Quantum channels, complex Stiefel manifolds, and optimization

TL;DR

This paper establishes a topological and metric relation between quantum channels and complex Stiefel manifolds, enabling new approaches to quantum optimization problems involving control objectives and quantum gate generation.

Contribution

It introduces a homeomorphism between quantum channels and Stiefel manifold quotients, and applies this to optimize quantum control objectives.

Findings

Defined a metric on quantum channels from Stiefel manifold metrics.

Analyzed extrema of quantum control functionals using the new framework.

Provided a geometric approach to quantum optimization problems.

Abstract

Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known are the representations of quantum channels by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quantum channels by points of a suitable quotient under the action of the unitary group of some complex Stiefel manifold. In this work, we establish a continuity relation (homeomorphism) between the topological space of quantum channels and the quotient of the complex Stiefel manifold. Then the metric on the set of quantum channels induced by the Riemannian metric on the Stiefel manifold is defined. The established relation can be applied to various quantum optimization problems. As an example, we apply it to the analysis of extrema points for a wide variety of quantum control objective functionals defined on the complex Stiefel manifolds, including mean value, generation of quantum gates, thermodynamic quantities involving entropy, etc.