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Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum information theory)
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum information theory)
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum information theory)
2005 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 72, no 4, 042317- p.Article in journal (Refereed) Published
Abstract [en]

In Phys. Rev. A 71, 060312(R) (2005), the robustness of the local adiabatic quantum search to decoherence in the instantaneous eigenbasis of the search Hamiltonian was examined. We expand this analysis to include the case of the global adiabatic quantum search. As in the case of the local search the asymptotic time complexity for the global search is the same as for the ideal closed case, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, we find that the time complexity of the global quantum adiabatic search scales like N3∕2, where N is the list length. We moreover extend the analysis to include success probabilities p<1 and prove bounds on the run time with the same scaling as in the conditions for the p→1 limit. We supplement the analytical results by numerical simulations of the global and local search.

Place, publisher, year, edition, pages
2005. Vol. 72, no 4, 042317- p.
Keyword [en]
Quantum computation, adiabatic approximation, search algorithm, open quantum systems
National Category
Physical Sciences
Research subject
Physics
Identifiers
URN: urn:nbn:se:uu:diva-93321DOI: 10.1103/PhysRevA.72.042317ISI: 000232931800051OAI: oai:DiVA.org:uu-93321DiVA: diva2:166765
Note
Also in Virtual Journal of Nanoscale Science and Technology, October 31 issue 2005 and Virtual Journal of Quantum Information, November issue 2005. Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2011-10-06
In thesis
1. Open Quantum Systems: Effects in Interferometry, Quantum Computation, and Adiabatic Evolution
Open this publication in new window or tab >>Open Quantum Systems: Effects in Interferometry, Quantum Computation, and Adiabatic Evolution
2005 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated.

Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can.

Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N3/2 , where N is the list length. We consider success probabilities p<1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit.

Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2005. vii + 52 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 77
Keyword
Physics, Open Systems, Completely Positive Maps, Channels, Decoherence, Quantum Information, Quantum Computing, Adiabatic Approximation, Quantum Search, Time-Complexity, Fysik
National Category
Physical Sciences
Identifiers
urn:nbn:se:uu:diva-5893 (URN)91-554-6307-X (ISBN)
Public defence
2005-09-22, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15
Opponent
Supervisors
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved

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Sjöqvist, Erik

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