nLab parameterized quantum system

1. 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.

2. 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.

3. 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.

4. Related concepts

• Dyson formula

• quantum adiabatic theorem

• controlled quantum gate

• parameterized quantum computation

• parameterized homotopy theory

• parameterized stable homotopy theory

• dependent linear type theory

• subsystem

• EPR phenomena

5. 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]