A computational approach to quantum encoder design for purity optimization
Abstract
In this paper, we address the problem of designing a quantum encoder that maximizes the minimum output purity of a given decohering channel, where the minimum is taken over all possible pure inputs. This problem is cast as a max-min optimization problem with a rank constraint on an appropriately defined matrix variable. The problem is very hard to solve because it is non-convex with respect to both the objective function (output purity) and the rank constraint. To obtain a tractable computational algorithm, we systematically relax both of these non-convex functions to convex linear matrix inequalities, and solve this new problem using semidefinite programming. Specifically, our approach consists of two stages: one that relaxes the objective function (using the Sum-of-Squares relaxation), and one that deals with the rank constraint (using the logarithm of determinant heuristic). We characterize conditions under which the first stage of the relaxation is in fact exact and yields the optimal solution. While in general optimality cannot be guaranteed, we show two typical quantum channels, where the relaxation works very well and the encoders designed are considered to be optimum. This practical success is due to the strong properties of both relaxations, which are also discussed.Keywords
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