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

Journal of Mathematical Physics

Volumn 50, Issue 10, 2009, Pages

  Lidar, Daniel A ^a,b   Rezakhani, Ali T ^a   Hamma, Alioscia ^a,c,d  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b Perimeter Institute for Theoretical Physics ^*  (United States)

^c PERIMETER INSTITUTE FOR THEORETICAL PHYSICS   (Canada)

^d MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350733237     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3236685     Document Type: Article

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51. Note that the g (2) factor present in Eq. [evaluated at N=1 and absent in Eq.] gives rise to a discrepancy between the two bounds, unless we set γ=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 γ 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..
52. k with k=1. This is required in order to obtain a bound involving just a single power of A and of Β. Failing to do this allows for the possibility that the two bounds will not agree.
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54. Note that σ (τ) may in general depend on n. One can see this through a simple example. Imagine a cylinder of gaseous particles with short-ranged interactions. Any particle will interact with all particles inside a sphere of radius rint -the range of the interaction-around it. If we add new particles to the cylinder, at some point (i.e., at some n) all the space inside the shell will be occupied (close packed); hence, the new particles cannot interact with the particle in the center. For such particles, the coupling strength of the interaction with the particle in the center is effectively zero. The other condition we mention, namely time-independence of the graph or lattice, is designed to exclude folding of the system lattice, for example, in the case of large polymer or protein molecules, as this would also potentially allow a dependence of σ (τ) on n.
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