nLab
parameterized quantum system
Contents
Contents
Idea
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 .

Properties
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 .

Examples
Time-dependent quantum mechanics
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 .

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 .

References
Discussion of classically-parameterized quantum circuits:

(…)

Discussion of quantumly-parameterized quantum circuits:

Evan Peters, Prasanth Shyamsundar, Qarameterized circuits: Quantum parameters for QML (2021) [web ]
following

Guillaume Verdon, Jason Pye, Michael Broughton, A Universal Training Algorithm for Quantum Deep Learning [arXiv:1806.09729 ]

Prasanth Shyamsundar, Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation [arXiv:2102.04975 ]

Last revised on March 29, 2023 at 16:52:40.
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