Quantum channel learning.

Area of Science:

• Quantum Information Theory
• Quantum Computing
• Mathematical Physics

Background:

• Current quantum state mapping often focuses on pure states and unitary transformations.
• Representing quantum states as density matrices and mapping as quantum channels offers a more general framework.
• Distinguishing between superpositions and probabilistic mixtures of quantum states is crucial for advanced quantum information processing.

Purpose of the Study:

• To formulate and solve the problem of optimal mapping between input (IN) and output (OUT) Hilbert spaces using density matrices.
• To develop an iterative algorithm for learning quantum channels based on fidelity maximization.
• To generalize unitary learning to mixed unitary quantum channels and density matrix mappings.

Main Methods:

• Formulating the mapping as an optimization problem maximizing total fidelity subject to probability preservation constraints.
• Developing an iterative algorithm for fidelity maximization when fidelity is a quadratic form.
• Representing quantum channels as a hierarchy of mixed unitary mappings.

Main Results:

• An iterative algorithm for learning quantum channels represented as mixed unitary channels.
• The ability to distinguish probabilistic mixtures of states from their superpositions.
• Demonstrated application to unitary learning of density matrix mappings and quantum channels.

Conclusions:

• The developed approach extends unitary learning to density matrices and quantum channels.
• The method provides a powerful tool for analyzing and learning quantum processes.
• Potential applications include quantum inverse problems, variational quantum algorithms, and quantum tomography.