Quantum computation as geometry

Science

Volumn 311, Issue 5764, 2006, Pages 1133-1135

  Nielsen, Michael A ^a   Dowling, Mark R ^a   Gu, Mile ^a   Doherty, Andrew C ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTERS; PROBLEM SOLVING;

QUANTUM CIRCUITS; QUANTUM COMPUTERS; RIEMANNIAN GEOMETRY;

QUANTUM THEORY;

PHYSICS;

ALGORITHM; ARTICLE; COMPUTER; GEOMETRY; MATHEMATICAL COMPUTING; PRIORITY JOURNAL; QUANTUM MECHANICS;

EID: 33644544118     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.1121541     Document Type: Article

References (19)

1. P. W. Shor, Proceedings of the 35th Annual Symposium on Fundamentals of Computer Science (IEEE Press, Los Alamitos, CA, 1994).
2. M. A. Nielsen, Quant. Inf. Comput., in press (preprint available at http://arxiv.org/abs/quant-ph/0502070).
3. V. Jurdjevic, Geometric Control Theory (Cambridge Univ. Press, Cambridge, 1996).
4. N. Khaneja, R. Brockett, S. J. Glaser, Phys. Rev. A 63, 032308 (2001).
5. N. Khaneja, S. J. Glaser, Chem. Phys. 267, 11 (2001).
6. N. Khaneja, S. J. Glaser, R. Brockett, Phys. Rev. A 65, 032301 (2002).
7. An alternate way of viewing this cost function is as a penalty metric of the kind used in sub-Riemannian geometry (18, p. 18).
8. The cost function has the property that F(αH) = |a|F(H), where a is any real number. This property and the chain rule imply invariance of the length with respect to reparameterization.
9. V. I. Arnold, B. A. Khesin, Topological Methods in Hydrodynamics, vol. 125 of Applied Mathematical Sciences (Springer, New York, 1998).
10. Our metric is superficially similar to the usual metric of Euclidean space, and it is tempting to suppose that geodesics must be straight lines, i.e., constant Hamiltonians. However, the Pauli coefficients for the Hamiltonian actually correspond to a changing local basis for the tangent space, not a fixed basis, and hence constant Hamiltonians do not, in general, give rise to geodesies.
11. J. Milnor, Morse Theory (Princeton Univ. Press, Princeton, NJ, 1969).
12. M. Berger, A Panoramic View of Kiemannian Geometry (Springer, Berlin, 2003).
13. Proofs are available as supporting material on Science Online.
14. 2 = 1.
15. The first inequality comes from the fact that there are 9n(n - 1)/2 + 3n one- and two-qubit terms.
16. M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).
17. The overhead factors in this theorem may be substantially improved, e.g., by making use of higher order analyses in lemmas 1 to 3. However, the key point-that U can be accurately approximated with a number of gates that scales polynomially with d(l, U)-remains the same.
18. R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, vol. 91 of Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 2002).
19. We thank S. Aaronson, M. de Burgh, J. Dodd, C. Hill, A. Hines, A. Lund, L. Noakes, M. Sarovar, and B. Toner for helpful discussions and the Australian Research Council for funding. We are especially grateful to L. Noakes for pointing out to us the simplified form of the geodesic equation for right-invariant metrics.