quantum algorithms:

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 $H$ is not just one fixed linear operator on a given Hilbert space of quantum states $\mathcal{H}$, but a continuous (or suitably smooth) map

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

of such, from some topological “parameter space” $P$.

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*.

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.

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

See also at *Dyson formula*.

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 12:37:38. See the history of this page for a list of all contributions to it.