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Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

J. Math. Phys. 50, 102106 (2009); doi:10.1063/1.3236685

Published 20 October 2009

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Daniel A. Lidar,1,2,3,4 Ali T. Rezakhani,1,4 and Alioscia Hamma1,4,5,6
1Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
2Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA
3Department of Physics, University of Southern California, Los Angeles, California 90089, USA
4Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089, USA
5Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario N2L 2Y5, Canada
6Massachusetts Institute of Technology, Research Laboratory of Electronics, 77 Massachusetts Ave., Cambridge, Massachusetts 02139, USA
We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap. ©2009 American Institute of Physics
History: Received 16 April 2009; accepted 2 September 2009; published 20 October 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/102106/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.67.Ac
    Quantum algorithms, protocols and simulations
  • 02.10.Ud
    Linear algebra
  • 03.65.Fd
    Algebraic methods in quantum mechanics
  • YEAR: 2009

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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  51. Note that the g(2) factor present in Eq. (91) [evaluated at N=1 and absent in Eq. (88)] gives rise to a discrepancy between the two bounds, unless we set gamma=1/14. This does not in fact impose a constraint on the family of Hamiltonians our proof applies to (recall Assumption 1), since in the application of Cauchy's theorem we are free to choose an arbitrarily small integration contour around the real-time axis. In spite of having thus fixed its value, we continue to write gamma rather than 1/14, as there is no fundamental importance to this value; it is merely an outcome of our rather loose bounds, e.g., as in Eq. (71).
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