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A parameterized quantum system is a quantum system depending on “external” or “classical” parameters.

In typical examples arising in quantum mechanics, this simply means that the Hamiltonian HH is not just one fixed linear operator on a given Hilbert space of quantum states \mathcal{H}, but a continuous (or suitably smooth) map

H():PEnd() H(-) \;\colon\; P \longrightarrow End(\mathcal{H})

of such, from some topological “parameter space” PP.

More abstractly, where a single quantum system is described by linear type theory (see also at quantum logic), a parameterized quantum system is described by dependent linear type theory.


Adiabatic Berry phases and adiabatic quantum computation

The quantum adiabatic theorem says that/how the evolution of a parameterized quantum system under sufficiently slow movement of the external parameters remains in an eigenstate for a given system of commuting quantum observables. The remaining transformations on these eigenspaces are known as (non-abelian) Berry phases. These are used as quantum gates in adiabatic quantum computation.


Time-dependent quantum mechanics

For example, time-dependent Hamiltonians/quantum systems are described this way for PP \subset \mathbb{R} interpreted as the space of time-parameters.

See also at Dyson formula.

Defect anyon braiding

In some models of braid group statistics (see there for more) the anyons are defects/solitons whose positions serve as external parameters of the system, so that the (non-abelian) Berry phases under their adibatic moverment constitutes a braid group representation on the quantum system‘s ground states.

Last revised on June 11, 2022 at 16:37:38. See the history of this page for a list of all contributions to it.