Hostname: page-component-77f85d65b8-45ctf Total loading time: 0 Render date: 2026-04-22T16:50:47.891Z Has data issue: false hasContentIssue false
  • English
  • Français

Probability in quantum computation and quantum computational logics: a survey

Published online by Cambridge University Press:  28 March 2014

MARIA LUISA DALLA CHIARA ,
ROBERTO GIUNTINI  and
GIUSEPPE SERGIOLI
MARIA LUISA DALLA CHIARA
Affiliation:
Dipartimento di Filosofia, Università di Firenze, Firenze, via Bolognese 52, I-50139 Firenze, Italy Email: dallachiara@unifi.it
ROBERTO GIUNTINI
Affiliation:
Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, via Is Mirrionis 1, I-09123 Cagliari, Italy Email: giuntini@unica.it; giuseppe.sergioli@gmail.com
GIUSEPPE SERGIOLI
Affiliation:
Dipartimento di Pedagogia, Psicologia, Filosofia, Università di Cagliari, via Is Mirrionis 1, I-09123 Cagliari, Italy Email: giuntini@unica.it; giuseppe.sergioli@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Quantum computation and quantum computational logics give rise to some non-standard probability spaces that are interesting from a formal point of view. In this framework, events represent quantum pieces of information (qubits, quregisters, mixtures of quregisters), while operations on events are identified with quantum logic gates (which correspond to dynamic reversible quantum processes). We investigate the notion of Shi–Aharonov quantum computational algebra. This structure plays the role for quantum computation that is played by σ-complete Boolean algebras in classical probability theory.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aharonov, D. (2003) A simple proof that Toffoli and Hadamard are quantum universal. arXiv:quant-ph/0301040.Google Scholar
Aharonov, D., Kitaev, A. and Nisan, N. (1998) Quantum circuits with mixed states. STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computing, ACM Press 20–30.CrossRefGoogle Scholar
Cattaneo, G., Dalla Chiara, M. L., Giuntini, R. and Leporini, R. (2004) Quantum computational structures. Mathematica Slovaca 54 87–108.Google Scholar
Cignoli, R., D'Ottaviano, I. M. L. and Mundici, D. (2000) Algebraic Foundations of Many-Valued Reasoning, Kluwer.CrossRefGoogle Scholar
Dalla Chiara, M. L., Giuntini, R. and Leporini, R. (2003) Quantum computational logics: A survey. In: Hendricks, V. F. and Malinowski, J. (eds.) Trends in Logic: 50 years of Studia Logica, Kluwer 213–255.Google Scholar
Dalla Chiara, M. L., Giuntini, R. and Leporini, R. (2005) Logics from Quantum Computation. International Journal of Quantum Information 3 293–337.CrossRefGoogle Scholar
Dalla Chiara, M. L., Giuntini, R., Freytes, H., Ledda, A. and Sergioli, G. (2009) The algebraic structure of an approximately universal system of quantum computational gates. Foundations of Physics 39 (6)559–572.CrossRefGoogle Scholar
Dalla Chiara, M. L., Giuntini, R., Ledda, R., Leporini, R. and Sergioli, G. (2010) Entanglement as a semantic resource. Foundations of Physics 40 (9/10)1494–1518.CrossRefGoogle Scholar
Dawson, C. M. and Nielsen, M. A. (2005) The Solovay–Kitaev algorithm. arXiv.org:quant-ph/0505030.Google Scholar
Deutsch, D. (1989) Quantum computational networks. Proceedings of the Royal Society of London A 425 73–90.Google Scholar
Gudder, G. (2003) Quantum computational logics. International Journal of Theoretical Physics 42 39–47.CrossRefGoogle Scholar
Kitaev, A. Y. (1997) Quantum Computations: Algorithms and Error correction. Russian Mathematical Surveys 52 (6)1191–1249.CrossRefGoogle Scholar
Kraus, K. (1983) States, effects and operations, Springer-Verlag.Google Scholar
Ledda, A., Konig, M., Paoli, F. and Giuntini, R. (2006) MV algebras and quantum computation. Studia Logica 82 (2)245–270.CrossRefGoogle Scholar
Nielsen, M. and Chuang, I. (2000) Quantum Computation and Quantum Information, Cambridge University Press.Google Scholar
Shi, Y. (2002) Both Toffoli and controlled-Not need little help to do universal quantum computation. arXiv:quant-ph/0205115.Google Scholar
Toffoli, T. (1980) Reversible computing. In: de Bakker, J. W. and van Leeuwen, J. (eds.) Automata, Languages and Programming, Springer 632–644.CrossRefGoogle Scholar