Introduction to quantum information and its mathematical language
Timothée Hoffreumon, Slovak Academy of Sciences
In this tutorial, we will learn how quantum systems are utilized to process and store information. Rather than focusing on their physical realization, we will focus on the mathematical model of ‘quantum information’.  

We will begin by covering the formalism of ‘pure states’ and their circuit representations. This foundation will help us introduce the postulates of quantum theory as well as some of the main aspects of the ‘quantum magic’, namely superposition, no-cloning, entanglement, and non-locality. To illustrate these concepts, we will examine a few practical quantum protocols that make use of these phenomena. 

Next, we will move on to the ‘mixed state’ formalism and its corresponding circuit representation. This more advanced presentation of quantum circuits allows to discuss the statistical properties of the quantum systems, which is necessary step to arrive at the theory of quantum information. If time permits, we will touch on the basic topics of quantum information, such as noise and error modelling, the entropy of states, and the capacity of channels. 

Causation in quantum theory: from Bell’s scenario to the general case
Roger Colbeck, Department of Mathematics, University of York
I will introduce causal structures starting with an in-depth treatment of the Bell causal structure, explaining how to distinguish classical correlations, quantum correlations and those beyond, and that some quantum correlations are self-testing. I will then move on to more general causal structures, explaining why their analysis is more difficult, and illustrating a few techniques that are useful for deciding whether given correlations are compatible with the causal structure or not.

Poster session
1 – Bayesian Inference in Quantum Programs, Christina Gehnen
2 – Higher order quantum transformations for known input state, Vanessa Brzić
3 – Dynamic Routing in Software-Defined QKD Networks: Overview and Challenges, Hamid Taramit
4 – How Likely Are You to Observe Non-locality with Imperfect Detection Efficiency and Random Measurement Settings?, Paweł Cieśliński
5 – Quantum-enhanced belief propagation, Sheila M. Pérez García
6 – Observational entropy of quantum correlations, Leonardo Rossetti
7 – Variational Inference for Quantum HyperNetworks, Alix Lhéritier
8 – Vacua in Discrete Spacetime, Chaitanya Gupta
9 – Does the Unclonable Bit Exist?, Pierre Botteron
10 – Quantum Algorithms for Optimization Problems, Raneem Madani
11 – Transtatistics: Beyond Bosons And Fermions, Tristan Maleville
12 – Discrete symetries in quantum circuits, Maximilian Mansky
13 – Witnessing PPT entanglement via rank properties of (sub)matrices, Aabhas Gulati
14 – Deriving Entanglement Generation and Swapping Policies in Quantum Networks, Álvaro Troyano Olivas
15 – Magic States for GKP Qubits, Sharon David

Causation in quantum theory: from Bell’s scenario to the general case
Roger Colbeck, Department of Mathematics, University of York
I will introduce causal structures starting with an in-depth treatment of the Bell causal structure, explaining how to distinguish classical correlations, quantum correlations and those beyond, and that some quantum correlations are self-testing.

I will then move on to more general causal structures, explaining why their analysis is more difficult, and illustrating a few techniques that are useful for deciding whether given correlations are compatible with the causal structure or not.

Higher Order Quantum Theory
Matthew Wilson, CentraleSupélec, Université Paris-Saclay
Higher order quantum operations can be motivated in at-least three ways. First, one might simply ask a mathematical question, “which operations can be applied to part of a quantum channel to return a new one”. Second, one might ask, “what are the most general causal correlations compatible with local quantum theory”. Third, one might ask, “what is the right generalisation of non-markovian processes to the quantum regime”. In either case, one lands on variants of the same mathematical objects called higher order quantum operations. In this lecture we will review the motivations and basic theory of higher order processes, along with current open questions in the field. 

Randomness, entropy and accumulation
Peter Brown, Télécom Paris, Institut Polytechnique de Paris.
Quantum random number generators are already a commercially viable quantum technology. However, to claim these random number generators are secure we need rigorous mathematical security proofs. In this lecture we’ll explore the question of what it means for a quantum random number generator to be « secure ». We’ll then relate security of the quantum random number generator to a problem of quantifying entropy and finally we’ll explore how entropy accumulation theorems help to solve these quantification problems.

