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References (70)
-
(1996)
Field Quantization, vol -
M. Postnikov (2001)
Geometry VI: Riemannian Geometry -
(2002)
of Mathematical Surveys and Monographs -
Roel Hospel, Ck -smooth (1999)
Morse Theory -
A. Cañada, P. Drábek, A. Fonda (2004)
Handbook of differential equations -
(1947)
Calculation of the coefficients in the Campbell-Hausdorff formula -
L. Debnath (1997)
Nonlinear Partial Differential Equations for Scientists and Engineers -
N. Varopoulos (1988)
Analysis on Lie groupsJournal of Functional Analysis, 76
-
P. Lax (1968)
Integrals of Nonlinear Equations of Evolution and Solitary Waves -
J. Milnor (1976)
Curvatures of left invariant metrics on lie groupsAdvances in Mathematics, 21
-
R. Abraham, J. Marsden (1987)
Foundations of Mechanics, Second Edition, 364
-
J. Campbell (1897)
On a Law of Combination of Operators (Second Paper)Proceedings of The London Mathematical Society
-
T. Miwa, M. Jimbo, E. Date (2000)
Solitons -
J. Jost (1995)
Riemannian geometry and geometric analysis -
(2002)
A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Vol -
M. Nielsen, M. Dowling, M. Gu, A. Doherty (2006)
Quantum Computation as GeometryScience, 311
-
F. Hausdorff (2001)
Die symbolische Exponentialformel in der Gruppentheorie -
J. Cornwell, W. Buck (1984)
Group Theory in Physics: An Introduction -
B. Hall (2003)
Lie Groups, Lie Algebras, and Representations -
J. Jost (2008)
Riemannian Geometry and Geometric Analysis, 5th Edition -
F. Murnaghan (1938)
The theory of group representations -
C. Moseley (2004)
Geometric control of quantum spin systems, 5436
-
J. Cornwell (1984)
Group theory in physics -
(2004)
Tensors and Manifolds, 2nd edn -
M.M. Postnikov (2001)
Geometry VI: Riemannian Geometry, Encyclopedia of Mathematical Sciences, vol. 91 -
N. Khaneja, S. Glaser, R. Brockett (2003)
Erratum: Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer [Phys. Rev. A 65, 032301 (2002)]Physical Review A, 71
-
J. Belinfante, B. Kolman (1969)
An Introduction to Lie Groups and Lie Algebras, with Applications. IIISiam Review, 11
-
M.R. Sepanski (2007)
Compact Lie Groups -
R. Malley, R. Osserman (2005)
A Panoramic View of Riemannian Geometry. -
H. Baker
Alternants and Continuous GroupsProceedings of The London Mathematical Society
-
John Lee (1997)
Riemannian Manifolds: An Introduction to Curvature -
F. Brown (1940)
The Theory of Group RepresentationsNature, 146
-
J. Faraut (2008)
Analysis on Lie Groups: The groups SU (2) and SO (3), Haar measures and irreducible representations -
N. Khaneja, S. Glaser, R. Brockett (2001)
Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transferPhysical Review A, 65
-
(2006)
Riemannian Geometry -
M. Nielsen (2005)
A geometric approach to quantum circuit lower boundsQuantum Inf. Comput., 6
-
Friedrich Sauvigny (2012)
Nonlinear Partial Differential Equations -
C. Reutenauer (1993)
Free Lie Algebras -
S. Weigert (1997)
BAKER-CAMPBELL-HAUSDORFF RELATION FOR SPECIAL UNITARY GROUPS SU(N)Journal of Physics A, 30
-
J.F. Cornwell (1984)
Group Theory in Physics, vols. 1 & 2 -
W. Pfeifer (2003)
The Lie algebras su(N) -
M. Nielsen, M. Dowling, M. Gu, A. Doherty (2006)
Optimal control, geometry, and quantum computingPhysical Review A, 73
-
E. Zeidler (1997)
Nonlinear Functional Analysis and its Applications: IV: Applications to Mathematical Physics -
E. Zeidler (1988)
Nonlinear functional analysis and its applications -
R. Wasserman (2004)
Tensors and Manifolds -
R. Horn, C. Johnson (1991)
Topics in Matrix Analysis, vol. 255 -
R. Kaushal, D. Parashar (2000)
Advanced Methods of Mathematical Physics -
L. Conlon (2001)
Differentiable Manifolds -
G. Giorgadze (2013)
Geometry of quantum computation -
J. Stillwell (2008)
Naive Lie Theory -
V. Arnold, B. Khesin (1998)
Topological methods in hydrodynamics -
B. Hou, B. Hou (1997)
Differential geometry for physicists -
V. Arnold (1974)
Mathematical Methods of Classical Mechanics -
M.A. Naimark, A.I. Stern (1982)
Theory of Group Representations -
H. Brandt (2009)
Riemannian geometry of quantum computationNonlinear Analysis-theory Methods & Applications, 71
-
Proceedings of the London Mathematical SocietyNature, 23
-
H. Brandt (2010)
Riemannian curvature in the differential geometry of quantum computationPhysica E-low-dimensional Systems & Nanostructures, 42
-
W. Steeb, Y. Hardy (2004)
Problems and Solutions in Quantum Computing and Quantum Information -
P. Fox (1918)
THE FOUNDATIONS OF MECHANICS.Science, 48 1240
-
C.G. Moseley (2004)
Quantum Information and Computation II, Proceedings of the SPIE vol. 5436 -
S. Weinberg (1995)
The Quantum Theory of Fields: THE CLUSTER DECOMPOSITION PRINCIPLE -
J. Clelland, C. Moseley (2004)
Sub-Finsler geometry in dimension threeDifferential Geometry and Its Applications, 24
-
R. Montgomery (2002)
A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Vol. 91 of Mathematical Surveys and Monographs -
R. Horn, Charles Johnson (1991)
Topics in Matrix Analysis -
A.A. Sagle, R.E. Walde (1973)
Introduction to Lie Groups and Lie Algebras -
(1973)
Gravitation, pp -
W. Greiner, J. Reinhardt (1996)
Field Quantization, vol. 219 -
Yazhen Wang (2012)
Quantum Computation and Quantum InformationStatistical Science, 27
-
S. Weinberg (1995)
The Quantum Theory of Fields I, vol. 143 -
J. Campbell (1896)
On a Law of Combination of Operators bearing on the Theory of Continuous Transformation GroupsProceedings of The London Mathematical Society
- Publisher
- Springer Journals
- Copyright
- Copyright © 2011 by Springer Science+Business Media, LLC (outside the USA)
- Subject
- Physics; Theoretical, Mathematical and Computational Physics; Mathematics, general; Quantum Physics; Physics, general; Computer Science, general
- ISSN
- 1570-0755
- eISSN
- 1573-1332
- DOI
- 10.1007/s11128-011-0290-6
- Publisher site
- See Article on Publisher Site
Abstract
An introduction is first given of recent developments in the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation, and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
Journal
Quantum Information Processing – Springer Journals
Published: Sep 16, 2011