Geometric Parameterization of Kraus Operators with Applications to Quasi Inverse Channels for Multi Qubit Systems

This work presents a differentiable geometric parameterization of quantum channels in Kraus representation, which can be efficiently probed to find an unknown quantum channel. We explore its feasibility in finding the quasi inverse channels, which can be a tedious analytically for complex noise processes and is often achievable only for a limited range of parameters. In this regard, machine learning based algorithms have been employed successfully to find quasi inverse of quantum channels. The space of quantum channels in this scheme is a unit hypersphere, and components of mutually constrained unit vectors residing in this space, are used to construct a physically valid quantum channel. Symplectic constraints, orthogonality, and unit length of the vectors suffice to maintain complete positivity and the trace-preserving property of the channels. By performing gradient descent on this parametric space with a fidelity-based loss function, this approach is found to optimize quasi inverse of a variety of quantum channels, not limited to single-qubits, proving its effectiveness.