The ANR project ESQuisses is glad to announce a summer school to be held from July 4th to July 8th in Porquerolles, France. The lectures will start on Monday (July 4th) morning, and will end on Friday (July 8th) at noon. Participants are expected to arrive on Sunday (July 3rd) and depart on Friday (July 8th).

This summer school aims at gathering mathematicians, physicists and computer scientists interested in stochastic methods in quantum mechanics.

Several introductory lectures will be given by leading researchers in the field. The topics covered include quantum machine learning, correlations in device-independent scenarios, quantum trajectories and feedback, random unitary circuits, and quantum cellular automata. The lectures shall be complemented by additional, more focused, research talks. Young researchers as well as confirmed researchers are welcome to participate. Poster sessions for Phd and post-doctoral students will be organized during the week.

Ample time for discussion and exchanges will be set aside, allowing for an interactive and inclusive in-person event.

The school is going to be held on-site, at the IGESA (Institution de gestion sociale des armées) center, on the island of Porquerolles, France, during the week 4-8 July 2022.

The conference fees, which cover local expenses (full board accommodation), are as follows:

• Masters and PhD students, postdocs: 300 EUR
• Faculty and industry participants: 450 EUR

We ask you to pay these fees (or chose the "Financial aid - 0 €" option if you received financial aid) here, before Friday, June 17th.

Antonio Acin (ICFO Barcelona)  - Correlations in physical theories and the device-independent scenario for quantum information processing

Physical principles impose limits on the statistics observed when measuring the particles of a correlated multi-partite system. In the lecture, we first describe the set of correlations obtained in classical, quantum and general non-signalling theories. Then, we show how the characterisation of these correlations is relevant for the construction of device-independent quantum information protocols, where devices are seen as black boxes processing a classical input to produce a classical output. We also present open questions in these topics and future research directions.

Vedran Dunjko (Leiden University) -  Quantum machine learning with parametrized quantum circuits

In recent times, machine learning has been often highlighted as one of the most promising applications of quantum computers, especially in the near term. Why is this the case?  In this series of lectures we will cover the following topics

1. Fundamentals of machine learning,
   i.e. what is it we want the quantum computer to do, and what does it mean to do well?
2. Parametrized quantum circuits, hybrid computing and NISQ
   i.e. how do we want to use quantum computers
3. Progress in supervised learning
   i.e. what have we learned about this flavour of QML and the possibility of advantages
4. Beyond supervised learning: snapshots of generative modelling and reinforcement learning with parameterized quantum circuits
   i.e. there is more to life than classifying cats from dogs

The level of the lecture will be adjusted to students familiar with the basics of quantum computing, and little to no background in machine learning.

Hachem Kadri (Aix-Marseille University) -  From Classical to Quantum Machine Learning

This course is an introduction to the field of quantum machine learning. The course begins with a general overview of the fundamental notions and practical applications of classical machine learning, with a particular focus on the most common machine learning algorithms. It then introduces the main concepts of quantum machine learning and provides an overall picture of the field. To illustrate these concepts with concrete examples, quantum versions of the perceptron and the linear regression algorithms are examined.

Adam Nahum (École Normale Supérieure) - Random quantum circuits

I will give an introduction to random quantum circuits as simple models for time-evolution in many-body systems. I will emphasise mappings of dynamical observables (involving for example two-point correlations, or entanglement entropy) to effective classical stat mech problems involving random paths. I will also describe how sufficiently strong monitoring by an external observer leads to a measurement-induced phase transition in the entanglement structure of the many-body state.

Lorenzo Piroli (École Normale Supérieure) - Quantum Cellular Automata

In these lectures we will introduce and discuss Quantum Cellular Automata. We will go through their definition and some fundamental results regarding their structure. Finally, we will focus on the simplest case of one-dimensional systems and discuss the corresponding index theory.

Pierre Rouchon (Mines ParisTech)  - Quantum Trajectories and feedback stabilisation

Quantum control is an emerging research subject with an increasing role in technologies related to high precision metrology, quantum simulation, quantum information processing and communication. Its development requires to reconsider how measurement, control, and interactions fundamentally affect a system — in particular, the intrinsic invasive character of measurements. This 6-hour course presents some mathematical methods for modeling and feedback stabilization of open quantum systems. The level will be that of a graduate course intended for a general control or applied-mathematics audience without any prerequisites in quantum mechanics.

Federico Girotti - Concentration Inequalities for Output Statistics of Quantum Markov Processes

We present new concentration bounds for time averages of measurement outcomes in quantum Markov processes. They generalize well-known bounds for classical Markov chains which provide constraints on finite time fluctuations of time-additive quantities around their averages. More precisely, we derived a Bernstein-type and a Hoeffding-type concentration bounds for time averages of the measurement outcomes of a quantum Markov chain and we generalized the Bernstein-type bound to counting processes of continuous time quantum Markov processes. If specialized to the classical setting, the bounds provide new concentration inequalities for empirical fluxes of classical Markov chains. If time allows, we present extensions and suggest potential applications of our results. The talk is based on joint work with Madalin Guta and Juan P. Garrahan.

Maria Jivulescu - Order preserving maps on quantum measurements

In this talk we present our studies about the partially ordered set of equivalence classes of quantum measurements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement outcomes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps. The results are joint work with Teiko Heinosaari and Ion Nechita (arXiv:2202.00725 [1]).

Ludwig Hruza - The Quantum Symmetric Simple Exclusion Process: A toy model for coherent fluctuations in out-of-equilibrium many-body quantum systems

In general, the difficulty to characterize non-equilibrium systems lies in the fact that there is no analog of the Boltzmann-distribution to describe thermodynamic variables and their fluctuations. Over the last 20 years, however, it was observed that there is a class of classical non-equilibrium systems with diffusive transport in which the statistics of density and current profiles show universal properties that do not depend on the microscopic details of the model. The general framework to characterise these systems from a macroscopic point of view is today called the  “Macroscopic Fluctuation Theory”. A natural question is whether this framework can be extended to quantum mechanics to describe the statistics of purely quantum mechanical effects such as entanglement in diffusive out-of-equilibrium systems. With this aim in mind, I will introduce the Quantum Simple Symmetric Exclusion Process (Q-SSEP), a microscopic toy model, from which we hope to gain insights in possible universal features of these quantum mechanical effects. I will present the results obtained so far and comment shortly on the recent observation that free cumulants, a tool from free probability theory, seems to play a role in the mathematical structure of the model.

Arthur Braida - Locality and Approximation in quantum annealing

Quantum Annealing (QA) is a computational framework where a quantum system's continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework. Eventually, we discuss theoretical and experimental arguments for further improvements.

Reda Chhaibi - Filtering and strong noise

Guillaume Aubrun - Monogamy of entanglement in cones

Satvik Singh - Detecting positive quantum capacities of quantum channels

Scientific organizers

• Denis Bernard (CNRS, ENS)
• Cecilia Lancien (CNRS, U Grenoble Alpes)
• Ion Nechita (CNRS, U Toulouse III)
• Clément Pellegrini (U Toulouse III)
• Denis Rochette (U Toulouse III)

Administration

• Malika Bentour (LPT Toulouse)
• Mathilde Rasolomalala (LPT Toulouse)

To contact the organizers, send an email to esquisses2022@sciencesconf.org.