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Consistency of the Adiabatic Theorem

M. H. S. Amin

The adiabatic theorem provides the basis for the adiabatic model of quantum computation. Recently the conditions required for the adiabatic theorem to hold have become a subject of some controversy. Here we show that the reported violations of the adiabatic theorem all arise from resonant transitions between energy levels. In the absence of fast driven oscillations the traditional adiabatic theorem holds. Implications for adiabatic quantum computation are discussed.

arXiv:0810.4335Physical Review LettersJun 3, 2009186 citations

Extending the Adiabatic Theorem

Rachel Berkowitz

PhysicsApr 1, 20260 citations

Validity of the quantum adiabatic theorem

Zhaoyan Wu, Hui Yang

The consistency of the quantum adiabatic theorem has been doubted recently. It is shown in the present paper that the difference between the adiabatic solution and the exact solution to the Schr\"odinger equation with a slowly changing driving Hamiltonian is small; while the difference between their time derivatives is not small. This explains why substituting the adiabatic solution back into the Schr\"odinger equation leads to ``inconsistency'' of the adiabatic theorem. Physics is determined completely by the state vector, and not by its time derivative. Therefore the quantum adiabatic theorem is physically correct.The consistency of the quantum adiabatic theorem has been doubted recently. It is shown in the present paper that the difference between the adiabatic solution and the exact solution to the Schr\"odinger equation with a slowly changing driving Hamiltonian is small; while the difference between their time derivatives is not small. This explains why substituting the adiabatic solution back into the Schr\"odinger equation leads to ``inconsistency'' of the adiabatic theorem. Physics is determined completely by the state vector, and not by its time derivative. Therefore the quantum adiabatic theorem is physically correct.

arXiv:quant-ph/0410118Physical Review AJul 22, 200555 citations

Proof of the Adiabatic Theorem

CRC Press eBooksMay 21, 20043 citations

Adiabatensätze mit und ohne Spektrallückenbedingung

Jochen Schmid

In this work we generalize some of the previously known adiabatic theorems to situations with non-unitary evolutions in Banach spaces. We prove adiabatic theorems with uniform gap condition (generalizing a theorem of Abou Salem), adiabatic theorems with non-uniform gap condition (generalizing a theorem of Kato) and qualitative as well as quantitative adiabatic theorems without gap condition (generalizing theorems of Avron and Elgart, and Teufel). Additionally, we give a generalized version of an adiabatic theorem of higher order due to Nenciu. In all these adiabatic theorems the considered spectral values need not lie on the imaginary axis and in the adiabatic theorems with spectral gap condition and the adiabatic theorem of higher order compact subsets of the spectrum are sufficient (in particular, these subsets need not consist of eigenvalues). We explore the strength of the presented adiabatic theorems in numerous examples. In particular, we show that the theorems of the present work are more general than the previously known theorems. This work was finished in October 2010 and handed in as a diploma thesis at the mathematics department of the University of Stuttgart. The more recent results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June 2011) are therefore not taken into consideration here. We point out, however, that the theorems of the present work are more general than the corresponding theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic theorem of higher order which is in no logical relation to the adiabatic theorem of higher order of Avron, Fraas, Graf and Grech. We are about to gather the most important theorems of the present work in an article and will upload it to arXiv soon.In this work we generalize some of the previously known adiabatic theorems to situations with non-unitary evolutions in Banach spaces. We prove adiabatic theorems with uniform gap condition (generalizing a theorem of Abou Salem), adiabatic theorems with non-uniform gap condition (generalizing a theorem of Kato) and qualitative as well as quantitative adiabatic theorems without gap condition (generalizing theorems of Avron and Elgart, and Teufel). Additionally, we give a generalized version of an adiabatic theorem of higher order due to Nenciu. In all these adiabatic theorems the considered spectral values need not lie on the imaginary axis and in the adiabatic theorems with spectral gap condition and the adiabatic theorem of higher order compact subsets of the spectrum are sufficient (in particular, these subsets need not consist of eigenvalues). We explore the strength of the presented adiabatic theorems in numerous examples. In particular, we show that the theorems of the present work are more general than the previously known theorems. This work was finished in October 2010 and handed in as a diploma thesis at the mathematics department of the University of Stuttgart. The more recent results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June 2011) are therefore not taken into consideration here. We point out, however, that the theorems of the present work are more general than the corresponding theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic theorem of higher order which is in no logical relation to the adiabatic theorem of higher order of Avron, Fraas, Graf and Grech. We are about to gather the most important theorems of the present work in an article and will upload it to arXiv soon.

arXiv:1112.6338Dec 29, 20110 citations

Remarks on the formulations of the adiabatic theorem

Mirjana Božić, R.J. Lombard, Z. Marić

The European Physical Journal DDec 1, 19917 citations

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