Relation between dual-based Choi-space projection and hyperplane intersection algorithm

Determine how the Lagrangian dual–based method for projecting a Hermitian operator onto the set of physical Choi operators (implemented by minimizing θ(Y) over Hermitian Y and applying the projector Π to P + I ⊗ Y) relates to the hyperplane intersection projection algorithm proposed for the same Choi-space projection task.

Background

To compute the nearest physical Choi operator in Frobenius norm, the paper adopts a dual formulation: solve an unconstrained convex optimization over a Hermitian variable Y to minimize θ(Y), then apply the projector Π to P + I ⊗ Y to obtain the projected Choi operator. This approach leverages standard nonlinear convex optimization methods (e.g., BFGS) and reduces parameter tracking compared to direct semidefinite programming.

An alternative method—the hyperplane intersection projection algorithm—has been developed in the quantum process tomography literature for the same projection problem. While preliminary evidence suggests the dual approach is competitive, the authors explicitly state that the relationship between these two algorithms remains unclear.

References

Up to now, it is unclear how this dual approach relates to the more recent hyperplane intersection projection algorithm, even though preliminary results can already be found in Ref., where it is shown that the dual approach is still competitive as a state-of-art method for solving~eq:projection-choi.

eq:projection-choi:

$\widetilde{P} \coloneqq \operatorname*{arg\,min}_{X \in \mathfrak{J}(\mathcal{H})} \norm{P-X}, $

— Recovering complete positivity of non-Markovian quantum dynamics with Choi-proximity regularization  (2309.16320 - D'Abbruzzo et al., 2023) in Appendix A (Algorithm for Choi-space projection)