nLab quantum adiabatic theorem



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In quantum physics, the quantum adiabatic theorem for a parameterized quantum system says that under a sufficiently slow motion of the external parameters along a path γ:[0,1]P\gamma \,\colon\, [0,1] \to P in parameter space PP, eigenstates for mutually commuting quantum observables of the quantum system at γ(0)\gamma(0) will approximately evolve to eigenstates at γ(1)\gamma(1), even if the corresponding eigenvalues change significantly.

For adiabatic transport along a loop, γ(0)=γ(1)\gamma(0) = \gamma(1), this implies that the eigenspaces of the system are acted on by unitary operators, which (at least for multiplicity/eigen-dimension 1) are known as Berry phases.

The adiabatic parameter-action on ground states of (topologically ordered) quantum materials is one model for quantum computation, see at adiabatic quantum computation. If these adiabatic quantum gates depend only on the isotopy class of the parameter path, then one speaks of topological quantum computation (made explicit in Arovas, Schrieffer, Wilczek & Zee 1985, p. 1, Freedman, Kitaev, Larsen & Wang 2003, pp. 6 and Nayak, Simon, Stern & Freedman 2008, §II.A.2 (p. 6), see also at braiding of anyonic defects).

The following graphics is meant to illustrate this general idea for the case of adiabatic transformation along the braiding of nodal points in the Brillouin torus of semi-metal quantum materials (see the discussion at braid group statisticsanyonic band nodes):

(graphics from SS22)



The original formulation:

Making explicit the unitary action on degenerate eigenspaces by non-abelian Berry phases:

See also:

On adiabatic transport subject to repeated quantum measurement:

  • Ludmila Viotti, Ana Laura Gramajo, Paula I. Villar, Fernando C. Lombardo, Rosario Fazio, Geometric phases along quantum trajectories [arXiv:2301.04222]

In condensed matter theory

Discussion in solid state physics and in view of gapped topological phases of matter (quantum Hall effect, topological insulators):


See also:

For topological quantum computation

References which make explicit that the standard model of topological quantum computation by braiding of anyons is a form of adiabatic quantum computation:

Last revised on February 1, 2024 at 11:22:42. See the history of this page for a list of all contributions to it.