Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T06:22:26.387Z Has data issue: false hasContentIssue false

5 - The Quantum Walk of F. Riesz

Published online by Cambridge University Press:  05 December 2012

F. A. Grünbaum
Affiliation:
University of California
L. Velázquez
Affiliation:
Universidad de Zaragoza
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2012

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.)

References

[1] A., Ambainis, Quantum walks and their algorithmic applications. International Journal of Quantum Information, 1, (2003) 507–518.Google Scholar
[2] A., Ambainis, E., Bach, A., Nayak, A., Vishwanath and J., Watrous, One dimensional quantum walks. in Proc. of the ACM Symposiumon Theory and Computation (STOC'01), July 2001, ACM, NY, 2001, 37–49.
[3] O., Bourget, J. S., Howland, and A., Joye, Spectral analysis of unitary band matrices. Commun. Math. Phys., 234, (2003) 191–227.Google Scholar
[4] M. J., Cantero, L., Moral and L., Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl., 362, (2003) 29–56.
[5] M. J., Cantero, L., Moral and L., Velázquez, Minimal representations of unitary operators and orthogonal polynomials on the unit circle. Linear Algebra Appl., 405, (2005) 40–65.
[6] M. J., Cantero, F. A., Grünbaum, L., Moral and L., Velázquez, Matrix valued Szego polynomials and quantum random walks. Commun. Pure Applied Math., 58, (2010) 464–507.Google Scholar
[7] Ya. L., Geronimus, On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur functions. Mat. Sb., 15, (1944) 99–130. [Russian]Google Scholar
[8] C. G., Graham and O. C., McGehee, Essays in Commutative Harmonic Analysis. Springer-Verlag, 1979, Chapter 7.
[9] F. A., Grünbaum, L., Velazquez, A. Werner and R., Werner, Recurrence for discrete time unitary evolutions. In preparation.
[10] F. A., Grünbaum and L., Velazquez, The Riesz quantum walk on the integers. In preparation.
[11] Y., Katznelson, An Introduction to Harmonic Analysis. John Wiley & Sons, 1968.
[12] J., Kempe, Quantum random walks-an introductory overview. Contemporary Physics, 44, (2003) 307–327.Google Scholar
[13] N., Konno, Quantum walks. in Quantum Potential Theory, U., Franz, M., Schürmann, editors, Lecture Notes in Mathematics 1954, Springer-Verlag, Berlin, Heidelberg, 2008.
[14] A., Magnus, Freund equation for Legendre polynomials on a circular arc and solution to the Grünbaum-Delsarte-Janssen-deVries problem. J. Approx. Theory, 139, (2006) 75–90.Google Scholar
[15] D., Meyer, From quantum cellular automata to quantum lattice gases. J. Stat. Physics, 85, (1996) 551–574, quant-ph/9604003.Google Scholar
[16] A., Nayak and A., Vishwanath, Quantum walk on the line. Center for Discrete Mathematics & Theoretical Computer Science, 2000, quant-ph/0010117.
[17] G., Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Annalen, 84, (1921) 149–160.Google Scholar
[18] F., Riesz, Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung. Math. Z., 18, (1918) 312–315.Google Scholar
[19] F., Riesz and B., Sz-Nagy, Functional Analysis. F. Ungar Publishing, New York, 1955.
[20] W., Rudin, Real and Complex Analysis, second edition. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974.
[21] I., Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math., 147, (1916) 205–232 and 148, (1917) 122–145.Google Scholar
[22] B., Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. AMS Colloq. Publ., vol. 54.1, AMS, Providence, RI, 2005.
[23] B., Simon, CMV matrices: Five years after. J. Comput. Appl. Math., 208, (2007) 120–154.Google Scholar
[24] M., Stefanak, T., Kiss and I., Jex, Recurrence properties of unbiased coined quantum walks on infinite d dimensional lattices. arXiv: 0805.1322v2 [quant-ph] 4 Sep 2008.
[25] G., Szego, Orthogonal Polynomials, 4th ed. AMS Colloq. Publ., vol. 23, AMS, Providence, RI, 1975.
[26] H., Wall, Continued fractions and bounded analytic functions. Bull. Amer. Math. Soc., 50, (1944) 110–119.Google Scholar
[27] D. S., Watkins, Some perspectives on the eigenvalue problem. SIAM Rev., 35, (1993) 430–471.Google Scholar
[28] A., Zygmund, Trigonometric Series, 2nd ed. Cambridge University Press, 1959.

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×