Network non-locality
Mirjam Weilenmann, INRIA Saclay
Building on the discussion of classical and quantum correlations earlier in the day, this lecture will introduce techniques to distinguish classical from quantum correlations in network scenarios. We will introduce different methods to derive compatibility constraints for network scenarios, where we will focus on the most successful technique or this purpose: the inflation technique. We will further look into the topic of memory attacks and see what these imply for the network topologies needed for applications.

An introduction to Stochastic Master Equation (SME) for open quantum systems
Pierre Rouchon, LPENS, Mines Paris-PSL, Inria, ENS-PSL, Université PSL, CNRS, Sorbonne Université

-1- SME of the photon box: wave-function/density-operator formulation, dispersive/resonant propagator, Markov model, quantum Monte-Carlo trajectories, (super)-martingales, Quantum Non-Demolition (QND) measurement of photons, Bayesian inference to include measurement imperfections and decoherence,   simulation and convergence analysis.

-2- Structure of dynamical models describing open quantum systems including measurement back-action and decoherence: discrete-time models based on quantum channels and left stochastic matrices;  continuous-time models driven by Wiener processes (weak measurement) or Poisson processes (quantum jump and counting measurement), and their time-discretization.

Poster session
1 – Overlap Gap Property limits limit swapping in QAOA, Mark Gosh
2 – Security of DIQKD from multipartite information causality, Lucas Pollyceno
3 – Evaluating different Quantum Hardwares for Graph Cut Optimization, Ali Abbassi
4 – Complexities of mixed Schatten norms of quantum maps, Jan Kochanowski
5 – Benchmarking quantum devices beyond classical capabilities, Marcin Rudziński
6 – Emulation Capacity between Idempotent Channels, Idris Delsol
7 – UniqueNESS: Graph Theory Approach to the Uniqueness of Non-Equilibrium Steady-States & Self-Similarity in the Thermodynamic Limit, Martin Seltmann
8 – Quantum oblivious transfer and coherent-one-way quantum key distribution, Juan José Romero
9 – Ergodic Properties of Quantum Markov Semigroups, Nicolas Mousset
10 – Efficient Classical Simulations via a Beyond-Quantum Many Body Representation, Peter Martin
11 – Quantifying the diabatic error in coupled photonic waveguides, Ankit Singh Bhadauriya
12 – Graphon Quantum Filtering system, Sofiane Chalal
13- Towards Efficient Resource Management in MadQCI, Leduin José Cuenca Macas
14- Orthogonal Faces in the CHSH Scenario, Andrea Zingarofalo
15- Resourcefulness of non-classical continuous-variable quantum gates, Antoine Debray

Zero-error communication under discrete-time Markovian dynamics
Nilanjana Datta, Faculty of Mathematics, University of Cambridge, Pembroke College
Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics ‘scrambles’ the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.

Quantum entropy power inequalities, recent developments and open problems
Salman Beigi, School of Mathematics, Institute for Research in Fundamental Sciences
Entropy power inequalities are significant tools in information theory, probability theory and geometric analysis. Generalization of these inequalities in the quantum realm goes back to more than fifteen years ago. This tutorial is devoted to a review of these inequalities and their significance, technical tools for proving them, and some open problems. 

Quantum Feedback Networks: Theory and Applications in Quantum Control Systems
John Gough, Aberystwyth University
We review the theory of quantum feedback networks and show how it gives a framework for a system theoretic description applicable to quantum engineering. The network rules are presented along with an overview of feedback based control. There are two main approaches: feedback by the observed output signal; and feedback by the actual quantum output itself. We will illustrate this with several current examples from quantum technology, in particular, quantum optics and super-conducting qubits. We introduce the  Hudson-Parthasarathy quantum stochastic calculus and the input-output theory for open quantum systems (known as the « SLH » formalism), and review the theory of quantum feedback networks. We will give an overview of feedback techniques, including current directions in quantum measurement-based and quantum coherent feedback control.

Optimal estimation of quantum Markov chains using coherent absorbers and displaced-null measurements
Madalin Guta, School of Mathematics, University of Nottingham
In this presentation I will discuss the problem of estimating dynamical parameters of a quantum Markov chain. The key tool will be the use of a coherent quantum absorber which transforms the problem into a simpler one pertaining to a system with a pure stationary state at a reference parameter value. Motivated by the proposal in [1] I will consider counting output measurements and show how the statistics of the counts can be used to compute a simple, asymptotically optimal estimator of the unknown parameter. For this, I will introduce translationally invariant modes (TIMs) of the output and show that these modes are Gaussian in the limit of large times and capture the entire quantum  Fisher information of the output. Moreover, the counting measurement provides an effective joint measurement of the TIMs number operators. The unknown parameter is estimated using a two stage estimation procedure. A rough estimator is first computed using a simple measurement, and is used to set the absorber parameter. Due to non-identifiability issues of the counting measurement the reference parameter needs to be shifted away from the initial rough estimator, as shown in the displaced-null measurements theory [2]. Finally, an optimal estimator is computed in terms of the total number of excitations of the TIMs, avoiding the need for expensive estimation procedures. Details can be found in [3].
 
