Abstract
Information processing in the physical world must be based on a physical system representing the information. If such a system has a quantum physical description, then one talks about quantum information processing. The mathematics needed to handle quantum information is somewhat more complicated than that used for classical information. The purpose of this chapter is to introduce the basic mathematical machinery used to describe quantum systems with finitely many potential observable values. A typical example of such a system is the quantum bit (or qubit, for short), which has at most two potential values for any physical observable. The mathematics for quantum information processing is based on vector spaces over complex numbers, and can be seen as a straightforward generalization of linear algebra of real vector spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aharonov A, Kitaev A, Nisan N (1998) Quantum circuits with mixed states. In: proceedings of the 30th annual ACM symposium on theory of computation, Dallas, TX, May 1998, pp 20–30
Axler S (1997) Linear algebra done right. Springer, New York, http://www.springer.com/mathematics/algebra/book/978-0-387-98259-5
Barenco A, Bennett CH, Cleve R et al. (1995) Elementary gates for quantum computation. Phys Rev A 52(5): 3457–3467
Benioff PA (1980) The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J Stat Phys 22(5):563–591
Benioff PA (1982) Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: application to Turing machines. Int J Theor Phys 21(3/4):177–202
Bennett CH (1973) Logical reversibility of computation. IBM J Res Dev 17:525–532
Bernstein E, Vazirani U (1997) Quantum complexity theory. SIAM J Comput 26(5):1411–1473
Cohn PM (1994) Elements of linear algebra. Chapman & Hall, Boca Raton, FL. CRC Press reprint (1999). http://www.amazon.com/Elements-Linear-Algebra-Chapman-Mathematics/dp/0412552809
Feynman RP (1982) Simulating physics with computers. Int J Theor Phys 21(6/7):467–488
Deutsch D (1985) Quantum theory, the Church-Turing principle and the universal quantum computer. Proc R Soc Lond Ser A Math Phys Sci 400:97–117
Deutsch D (1989) Quantum computational networks. Proc R Soc Lond A 425:73–90
Gleason AM (1957) Measures on the closed subspaces of a Hilbert space. J Math Mech 6:885–893
Grover LK (1996) A fast quantum-mechanical algorithm for database search. In: Proceedings of the 28th annual ACM symposium on the theory of computing, Philadelphia, PA, May 1996, pp 212–219
Hirvensalo M (2004) Quantum computing, 2nd edn. Springer, Heidelberg
Hirvensalo M (2008) Various aspects of finite quantum automata. In: Proceedings of the 12th international conference on developments in language theory, Kyoto, Japan, September 2008. Lecture notes in computer science, vol 5257. Springer, Berlin, pp 21–33
Kondacs A, Watrous J (1997) On the power of quantum finite state automata. In: Proceedings of the 38th annual symposium on foundations of computer science, Miami Beach, FL, October 1997, pp 66–75
Moore C, Crutchfield JP (2000) Quantum automata and quantum grammars. Theor Comput Sci 237(1–2): 275–306
Nielsen MA, Chuang IL (2000) Quantum computation and quantum information. Cambridge University Press
Ozawa M (1984) Quantum measuring processes of continuous observables. J Math Phys 25:79–87
Papadimitriou CH (1994) Computational complexity. Addison-Wesley, Reading, MA
Paz A (1971) Introduction to probabilistic automata. Academic, New York
Shi Y (2003) Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Info Comput 3(1):84–92
Shor PW (1994) Algorithms for quantum computation: discrete log and factoring. In: Proceedings of the 35th annual IEEE symposium on foundations of computer science, Santa Fe, NM, November 1994, pp 20–22
Simon DR (1994) On the power of quantum computation. In: Proceedings of the 35th annual IEEE symposium on foundations of computer science, Santa Fe, NM, November 1994, pp 116–123
Stinespring WF (1955) Positive functions on C *-algebras. Proc Am Math Soc 6:211–216
Stone MH (1932) On one-parameter unitary groups in Hilbert space. Ann Math 33:643–648
Turing AM (1936) On computable numbers, with an application to the Entscheidungsproblem. Proc Lond Math Soc 2(42):230–265
von Neumann J (1927) Thermodynamik quantummechanischer Gesamheiten. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen 1:273–291
von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Springer, Berlin
Yu S (1997) Regular languages. In: Rozenberg G, Salomaa A (eds) Handbook of formal languages. Springer, Berlin
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Hirvensalo, M. (2012). Mathematics for Quantum Information Processing. In: Rozenberg, G., Bäck, T., Kok, J.N. (eds) Handbook of Natural Computing. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92910-9_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-92910-9_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92909-3
Online ISBN: 978-3-540-92910-9
eBook Packages: Computer ScienceReference Module Computer Science and Engineering