Quantum geometry of the Cartan control problem

TL;DR

This paper explores the quantum control problem using differential geometry, focusing on the Cartan decomposition of Lie algebras related to quantum circuits and their realization in the state space.

Contribution

It introduces a geometric framework for the Cartan control problem in quantum circuits, emphasizing the state-dependent nature of algebra decomposition in the projective Hilbert space.

Findings

Cartan decomposition is state-dependent in quantum control.

Differential geometry provides insights into quantum circuit transformations.

The algebraic structure influences control strategies in quantum systems.

Abstract

The Cartan control problem of the quantum circuits discussed from the differential geometry point of view. Abstract unitary transformations of $SU(2^n)$ are realized physically in the projective Hilbert state space $CP(2^n-1)$ of the n-qubit system. Therefore the Cartan decomposition of the algebra $AlgSU(2^n-1)$ into orthogonal subspaces $h$ and $b$ such that $[h,h] \subseteq h, [b,b] \subseteq h, [b,h] \subseteq b$ is state-dependent and thus requires the representation in the local coordinates.