Adiabatic Theorem without a Gap Condition

Avron, J. E.; Elgart, Alexander

doi:10.1007/s002200050620

Cited by 215 publications

(257 citation statements)

References 42 publications

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“…The relevant gap is now the gap between the discs. In this way, we showed that the relevant gap to consider is the gap between eigenmanifolds and not simply between the ground state and an excited state [2,11,12]. This is so, since these types of problems have large degeneracies in the ground state of H P , and the standard adiabatic theorem of quantum mechanics is not designed to handle these cases.…”

Section: Discussionmentioning

confidence: 98%

A quantum differentiation ofk-SAT instances

Tamir¹,

Ortíz

2010

New J. Phys.

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Abstract. We present a quantum adiabatic algorithm to differentiate between k-SAT instances, those with no solutions and those that have many solutions. The time complexity of the algorithm is a function of the energy gap between the subspace of all 0-eigenvectors (ground states) and the first excited states manifold, and scales polynomially with the number of resources. The idea of gaps between subspaces suggests a new tool to analyze time complexity in adiabatic quantum machines. Contents

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Section: Discussionmentioning

confidence: 98%

A quantum differentiation ofk-SAT instances

Tamir¹,

Ortíz

2010

New J. Phys.

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“…32 For this to be true, the changes must occur on a time scale that is long compared to the characteristic time associated with the gap separating the eigenvalue of interest from the rest of the energy spectrum. Recently, Avron and Elgart have formulated an extension of the QAT to cover systems with no spectral gaps, 33 which is the case presented by the quasibound states described here. As these authors have shown, all one really needs for adiabatic evoultion is a finite-dimensional spectral projection for the Hamiltonian that depends smoothly on time.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…As noted in Ref. 33, such queries require painstaking analysis to elucidate the optimal criteria for each particular case, a task we leave to future study.…”

Section: Summary and Discussionmentioning

confidence: 99%

A unified theory of quasibound states

Moyer

2014

AIP Advances

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We have developed a formalism that includes both quasibound states with real energies and quantum resonances within the same theoretical framework, and that admits a clean and unambiguous distinction between these states and the states of the embedding continuum. States described broadly as 'quasibound' are defined as having a connectedness (in the mathematical sense) to true bound states through the growth of some parameter. The approach taken here builds on our earlier work by clarifying several crucial points and extending the formalism to encompass a variety of continuous spectra, including those with degenerate energy levels. The result is a comprehensive framework for the study of quasibound states. The theory is illustrated by examining several cases pertinent to applications widely discussed in the literature.

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“…Although outside the context of scattering theory, adiabatic switching and the Gell-Mann-Low formula have been subject of active research recently; see, e.g., [12][13][14][15][16][17][18] and references therein.…”

Section: Adiabatic Switchingmentioning

confidence: 99%

On some aspects of the definition of scattering states in quantum field theory

Tóth

2012

Open Physics

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Abstract:The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the applications and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.

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Adiabatic Theorem without a Gap Condition

Cited by 215 publications

References 42 publications

A quantum differentiation ofk-SAT instances

A quantum differentiation ofk-SAT instances

A unified theory of quasibound states

On some aspects of the definition of scattering states in quantum field theory

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