birkhoff and John von Neumann. "The logic of quantum mechanics". In: Annals of Mathematics 37.823 (1936).

[5]

P. Oscar Boykin and Vwani Roychowdhury. "Optimal encryption of quantum bits". In: Phys. Rev. A 67 (4 2003), p. 042317.

[6]

R. Chadha, L. Cruz-Filipe, P. Mateus, and A. Sernadas. "Reasoning About Probabilistic Sequential Programs". In: Theor. Comput. Sci. 379.1-2 (July 2007), pp. 142--165. issn: 0304-3975.

Digital Library

[7]

Rohit Chadha, Paulo Mateus, and Amílcar Sernadas. "Reasoning About Imperative Quantum Programs". In: ENTCS 158 (May 2006), pp. 19--39. issn: 1571-0661.

Digital Library

[8]

Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. isbn: 110710422X.

[9]

Ellie D'Hondt and Prakash Panangaden. "Quantum Weakest Preconditions". In: Mathematical. Structures in Comp. Sci. 16.3 (June 2006), pp. 429--451. issn: 0960-1295.

Digital Library

[10]

Edsger W. Dijkstra. "Guarded commands, nondeterminacy and formal derivation of programs". In: Communications of the ACM 18.8 (Aug. 1975), pp. 453--457. issn: 0001-0782.

Digital Library

[11]

Yuan Feng, Runyao Duan, Zhengfeng Ji, and Mingsheng Ying. "Proof rules for the correctness of quantum programs". In: Theoretical Computer Science 386.1 (2007), pp. 151--166. issn: 0304-3975.

Digital Library

[12]

Lov K. Grover. "A Fast Quantum Mechanical Algorithm for Database Search". In: STOC. 1996, pp. 212--219.

Digital Library

[13]

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. "Quantum Algorithm for Linear Systems of Equations". In: Physical Review Letters 103.15 (2009).

[14]

J. I. den Hartog and E. P. de Vink. "Verifying probabilistic programs using a Hoare like logic". In: International Journal of Foundations of Computer Science 13.03 (2002). Special Issue: 5th Asian Computing Science Conference (ASIAN'99), pp. 315--340.

[15]

Charles Antony Richard Hoare. "An axiomatic basis for computer programming". In: Communications of the ACM 12.10 (1969), pp. 576--580. ISSN: 0001-0782.

Digital Library

[16]

Martin Hofmann and Mariela Pavlova. "Elimination of Ghost Variables in Program Logics". In: Trustworthy Global Computing. Ed. by Gilles Barthe and Cédric Fournet. Springer, 2008, pp. 1--20. ISBN: 978-3-540-78663-4.

Digital Library

[17]

Yoshihiko Kakutani. "A Logic for Formal Verification of Quantum Programs". In: ASIAN 2009. Ed. by Anupam Datta. Berlin, Heidelberg: Springer, 2009, pp. 79--93. ISBN: 978-3-642-10622-4.

Digital Library

[18]

Dexter Kozen. "A Probabilistic PDL". In: STOC '83. New York, NY, USA: ACM, 1983, pp. 291--297. ISBN: 0-89791-099-0. URL: http://doi.acm.org/10.1145/800061.808758.

Digital Library

[19]

Annabelle McIver and Carroll Morgan. Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer, 2005.

Digital Library

[20]

Michele Mosca, Alain Tapp, and Ronald de Wolf. Private Quantum Channels and the Cost of Randomizing Quantum Information. arXiv:quant-ph/0003101. Mar. 2000.

[21]

Tobias Nipkow, Larry Paulson, and Markus Wenzel. Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Vol. 2283. LNCS. Springer, 2002.

Digital Library

[22]

Lyle Harold Ramshaw. "Formalizing the Analysis of Algorithms". https://apps.dtic.mil/dtic/tr/fulltext/u2/a086916. PhD thesis. Stanford University, 1979.

