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

TL;DR

This work introduces a differentiable geometric parameterization of quantum channels using Kraus vectors on a Kraus sphere, enforcing CPTP constraints by construction through orthogonal–symplectic relations. By deriving the associated Lie algebra and finite transformations, the authors enable controlled navigation of the CPTP landscape and formulate quasi-inverse learning as a fidelity-maximization problem conducted with gradient-based optimization on the Kraus sphere. Empirical results for single- and two-qubit channels (bit flip, phase flip, and bit-phase flip) show high-fidelity quasi inverses that are effectively unitary, with depolarizing channels yielding the identity as expected when no quasi-inverse exists. The framework is well-suited for variational algorithms, Monte Carlo sampling, and channel-learning tasks, offering a principled, differentiable, and physically valid approach to approximate quantum error reversal and channel analysis.

Abstract

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.

Figures (8)

• Figure 1: Original states $\rho$ are transformed by $\mathcal{E}$ into $\rho'$ and then fed into $\mathcal{E}_q$ to produce $\rho"$ which are the recovered states.
• Figure 2: The process starts from original states $\rho$ which are transformed by $\mathcal{E}$ into $\rho'$ and then fed into Kraus parameterization $\mathcal{E}_{\theta_i}$ to recover to $\rho"$. The loss function is computed using $\rho$ and $\rho"$, and its gradient gives the optimized angles which are backpropagated for further optimization, finally giving the quasi inverse channel $\mathcal{E}_q$.
• Figure 3: Fidelity recovery curve for the single-qubit bit flip channel generated with varying noise strength $p$. High recovery occurs for strong noise after $p=0.5$.
• Figure 4: Fidelity recovery curve for the single-qubit phase flip channel generated with varying noise strength $p$.
• Figure 5: Fidelity recovery curve for the single-qubit bit-phase flip channel generated with varying noise strength $p$.