Abstract
Entanglement is a non local property of quantum states which has no classical counterpart and plays a decisive role in quantum information theory. Several protocols, like the teleportation, are based on quantum entangled states. Moreover, any quantum algorithm which does not create entanglement can be efficiently simulated on a classical computer. The exact role of the entanglement is nevertheless not well understood. Since an exact analysis of entanglement evolution induces an exponential slowdown, we consider approximative analysis based on the framework of abstract interpretation. In this paper, a concrete quantum semantics based on superoperators is associated with a simple quantum programming language. The representation of entanglement, i.e. the design of the abstract domain is a key issue. A representation of entanglement as a partition of the memory is chosen. An abstract semantics is introduced, and the soundness of the approximation is proven.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramsky, S.: A Cook’s tour of a simple quantum programming language. In: 3rd International Symposium on Domain Theory, Xi’an, China (May 2004)
Aspect, A., Grangier, P., Roger, G.: Experimental tests of realistic local theories via Bell’s theorem. Phys. Rev. Lett. 47, 460 (1981)
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)
Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 53–85 (1999)
Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL, pp. 238–252 (1977)
Eggeling, T., Werner, R.F.: Separability properties of tripartite states with uuu -symmetry. Phys. Rev. A 63(0421111) (2001)
Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of reality be considered complete? Phys. Rev. 47(10), 777–780 (1935)
Gay, S.J.: Quantum programming languages: Survey and bibliography. Mathematical Structures in Computer Science 16(4) (2006)
Gay, S.J., Rajagopal, A.K., Papanikolaou, N.: Qmc: A model qmc: A model checker for quantum systems. arxiv:0704.3705 (2007)
Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In: Proceedings of the 35-th ACM Symposium on Theory of Computing, p. 10. ACM Press, New York (2003)
Jorrand, P., Mhalla, M.: Separability of pure n-qubit states: two characterizations. IJFCS 14(5), 797–814 (2003)
Kashefi, E.: Quantum domain theory - definitions and applications. In: Proceedings of Computability and Complexity in Analysis (CCA 2003) (2003)
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, New York (2000)
Perdrix, S.: Formal models of quantum computation: resources, abstract machines and measurement-based quantum computation (in french). PhD thesis, Institut National Polytechnique de Grenoble (2006)
Perdrix, S.: A hierarchy of quantum semantics. In: The Proceedings of the 3rd International Workshop on Development of Computational Models (to appear, 2007)
Prost, F., Zerrari, C.: A logical analysis of entanglement and separability in quantum higher-order functions. arXiv.org:0801.0649 (2008)
Selinger, P.: A brief survey of quantum programming languages. In: Kameyama, Y., Stuckey, P.J. (eds.) FLOPS 2004. LNCS, vol. 2998, pp. 1–6. Springer, Heidelberg (2004)
Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14(4), 527–586 (2004)
Shor, P.: Algorithms for quantum computation: Discrete logarithms and factoring. In: Goldwasser, S. (ed.) Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society Press, Los Alamitos (1994)
Van den Nest, M., Miyake, A., Dür, W., Briegel, H.J.: Universal resources for measurement–based quantum computation (2006)
White, A.G., Gilchrist, A., Pryde, G.J., O’Brien, J.L., Bremner, M.J., Langford, N.K.: Measuring two-qubit gates. J. Opt. Soc. Am. B 24(2), 172–183 (2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Perdrix, S. (2008). Quantum Entanglement Analysis Based on Abstract Interpretation. In: Alpuente, M., Vidal, G. (eds) Static Analysis. SAS 2008. Lecture Notes in Computer Science, vol 5079. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69166-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-69166-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69163-1
Online ISBN: 978-3-540-69166-2
eBook Packages: Computer ScienceComputer Science (R0)