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Habilitation À Diriger Des Recherches Year : 2020

Habilitationà Diriger des Recherches (HDR)

Alain Sarlette
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This manuscript is meant to provide an overview of most of my research activities, taking a step back to highlight the links between my contributions and suggest perspectives for future research directions. A central element on which I have focussed turns out to be controlled interactions, with an approach at the boundary between applied mathematics, theoretical physics, and control theory. Other recurrent elements are nonlinear manifolds, and most importantly tools towards developing quantum technology. The overarching theme is the interplay between information and physical systems modeled by differential equations. A first type of work establishes some fundamental limitations, as a help to guide further research focus. These results include - fundamental bounds on rejecting disturbances in linear systems due to distributed sensing (with European Extremely Large Telescope primary mirror application); - impossibility results on achieving so-called \string stability" in long chains of systems, with the first results in presence of strong nonlinearities (including any form of digital communication); - a structural bound, showing that a rst-order memory already achieves optimal convergence speed in consensus algorithms when the network spectrum can span a given interval (with an opening to accelerate the convergence for clustered spectra) - comprehensive and tight results on the ultimately achievable mixing speed of Quantum Random Walks (QRWs), compared to Markov Chains and so-called Lifted Markov Chains: once memory is allowed, in principle QRWs provide no further speedup. A second type of work has been developing tools for the analysis of few-body, controlled quantum systems, like those built in the experiments of my physics colleagues. Results along these lines include: - results on Lyapunov functions for proving convergence of dissipative quantum systems towards a unique steady state, with an aim for general results and for covering infinite-dimensional systems; - procedures for high-accuracy model reduction in open quantum systems, on the basis of series expansions, extending the usual adiabatic elimination method; - characterizing the presence of low-dimensional deterministic manifolds on which some typical quantum stochastic di erential equations remain con ned with probability 1. A third type of work has been the proposal of particular designs or algorithmic procedures. - In the sense of generalizing simple linear procedures, we have proposed principles for applying integral control on nonlinear manifolds; and to formulate \consensus" as an abstract symmetrization procedure that can be extended among others to quantum systems; - Towards stabilizing quantum systems, we have proposed a way to tune the repeated interaction with a stream of auxiliary systems in order to stabilize a target quantum system into a desired state, like so-called Schroedinger cat states of harmonic oscillators, Fock states or squeezed sates - For the same goal, we have developed some improved quantum feedback schemes based on Quantum Non-Demolition measurements: stabilizing highly entangled states between remote systems despite decoherence along the line; and proving how injecting noise of controlled amplitude can achieve the so-far elusive exponential stabilization of eigenstates under weak continuous-time measurement. - While working towards a better understanding of quantum algorithms, in particular QRWs, we have proposed a Quantum Fast-Forwarding algorithm (QFF) which provides an accelerated evolution of a classical Markov chain, without requiring to compute the associated lifted Markov chain as would be needed for designing a classical forwarding algorithm. This algorithm is shown to accelerate primitives on graphs provided via quantum query. The main drive behind all this research has been to understand the essence and limitations of various procedures towards taking decisions in complex dynamical systems, on the basis of continuous physical variables. In this sense, the extraction of some most fundamental elements of quantum technology, combining both hardware level and logical level into a consistent and robust system-theoretic picture, remains a guiding objective for the years to come. This shall involve further intense collaboration with quantum physics experimentalists, and also with algorithm experts as I have started doing more recently. On the way, we plan to develop fundamental tools for approximate analysis and control design guidelines, tailored to the needs of interacting quantum systems. Much of this work has been carried out in collaboration with outstanding and friendly scientific colleagues, whom I most warmly thank for this experience. Those whose work is missing here should please just consider that I had to leave out some things in the process of keeping the present document precise and concise.
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tel-03160258 , version 1 (05-03-2021)


  • HAL Id : tel-03160258 , version 1


Alain Sarlette. Habilitationà Diriger des Recherches (HDR). Optimization and Control [math.OC]. Sorbonne Universites, UPMC University of Paris 6, 2020. ⟨tel-03160258⟩
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