 [1] D. Yang, S. F. Huelga, and M. B. Plenio PRX Quantum 13, 031012 (2023) 
 [2] F. Girotti, A. Godley and M. Guta, arXiv:J. Phys. A 57 245304 (2024)
 [3] F. Girotti, A. Godley and M. Guta, arXiv: 2408.00626

Quantum simulations of quantum many-body systems
Cambyse Rouzé, Inria, Télécom Paris, Institut Polytechnique de Paris
Envisioned by Richard Feynman forty years ago, one of the most promising applications where quantum computers are expected to outperform classical ones is the prediction of physical properties in complex quantum systems. Such advancements have the potential to transform industries like pharmaceuticals and semiconductor chip design, where the expected runtime of classical simulations currently poses a major bottleneck.

In this tutorial, I will discuss key simulation tasks centered on time-evolved, ground, and Gibbs states of geometrically local quantum Hamiltonians, with particular emphasis on their comparison to classical simulation methods. I will then provide a concise overview of the main quantum algorithms used for preparing these states. If time allows, I will conclude with a recent proposal for ground and Gibbs state preparation that promises both quantum advantage and enhanced implementation robustness.

Quantum Query Complexity
Titouan Carette, LIX, Ecole Polytechnique, Institut Polytechnique de Paris
Quantifying quantum advantage is a necessary task that can be accomplished in many ways, different definitions leading to more or less optimist views on quantum computing. From the theoretical point of view, the most successful approach so far is query complexity, where strict separation between classical and quantum complexity are known. In this tutorial, we’ll outline the main definitions and results in this field as well as interesting recent results on more exotic quantum computational models.

Quantum Programming languages with classical control
Benoit Valiron, LMF, Université Paris-Saclay
This lecture is devoted to the programming model of circuit description languages. We first overview a few typical structures found in quantum algorithms and derive the necessary constructors for implementing them. We then analyse several standard approaches found both in the literature and in concrete programming languages. We finally focus on the typical type systems used for quantum circuit description languages.

Quantum programming languages with quantum control
Alejandro Diaz-Caro, INRIA, LORIA
In this lecture, we explore an alternative approach to defining quantum programming languages, in which programs—just like data—can be superposed. This perspective enables the natural representation of interesting configurations, such as the quantum switch. Moreover, such languages are natural candidates for serving as proof languages for substructural logics, such as linear logic, or even for uncovering new kinds of logic.

Graphical languages for quantum computation
Marc de Visme, INRIA, LMF, Université Paris-Saclay
In this lecture, we present graphical languages for quantum computation. While we focus on the most used ones, that is quantum circuits and the ZX calculus, we also give an overview of the diversity existing in graphical languages, the reasons why one might use graphical languages over textual languages, and the theoretical framework underlying.

Emulating Quantum Computation
Simon Martiel, IBM Quantum, IBM France Lab,
In this lecture, we will investigate the different known quantum simulation algorithms, ranging from direct linear algebra simulation and tensor network contraction, to simulation of Clifford circuits through Tableau formalism. We will attempt to list classes of circuits for which efficient simulation techniques are known, and prove hardness of some other classes of circuits. The goal is to provide a collection of tools that can be used to explore quantum algorithms as efficiently as possible, and enough knowledge to discriminate between « hard » and « easy » quantum circuits.

Quantum algorithms for high performance computing
Marc Baboulin, LMF, ENS Paris-Saclay, CNRS, Université Paris-Saclay
In this lecture we explain how core tasks in scientific computing can be addressed by quantum algorithms, possibly combined with classical ones. In particular we describe recent advances in algorithms for decomposing and handling matrices (generic, or coming from PDE’s) in quantum computers. We also present promising methods for the solution of linear systems of equations with improvement in terms of accuracy and cost for the solution.