Optimal Probabilistic Quantum Control Theory

Universitadi Padova

Randa Herzallah, Abdessamad Belfakir

There is a fundamental limit to what is knowable about atomic and molecular scale systems. This fuzziness is not always due to the act of measurement. Other contributing factors include system parameter uncertainty, functional uncertainty that originates from input functions, and sensors noises to mention a few. This indeterminism has led to major challenges in the development of accurate control methods for atomic scale systems. To address the probabilistic and uncertain nature of these systems, this work proposes a novel control framework that considers the representation of the quantum system states and the quantification of its physical properties following a probabilistic approach. Our framework is fully probabilistic. It uses the Shannon relative entropy from information theory to design optimal randomised controllers that can achieve a desired outcome of an atomic scale system. Two experiments are carried out to illustrate the applicability and effectiveness of the proposed approach.

Victor Mukherjee, Vittorio Giovannetti, Rosario Fazio +3 more

Abderrahim Benmachiche, Alan C. Santos, Azeddine Messikh +1 more

\n\nAdiabatic and superadiabatic controlled evolutions are used as universal models for quantum computations, where an auxiliary system is used as the main resource in the theory. While the adiabatic case demands a slow dynamics (so that the adiabatic theorem is satisfied), the superadiabatic model can speed up such process by spending more energy than the adiabatic one. In this paper, we explore a hybrid scheme of controlled evolutions and measurement in order to optimize the thermodynamic cost of implementing quantum gates. By adjusting a free parameter of the theory of controlled evolutions, we show that we can improve the energy cost of the evolution. To this end, we used a model of computation based on adiabatic/superadiabatic evolution and post-selection measurement on the auxiliary qubit. In both cases, adiabatic and superadiabatic, the probabilistic model allows for optimizing the thermodynamics cost of implementing gates. When the system reservoir interaction is taken into account, we show that the superadiabatic probabilistic model is more robust than the adiabatic one.\n

V. P. Belavkin, Simon Edwards

Julius A. Zeiss, Gereon Koßmann, René Schwonnek +1 more

Leveraging a recently proposed notion of relative entropy in general probabilistic theories (GPT), we prove a finite de Finetti representation theorem for general convex bodies. We apply this result to address a fundamental question in polynomial optimization: the existence of a convergent outer hierarchy for problems with inequality constraints and analytical convergence guarantees. Our strategy generalizes a quantitative monogamy-of-entanglement argument from quantum theory to arbitrary convex bodies, establishing a uniform upper bound on mutual information in multipartite extensions. This leads to a finite de Finetti theorem and, subsequently, a convergent conic hierarchy for a wide class of polynomial optimization problems subject to both equality and inequality constraints. We further provide a constructive rounding scheme that yields certified interior points with controlled approximation error. As an application, we express the optimal GPT value of a two-player non-local game as a polynomial optimization problem, allowing our techniques to produce approximation schemes with finite convergence guarantees.