Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition

TL;DR

This paper develops a linear-algebraic framework and a QR-based control algorithm to stabilize specific subspaces in discrete-time quantum systems, enhancing quantum information processing capabilities.

Contribution

It introduces a canonical QR decomposition method and a feedback control design algorithm for stabilizing quantum subspaces, advancing quantum control techniques.

Findings

The control algorithm effectively stabilizes target subspaces when feasible.

A canonical QR decomposition is derived for analyzing quantum dynamics.

The method can simulate open-system quantum dynamics.

Abstract

We analyze the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest. Global asymptotic stability of subspaces is relevant to quantum information processing, in particular for initializing the system in pure states or subspace codes. We provide a linear-algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace. We then construct a design algorithm for discrete-time feedback control that allows to stabilize a target subspace, proving that if the control problem is feasible, then the algorithm returns an effective control choice. In order to prove this result, a canonical QR matrix decomposition is derived, and also used to establish the control scheme potential for the simulation of open-system dynamics.