EXPLORING STRUCTURE AND QUANTUM ADVANTAGE IN COMPLEX SYSTEMS

THE TRANSFORMATIVE ADVANCES ACHIEVED AND EXPECTED BY QUANTUM TECHNOLOGIES RELY ON A SET OF PREMISES THAT CONCERN BOTH FOUNDATIONAL ASPECTS AS WELL AS APPLICATIONS. THE PRESENT PROPOSAL ADDRESSES WHAT ARE THE ADVANTAGES, BOUNDS, STRATEGIES AND LIMITATIONS FOR TASKS CONCERNING COMPLEX QUANTUM SYSTEMS. IT IS BROAD IN SCOPE RANGING FROM THE STATISTICAL INFERENCE OF THE QUANTUM PROPERTIES OF STATES, PROCESSES AND OBSERVABLES, TO THE THERMODYNAMICAL CONSEQUENCES DERIVED BY PRESENT MINIATURIZATION OF QUANTUM DEVICES. IT ADDRESSES AS WELL THE INTRINSIC COMPLEXITY INVOLVED IN COMPUTATIONAL MODELS, IN SIMULATION OF MANY-BODY SYSTEMS OR NOVEL CONSEQUENCES DERIVING FROM INFINITE-DIMENSIONAL HILBERT SPACE. THE PROPOSAL IS ORGANIZED ALONG FOUR DISTINCT BUT INTERCONNECTED RESEARCH LINES WHICH, SHARING CONCEPTUAL PRINCIPLES AND GOALS. RELYING IN THE MINIMAL SET OF ASSUMPTIONS, OUR PROJECT WILL DEVELOP UNIVERSAL QUANTUM LEARNING PROTOCOLS WHICH SHOULD ALLOW TO ANALYZE ANY INPUT QUANTUM DATA FOR VERIFICATION AND CERTIFICATION OF QUANTUMNESS. TO VERIFY AND CERTIFY CORRELATIONS IN SPACE AND TIME DEMANDS A NOVEL FORMULATION. WHILE CLASSICALLY, THE MATHEMATICAL STRUCTURE THAT GOVERNS CORRELATIONS BETWEEN SIMULTANEOUS EVENTS ACROSS SPACE AND CORRELATIONS GENERATED BY STOCHASTIC PROCESSES IN TIME ARE BOTH DESCRIBED USING JOINT PROBABILITY DISTRIBUTIONS, THIS IS NOT ANYMORE THE CASE IN THE QUANTUM PARADIGM. WE NEED TO DERIVE A PROPER MATHEMATICAL CHARACTERIZATION AND FORMULATION OF SPATIAL AND TEMPORAL QUANTUM CORRELATIONS WITH THE OBJECTIVE OF HARNESSING THEM FOR INFORMATION PROCESSING TASKS. COMPLEXITY NATURALLY ARISES WHEN THE NUMBER OF PARTIES, CORRELATIONS, OPERATIONS AND/OR MEASUREMENTS INCREASES. FOR INSTANCE, THE SPACE OF ALL POSSIBLE QUANTUM OPERATIONS THAT CAN BE PERFORMED ON A PHYSICAL SYSTEM IS ENORMOUS (TYPICALLY EXPONENTIAL IN THE NUMBER OF SUBSYSTEMS) AS IT IS THE SPACE OF ALL QUANTUM STATES OF INTERACTING MANY-BODY SYSTEMS. HOWEVER, IN BOTH CASES, THE SET OF PHYSICAL OPERATIONS (I.E., THOSE THAT CAN BE IMPLEMENTED), AND THE SET OF PHYSICAL STATES (THOSE THAT ARISE FROM A PHYSICAL HAMILTONIAN) IS MUCH MORE RESTRICTED. WE WANT TO CHARACTERIZE SUCH SETS USING QUANTUM INFORMATION PROTOCOLS AND CONVEX GEOMETRY. THE RAPIDLY DEVELOPMENT FIELD OF QUANTUM TECHNOLOGIES WITH A MINIATURIZATION OF QUANTUM DEVICES DEMANDS TO LIFT ALSO TYPICAL ASSUMPTIONS REGARDING THE INTERACTION OF A QUANTUM SYSTEM WITH ITS SURROUNDING ENVIRONMENT AND THE EQUILIBIRIUM HYPOTHESIS. DESCRIBING OPEN QUANTUM SYSTEMS FAR FROM EQUILIBRIUM IS CHALLENGING, IN PARTICULAR WHEN THE ENVIRONMENT IS MESOSCOPIC, WHEN IT DEVELOPS NON-EQUILIBRIUM FEATURES DURING THE EVOLUTION, OR WHEN THE MEMORY EFFECTS CANNOT BE DISREGARDED. FOLLOWING PREVIOUS RESULTS FROM OUR GROUP, WE WANT TO STUDY OBSERVATIONAL ENTROPIES AS A KEY CONCEPT TO CLOSE THE GAP BETWEEN STANDARD QUANTUM THERMODYNAMICS/STATISTICAL MECHANICS AND OUT-OF-EQUILIBRIUM PHYSICS, BUT ALSO ITS USE TO CHARACTERIZE COMPLEX CORRELATIONS. OUR PROPOSAL IS FUNDAMENTAL RESEARCH, AIMING AT GENERATING NEW KNOWLEDGE BUT CONSIDERING ALSO CONCRETE TASKS AND USES WHERE QUANTUM TECHNOLOGIES OFFER AN ADVANTAGE IN PERFORMANCE IN COMPARISON WITH STANDARD TECHNOLOGIES.