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

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.