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Science 24 February 2006:
Vol. 311. no. 5764, p. 1068
DOI: 10.1126/science.311.5764.1068f
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This Week in Science
Quantum computers hold great promises for solving difficult problems otherwise intractable on classical computers. However, actually finding algorithms, or the quantum circuitry on which the algorithms can be implemented, is challenging because the number of components in the quantum circuits should grow only polynomially with the complexity of the problem you want to solve. While manipulation of a single qubit can be thought of as the rotation of a unit vector in a sphere, a quantum computer will typically have n interacting qubits, giving rise to a 2n-dimensional space, Thus Nielsen et al. (p. 1133; see the Perspective by Oppenheim) recast the problem of finding an efficient quantum algorithm in terms determining the shortest path between two points in a certain curved, or Riemannian, geometry. The mathematical tools of Riemannian geometry can then be used to provide an understanding of quantum computation and a possible route to determine efficient quantum algorithms.
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Science. ISSN 0036-8075 (print), 1095-9203 (online)