[23]

Peter W. Shor. "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". In: SIAM Review 41.2 (1999), pp. 303--332. URL: https://doi.org/10.1137/s0036144598347011.

Digital Library

[24]

Dominique Unruh. Quantum Hoare Logic with Ghost Variables. arXiv:1902.00325 [quant-ph]. Full version of this paper. 2019.

Digital Library

[25]

Dominique Unruh. "Quantum relational Hoare logic". In: Proc. ACM Program. Lang. (POPL proceedings) 3 (2019). Full version is arXiv:1802.03188 [quant-ph], 33:1--33:31. ISSN: 2475-1421.

Digital Library

[26]

M. S. Ying, R. Y. Duan, Y. Feng, and Z. F. Ji. In: Semantic Techniques in Quantum Computation. Ed. by S. Gay and I. Mackie. Cambridge University Press, 2010. Chap. Predicate transformer semantics of quantum programs, pp. 311--360.

[27]

Mingsheng Ying. "Floyd-Hoare Logic for Quantum Programs". In: ACM Trans. Program. Lang. Syst. 33.6 (2012), 19:1--19:49. ISSN: 0164-0925.

Digital Library

Cited By

Feng YYing M(2021)Quantum Hoare Logic with Classical VariablesACM Transactions on Quantum Computing10.1145/34568772:4(1-43)Online publication date: 21-Dec-2021
https://dl.acm.org/doi/10.1145/3456877
Zhou LBarthe GHsu JYing MYu NGorla D(2021)A quantum interpretation of bunched logic & quantum separation logicProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470673(1-14)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470673

Index Terms

Quantum Hoare logic with ghost variables
1. Software and its engineering
2. Theory of computation
  1. Logic
  2. Semantics and reasoning
    1. Program reasoning

Index terms have been assigned to the content through auto-classification.

Recommendations

An applied quantum Hoare logic
PLDI 2019: Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation

We derive a variant of quantum Hoare logic (QHL), called applied quantum Hoare logic (aQHL for short), by: 1. restricting QHL to a special class of preconditions and postconditions, namely projections, which can significantly simplify verification of ...
Quantum Hoare Logic with Classical Variables
Hoare logic provides a syntax-oriented method to reason about program correctness and has been proven effective in the verification of classical and probabilistic programs. Existing proposals for quantum Hoare logic either lack completeness or support ...
Quantum relational Hoare logic

We present a logic for reasoning about pairs of interactive quantum programs – quantum relational Hoare logic (qRHL). This logic follows the spirit of probabilistic relational Hoare logic (Barthe et al. 2009) and allows us to formulate how the outputs of ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science

June 2019

784 pages

Conference Chair:
Patricia Bouyer
CNRS, France

Sponsors

SIGLOG: ACM Special Interest Group on Logic and Computation

In-Cooperation

EACSL: European Association for Computer Science Logic
IEEE-CS\DATC: IEEE Computer Society

Publisher

IEEE Press

Publication History

Published: 08 June 2021

Check for updates

Qualifiers

Research-article

Conference

LICS '19

Sponsor:

SIGLOG

LICS '19: 34th Annual ACM/IEEE Symposium on Logic in Computer Science

June 24 - 27, 2019

Vancouver, Canada

Acceptance Rates

Overall Acceptance Rate 215 of 622 submissions, 35%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
16
Total Downloads

Downloads (Last 12 months)7
Downloads (Last 6 weeks)3

Reflects downloads up to 05 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Feng YYing M(2021)Quantum Hoare Logic with Classical VariablesACM Transactions on Quantum Computing10.1145/34568772:4(1-43)Online publication date: 21-Dec-2021
https://dl.acm.org/doi/10.1145/3456877
Zhou LBarthe GHsu JYing MYu NGorla D(2021)A quantum interpretation of bunched logic & quantum separation logicProceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science10.1109/LICS52264.2021.9470673(1-14)Online publication date: 29-Jun-2021
https://dl.acm.org/doi/10.1109/LICS52264.2021.9470673

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents