2025Activity‌​‌ reportProject-TeamQUACS

4.1 Digital​​ quantum simulation

Feynman’s invention​​​‌ of Quantum Computing really‌ came out of a‌​‌ frustration: that of seeing​​ classical computers take such​​​‌ a long time to‌ simulate quantum systems. His‌​‌ intuition was that “quantum​​ computers”' would do a​​​‌ better job at simulating‌ quantum systems. There is‌​‌ not the slightest doubt​​ indeed that quantum simulation​​​‌ will have major outcomes‌ for society. Thinking about‌​‌ it, most of the​​ objects that surround us​​​‌ (cars, computers, furniture…) are‌ designed on computers, thanks‌​‌ to the fact that​​ we can prototype and​​​‌ simulate them on classical‌ computers. That is, up‌​‌ to a certain scale.​​ Below that we are​​​‌ left in the dark‌ as quantum effects come‌​‌ into play, yielding an​​ exponential blow up of​​​‌ the cost or simulation.‌ For now. But, the‌​‌ day we will have​​ good quantum computers and​​​‌ good quantum simulation algorithms‌ to run upon them,‌​‌ we will be able​​ to simulate these particles,​​​‌ atoms, molecules and the‌ way they interact. Consequently‌​‌ we will be able​​ to design specific-purpose molecules,​​​‌ materials, nanotechnologies, with applications‌ in chemistry, biochemistry, electronics,‌​‌ mechanics. At QuaCS we​​ focus on the bottom​​​‌ layer: the quantum-simulation algorithms‌ for fundamental particles. After‌​‌ all, to be able​​ to efficiently simulate fundamental​​​‌ interactions is to be‌ able to simulate virtually‌​‌ everything, from first principles.​​ An added bonus of​​​‌ this strand of research‌ is that usually when‌​‌ we express some physics​​ as a quantum algorithm,​​​‌ it becomes way simpler,‌ more explanatory.

Quantum simulation‌​‌ comes in two flavours:​​ continuous-time quantum simulation, which​​​‌ is very physicky and‌ consists of ad hoc‌​‌ emulation of one Hamiltonian​​ by another, and discrete-time​​​‌ quantum simulation, which is‌ much closer to quantum‌​‌ algorithmic: this is where​​ we stand. In particular,​​​‌ we focus on the‌ provision of a quantum-circuit‌​‌ description of the dynamics​​ of fundamental particles. Moreover,​​​‌ when we design these‌ quantum simulation schemes, our‌​‌ focus is on retaining​​ the symmetries of the​​​‌ simulated model. This is‌ both a matter of‌​‌ efficiency and correctness. For​​ instance, our discretizations have​​​‌ a maximum speed of‌ propagation of the information,‌​‌ which coincides with the​​ speed of light in​​​‌ the simulated system, as‌ a first step towards‌​‌ retaining Lorentz symmetry. Similarly,​​ our discretizations exhibit the​​​‌ gauge symmetries that motivate‌ the different fundamental particles.‌​‌ The long term goal​​ of this program is​​​‌ to provide a satisfactory‌ quantum-circuit descriptions of the‌​‌ whole standard model of​​ particle physics.

4.2 Semantics​​​‌

In the research program‌ on Semantics, the QuaCS‌​‌ team is working on​​​‌ developing mathematical methods and​ tools that formulate the​‌ precise meaning and behavior​​ of (quantum) systems, processes,​​​‌ type systems and programming​ languages, other formal languages​‌ and computational models. This​​ includes, but is not​​​‌ limited, to the following:​

• Operational semantics: a mathematically​‌ precise description of the​​ dynamics of quantum programs​​​‌ and other computational models​ (e.g., the small-step semantics​‌ of quantum lambda calculi,​​ token-machine semantics of quantum​​​‌ diagrammatic calculi).
• Mathematical and​ denotational semantics: a mathematical​‌ interpretation of a quantum​​ programming language, process theory,​​​‌ diagrammatic calculus, etc., which​ is always expected to​‌ be sound and often​​ expected to be adequate​​​‌ or complete.

This line​ of research is focused​‌ on identifying fundamental connections​​ between the static specification​​​‌ (e.g. syntax) of quantum​ languages, their dynamic behavior​‌ (e.g. operational semantics) and​​ their mathematical interpretation (e.g.​​​‌ denotational semantics) with the​ intention of developing each​‌ of these components further.​​

This line of research​​​‌ can reveal interesting connections​ between mathematical structures, computational​‌ models, type systems and​​ other formal languages. Ideally,​​​‌ one endpoint of such​ a connection can be​‌ used to influence the​​ design and development of​​​‌ the other endpoint, because​ these connections can allow​‌ us to improve our​​ understanding of the different​​​‌ aspects of the (quantum)​ systems and computational models​‌ under consideration.

For instance,​​ monads in category theory​​​‌ were the inspiration for​ introducing monads in programming​‌ languages. Another example includes​​ categorical quantum mechanics which​​​‌ lead to the development​ of the ZX-calculus along​‌ with other useful tools,​​ such as PyZX/QuiZX, which​​​‌ may be used for​ optimisation of quantum circuits​‌ and classical simulation of​​ quantum processes.

4.3 Graphical​​​‌ languages and rewriting

One​ of the main features​‌ of graphical languages is​​ that they can be​​​‌ made abstract enough to​ remove unnecessary clutter and​‌ ease reasoning on quantum​​ operators. This has several​​​‌ consequences : They are​ rather intuitive to work​‌ with, while at the​​ same time being completely​​​‌ formal. They can provide​ an intermediate representation of​‌ quantum programs, with enough​​ abstraction to reason about​​​‌ and modify the program​ during compilation. The most​‌ illustrative example of such​​ modification is circuit optimisation,​​​‌ where the goal is​ to reduce the number​‌ of "expensive" quantum gates​​ in the circuit, which​​​‌ can be achieved by​ turning the circuit into​‌ a ZX-diagram, then using​​ its equational theory to​​​‌ perform the reduction. Together​ with the simplification heuristic,​‌ it is possible to​​ exploit this "uncluttering" effect​​​‌ to perform more efficient​ classical simulation of quantum​‌ programs. It can be​​ exploited to perform automated​​​‌ verification of quantum programs.​

The QuaCS team is​‌ involved in the development​​ and study of graphical​​​‌ calculi such as quanctum​ circuits, ZX-, ZW-, ZH-calculi,​‌ but also languages for​​ linear optics, such as​​​‌ the LOv-calculus. These languages​ are supposed to represent​‌ particular features of quantum​​ computing, and hence are​​​‌ designed with a particular​ semantics in mind. A​‌ question of interest in​​ the field is that​​​‌ of completeness with respect​ to that semantics :​‌ the ability to graphically​​ turn any two equivalent​​ diagrams into one another,​​​‌ making it possible to‌ entirely reason within the‌​‌ language. The team is​​ interested in the structure​​​‌ quantum operators have, that‌ can be exhibited by‌​‌ the graphical approach, and​​ depending on the model​​​‌ of computation at hand.‌ It then becomes possible‌​‌ to study the links​​ between the graphical languages,​​​‌ and hence, between the‌ different models of computation.‌​‌ Recently, some focus has​​ been put in the​​​‌ use of graphical languages‌ for the study of‌​‌ indefinite causal orders, a​​ extension to the usual​​​‌ quantum computation model, where‌ not only data is‌​‌ quantum, but also the​​ control flow of the​​​‌ program, which is allowed‌ by the theory but‌​‌ still not well understood.​​

4.4 Programming for quantum​​​‌ computation

The hybrid quantum-classical‌ algorithms that have been‌​‌ developed in the team​​ concern more specifically the​​​‌ solution of linear systems,‌ which is a key‌​‌ function for most scientific​​ and data science applications.​​​‌ For example we have‌ developed a new mixed-precision‌​‌ solver that combines the​​ Quantum Singular Value Transformation​​​‌ with iterative improvement to‌ produce accurate linear system‌​‌ solutions. Furthermore, if we​​ want that quantum computing​​​‌ can address real-world applications‌ then we need to‌​‌ have programming models in​​ order to facilitate the​​​‌ integration of hybrid quantum-classical‌ computing into HPC environments.‌​‌ This implies to extend​​ standard C++ to enable​​​‌ dynamic interaction between classical‌ and quantum resources. On‌​‌ top of the quantum-HPC​​ hybridation aspects, this research​​​‌ focus includes everything constituting‌ tools for the quantum‌​‌ programmer: libraries, synthesis techniques,​​ classical simulation of quantum​​​‌ algorithms, etc.

The current‌ research targets projects with‌​‌ industrial partners: AQED with​​ C12, AeroQat with Alice&Bob​​​‌ and Thales, and a‌ PhD thesis funded by‌​‌ AID (in collaboration with​​ Thales and the laboratory​​​‌ SONDRA). All of these‌ projects are related to‌​‌ compiling, optimizing and analyzing​​ quantum algorithms for the​​​‌ context of numerical computation:‌ not only linear systems,‌​‌ but also differential equations,​​ and matrix inversion and​​​‌ manipulation for radar applications.‌ The team has also‌​‌ a more theoretical approach​​ to quantum programming languages.​​​‌ In particular, one of‌ the current topic of‌​‌ research is concerned with​​ the unification of quantum​​​‌ and classical computation in‌ a single programming framework.‌​‌ If a long-term goal​​ could be the programming​​​‌ of hybrid quantum computation,‌ the near-term goal is‌​‌ the understanding of the​​ capabilities of a model​​​‌ supporting superposition of executions,‌ and how this can‌​‌ be used to expand​​ our understanding of the​​​‌ possibilities offered by quantum‌ computation.

5.1 Quantum Linear Algebra​​ Solver

Participant: Marc Baboulin​​​‌.

10 We address‌ the problem of solving‌​‌ a system of linear​​ equations via the Quantum​​​‌ Singular Value Transformation (QSVT).‌ One drawback of the‌​‌ QSVT algorithm is that​​ it requires huge quantum​​​‌ resources if we want‌ to achieve an acceptable‌​‌ accuracy. To reduce the​​ quantum cost, we propose​​​‌ a hybrid quantum-classical algorithm‌ that improves the accuracy‌​‌ and reduces the cost​​ of the QSVT by​​​‌ adding iterative refinement in‌ mixed-precision A first quantum‌​‌ solution is computed using​​​‌ the QSVT, in low​ precision, and then refined​‌ in higher precision until​​ we get a satisfactory​​​‌ accuracy. For this solver,​ we present an error​‌ and complexity analysis, and​​ first experiments using the​​​‌ quantum software stack myQLM.​

5.2 Tensor Computations

Participant:​‌ Marc Baboulin.

27​​ We present a new​​​‌ mixed precision algorithm to​ compute low-rank matrix and​‌ tensor approximations, a fundamental​​ task in numerous applications​​​‌ in scientific computing and​ data analysis. Our algorithm​‌ is reminiscent of the​​ iterative refinement framework for​​​‌ linear systems: we first​ compute a low-rank approximation​‌ in low precision and​​ then refine its accuracy​​​‌ by iteratively updating it.​ We carry out an​‌ error analysis of our​​ algorithm which proves that​​​‌ we can reach a​ high accuracy while performing​‌ most of the operations​​ in low precision. We​​​‌ measure the computational cost​ of the algorithm, which​‌ depends on the numerical​​ rank of the input​​​‌ (matrix or tensor) as​ well as the speed​‌ ratio between low and​​ high precision arithmetic. We​​​‌ identify two situations where​ our method has a​‌ strong potential: when the​​ hardware provides fast low​​​‌ precision matrix multiply–accumulate units,​ and when the numerical​‌ rank of the input​​ is small at low​​​‌ accuracy levels. We confirm​ experimentally the potential of​‌ our algorithm for computing​​ various low-rank matrix and​​​‌ tensor decompositions such as​ SVD, QR, Tucker, hierarchical​‌ Tucker, and tensor-train.

5.3​​ Quantum Petri Nets

Participant:​​​‌ Marc de Visme.​

28 One of the​‌ main limitation of the​​ model of quantum event​​​‌ structures is the fact​ that even finite systems​‌ can have an infinite​​ representation. This is because​​​‌ event structures are by​ nature an unfolded model.​‌ In the same way​​ that one can unfold​​​‌ a finite automta into​ an infinite tree, one​‌ can unfold a finite​​ Petri Net into an​​​‌ infinite event structure. As​ such, formalising a quantum​‌ version of the Petri​​ Nets is a natural​​​‌ extension of the event​ structure work. Interestingly, the​‌ folded nature of Petri​​ Nets forced us toward​​​‌ a much more local​ approach to representing quantum​‌ computation compared to the​​ one used in quantum​​​‌ event structures, yielding a​ model which is in​‌ many ways simpler than​​ the unfolded model.

5.4​​​‌ Operator Spaces, Linear Logic​ and the Heisenberg-Schrödinger Duality​‌ of Quantum Theory

Participant:​​ Vladimir Zamdzhiev.

12​​​‌ We show that the​ category OS of operator​‌ spaces, with complete contractions​​ as morphisms, is locally​​​‌ countably presentable and a​ model of Intuitionistic Linear​‌ Logic in the sense​​ of Lafont. We then​​​‌ describe a model of​ Classical Linear Logic, based​‌ on OS, whose duality​​ is compatible with the​​​‌ Heisenberg-Schrödinger duality of quantum​ theory. We also show​‌ that OS provides a​​ good setting for studying​​​‌ pure state and mixed​ state quantum information, the​‌ interaction between the two,​​ and even higher-order quantum​​​‌ maps such as the​ quantum switch. Joint work​‌ with Bert Lindenhovius (Johannes​​ Kepler Universität).

5.5 Quantum​​​‌ Coherence Spaces Revisited: A​ von Neumann (Co)Algebraic Approach​‌

Participants: Thea Li,​​ Vladimir Zamdzhiev.

11​​ We describe a categorical​​​‌ model of MALL (Multiplicative‌ Additive Linear Logic) inspired‌​‌ by the Heisenberg-Schrödinger duality​​ of finite-dimensional quantum theory.​​​‌ Proofs of formulas with‌ positive logical polarity correspond‌​‌ to CPTP (completely positive​​ trace-preserving) maps in our​​​‌ model, i.e. the quantum‌ operations in the Schrödinger‌​‌ picture, whereas proofs of​​ formulas with negative logical​​​‌ polarity correspond to CPU‌ (completely positive unital) maps,‌​‌ i.e. the quantum operations​​ in the Heisenberg picture.​​​‌ The mathematical development is‌ based on noncommutative geometry‌​‌ and finite-dimensional von Neumann​​ (co)algebras, which can be​​​‌ defined as special kinds‌ of (co)monoid objects internal‌​‌ to the category of​​ finite-dimensional operator spaces.

5.6​​​‌ Combining Quantum and Classical‌ Control

Participants: Kinnari Dave‌​‌, Vladimir Zamdzhiev.​​

820 The two​​​‌ main notions of control‌ in quantum programming languages‌​‌ are often referred to​​ as “quantum” control and​​​‌ “classical” control. With the‌ latter, the control flow‌​‌ is based on classical​​ information, potentially resulting from​​​‌ a quantum measurement, and‌ this paradigm is well-suited‌​‌ to mixed state quantum​​ computation. Whereas with quantum​​​‌ control, we are primarily‌ focused on pure quantum‌​‌ computation and there the​​ “control” is based on​​​‌ superposition. The two paradigms‌ have not mixed well‌​‌ traditionally and they are​​ almost always treated separately.​​​‌ In this work, we‌ show that the paradigms‌​‌ may be combined within​​ the same system. The​​​‌ key ingredients for achieving‌ this are: (1) syntactically:‌​‌ a modality for incorporating​​ pure quantum types into​​​‌ a mixed state quantum‌ type system; (2) operationally:‌​‌ an adaptation of the​​ notion of “quantum configuration”​​​‌ from quantum lambda-calculi, where‌ the quantum data is‌​‌ replaced with pure quantum​​ primitives; (3) denotationally: suitable​​​‌ (sub)categories of Hilbert spaces,‌ for pure computation and‌​‌ von Neumann algebras, for​​ mixed state computation in​​​‌ the Heisenberg picture of‌ quantum mechanics. Joint work‌​‌ with Louis Lemonnier (University​​ of Edinburgh) and Romain​​​‌ Péchoux (Mocqua team).

5.7‌ IMALL with a Mixed-State‌​‌ Modality: A Logical Approach​​ to Quantum Computation

Participants:​​​‌ Kinnari Dave, Vladimir‌ Zamdzhiev.

7 We‌​‌ introduce a proof language​​ for Intuitionistic Multiplicative Additive​​​‌ Linear Logic (IMALL), extended‌ with a modality $\mathrm{B‌‌}$ to capture mixed-state quantum​​ computation. The language supports​​​‌ algebraic constructs such as‌ linear combinations, and embeds‌​‌ pure quantum computations within​​ a mixed-state framework via​​​‌ $B$, interpreted categorically‌ as a functor from‌​‌ a category of Hilbert​​ Spaces to a category​​​‌ of finite-dimensional C*-algebras. Measurement‌ arises as a definable‌​‌ term, not as a​​ constant, and the system​​​‌ avoids the use of‌ quantum configurations, which are‌​‌ part of the theory​​ of the quantum lambda​​​‌ calculus. Cut-elimination is defined‌ via a composite reduction‌​‌ relation, and shown to​​ be sound with respect​​​‌ to the denotational interpretation.‌ We prove that any‌​‌ linear map on ${\mathrm{ℂ}}^{2^n}$ can be​​​‌ represented within the system,‌ and illustrate this expressiveness‌​‌ with examples such as​​ quantum teleportation and the​​​‌ quantum switch. Joint work‌ with Alejandro Díaz-Caro (Mocqua).‌​‌

5.8 Resource-Efficient Synthesis of​​ Sparse Quantum States

Participants:​​​‌ Sunheang Ty, Renaud‌ Vilmart.

25 Preparing‌​‌ a quantum circuit that​​​‌ implements a given sparse​ state is an important​‌ building block that is​​ necessary for many different​​​‌ quantum algorithms. In the​ context of fault-tolerant quantum​‌ computing, the so-called non-Clifford​​ gates are much more​​​‌ expensive to perform than​ the Clifford ones. We​‌ hence provide an algorithm​​ for synthesizing sparse quantum​​​‌ states with a special​ care for quantum resources.​‌ The circuit depth, ancilla​​ count, and crucially non-Clifford​​​‌ count of the circuit​ produced by the algorithm​‌ are all linear in​​ the sparsity. We conjecture​​​‌ that the non-Clifford count​ complexity is tight, and​‌ show a weakened version​​ of this claim. The​​​‌ first key component of​ the algorithm is the​‌ synthesis of a generalized​​ W-state. We provide a​​​‌ tree-based circuit construction approach,​ and the relationship between​‌ the tree's structure and​​ the circuit's complexity. The​​​‌ second key component is​ a classical reversible circuit​‌ implementing a permutation that​​ maps the basis states​​​‌ of the W-state to​ those of the sparse​‌ quantum state. We reduce​​ this problem to the​​​‌ diagonalization of a binary​ matrix, using a specific​‌ set of elementary matrix​​ operations corresponding to the​​​‌ classical reversible gates. We​ then solve this problem​‌ using a new version​​ of Gauss-Jordan elimination, that​​​‌ minimizes the circuit complexities​ including circuit depth using​‌ parallel elimination steps.

5.9​​ Numerical Experiments Using Block-Diagonalization​​​‌ Technique for Solving Poisson’s​ Equation

Participants: Sunheang Ty​‌, Renaud Vilmart.​​

14 This work presents​​​‌ numerical experiments aimed at​ verifying solutions of Poisson’s​‌ equation using two existing​​ methodologies. First, block-diagonalization is​​​‌ employed to block-encode the​ matrix derived from Poisson’s​‌ equation through the finite​​ difference method (FDM), significantly​​​‌ improving computational complexity from​ N to log(N), where​‌ N is the matrix​​ size. Second, the Quantum​​​‌ Singular Value Transformation (QSVT)​ algorithm is applied to​‌ invert the matrix. However,​​ while block-diagonalization improves the​​​‌ complexity in N, QSVT​ introduces a bottleneck due​‌ to its linear dependency​​ on the condition number​​​‌ κ, which grows exponentially​ with N, posing challenges​‌ for large-scale problems. As​​ far as we know,​​​‌ this is the first​ numerical experiments solving problems​‌ with matrix size N​​ = 1024 and condition​​​‌ number κ = 500000;​ the largest matrix size​‌ and condition number from​​ existing works are 16​​​‌ and < 100, respectively.​

5.10 The Many-Worlds Calculus​‌

Participants: Kostia Chardonnet,​​ Marc de Visme,​​​‌ Benoît Valiron, Renaud​ Vilmart.

6 In​‌ this paper, we explore​​ the interaction between two​​​‌ monoidal structures: a multiplicative​ one, for the encoding​‌ of pairing, and an​​ additive one, for the​​​‌ encoding of choice. We​ propose a colored PROP​‌ to model computation in​​ this framework, where the​​​‌ choice is parameterized by​ an algebraic side effect:​‌ the model can support​​ regular tests, probabilistic and​​​‌ non-deterministic branching, as well​ as quantum branching, i.e.​‌ superposition. The graphical language​​ comes equipped with a​​​‌ denotational semantics based on​ linear applications, and an​‌ equational theory. We prove​​ the language to be​​​‌ universal, and the equational​ theory to be complete​‌ with respect to this​​ semantics.

5.11 The Tensor-Plus​​ Calculus

Participants: Kostia Chardonnet​​​‌, Marc de Visme‌, Benoît Valiron,‌​‌ Renaud Vilmart.

19​​ We propose a graphical​​​‌ language that accommodates two‌ monoidal structures: a multiplicative‌​‌ one for pairing and​​ an additional one for​​​‌ branching. In this colored‌ PROP, whether wires in‌​‌ parallel are linked through​​ the multiplicative structure or​​​‌ the additive structure is‌ implicit and determined contextually‌​‌ rather than explicitly through​​ tapes, world annotations, or​​​‌ other techniques, as is‌ usually the case in‌​‌ the literature. The diagrams​​ are used as parameter​​​‌ elements of a commutative‌ semiring, whose choice is‌​‌ determined by the kind​​ of computation we want​​​‌ to model, such as‌ non-deterministic, probabilistic, or quantum.‌​‌ Given such a semiring,​​ we provide a categorical​​​‌ semantics of diagrams and‌ show the language as‌​‌ universal for it. We​​ also provide an equational​​​‌ theory to identify diagrams‌ that share the same‌​‌ semantics and show that​​ the theory is sound​​​‌ and complete and captures‌ semantical equivalence. In categorical‌​‌ terms, we design an​​ internal language for semiadditive​​​‌ categories (C,+,0) with a‌ symmetric monoidal structure (C,x,1)‌​‌ distributive over it, and​​ such that the homset​​​‌ C(1,1) is isomorphic to‌ a given commutative semiring,‌​‌ e.g., the semiring of​​ non-negative real numbers for​​​‌ the probabilistic case.

5.12‌ The Decohered ZX-calculus

Participant:‌​‌ Renaud Vilmart.

The​​ discard ZX-calculus is known​​​‌ to be complete and‌ universal for mixed-state quantum‌​‌ mechanics, allowing for both​​ quantum and classical processes.​​​‌ However, if the quantum‌ aspects of ZX-calculus have‌​‌ been explored in depth,​​ little work has been​​​‌ done on the classical‌ side. In this paper,‌​‌ we investigate a fragment​​ of discard ZX-calculus obtained​​​‌ by decohering the usual‌ generators of ZX-calculus. We‌​‌ show that this calculus​​ is universal and complete​​​‌ for affinely supported probability‌ distributions over ${𝔽}_{\mathrm{2‌‌}}^n$ . To do​​ so, we exhibit a​​​‌ normal form, mixing ideas‌ from the graphical linear‌​‌ algebra program and diagrammatic​​ Fourier transforms. Our results​​​‌ both clarify how to‌ handle hybrid classical-quantum processes‌​‌ in the discard ZX-calculus​​ and pave the way​​​‌ to the picturing of‌ more general random variables‌​‌ and probabilistic processes.

5.13​​ Minimality in Finite-Dimensional ZW-Calculi​​​‌

Participants: Marc de Visme‌, Renaud Vilmart.‌​‌

16 The ZW-calculus is​​ a graphical language capable​​​‌ of representing 2-dimensional quantum‌ systems (qubit) through its‌​‌ diagrams, and manipulating them​​ through its equational theory.​​​‌ We extend the formalism‌ to accommodate finite dimensional‌​‌ Hilbert spaces beyond qubit​​ systems. First we define​​​‌ a qudit version of‌ the language, where all‌​‌ systems have the same​​ arbitrary finite dimension d,​​​‌ and show that the‌ provided equational theory is‌​‌ both complete - i.e.​​ semantical equivalence is entirely​​​‌ captured by the equations‌ - and minimal -‌​‌ i.e. none of the​​ equations are consequences of​​​‌ the others. We then‌ extend the graphical language‌​‌ further to allow for​​ mixed-dimensional systems. We again​​​‌ show the completeness and‌ minimality of the provided‌​‌ equational theory.

5.14 Causal​​ Decompositions of 1D Quantum​​​‌ Cellular Automata

Participants: Pablo‌ Arrighi, Octave Mestoudjian‌​‌.

23 Understanding quantum​​​‌ theory's causal structure stands​ out as a major​‌ matter, since it radically​​ departs from classical notions​​​‌ of causality. We present​ advances in the research​‌ program of causal decompositions,​​ which investigates the existence​​​‌ of an equivalence between​ the causal and the​‌ compositional structures of unitary​​ channels. Our results concern​​​‌ one-dimensional Quantum Cellular Automata​ (1D QCAs), i.e. unitary​‌ channels over a line​​ of N quantum systems​​​‌ (with or without periodic​ boundary conditions) that feature​‌ a causality radius r:​​ a given input cannot​​​‌ causally influence outputs at​ a distance more than​‌ r. We prove that,​​ they admit a causal​​​‌ decompositions: a unitary channel​ is a 1D QCA​‌ if and only if​​ it can be decomposed​​​‌ into a unitary routed​ circuit of nearest-neighbour interactions,​‌ in which its causal​​ structure is compositionally obvious.​​​‌

5.15 On Quantum Superpositions​ of Graphs, No-Signalling and​‌ Covariance

Participant: Pablo Arrighi​​.

26 We provide​​​‌ a mathematically and conceptually​ robust notion of quantum​‌ superpositions of graphs. We​​ argue that, crucially, quantum​​​‌ superpositions of graphs require​ node names for their​‌ correct alignment, which we​​ demonstrate through a no-signalling​​​‌ argument. Nevertheless, node names​ are a fiducial construct,​‌ serving a similar purpose​​ to the labelling of​​​‌ points through a choice​ of coordinates in continuous​‌ space. Graph renamings are​​ understood as a change​​​‌ of coordinates on the​ graph and correspond to​‌ a natively discrete analogue​​ of diffeomorphisms. We postulate​​​‌ renaming invariance as a​ symmetry principle in discrete​‌ topology of similar weight​​ to diffeomorphism invariance in​​​‌ the continuous. We show​ how to impose renaming​‌ invariance at the level​​ of quantum superpositions of​​​‌ graphs.

5.16 A Large-Scale​ Distributed Framework for Quantum​‌ Irregular Dynamics Simulations

Participant:​​ Pablo Arrighi.

15​​​‌ In traditional quantum computing,​ e.g. in the quantum​‌ circuit model, the size​​ of the data structure​​​‌ describing basis elements is​ well known, because the​‌ dimensionality is fixed. General​​ quantum systems, however, exhibit​​​‌ basis elements of variable​ size, and state spaces​‌ having dynamically unbounded, possibly​​ infinite dimensionality, e.g. for​​​‌ quantum Turing machines or​ quantum field theories. When​‌ seeking to simulate them​​ classically, this imposes an​​​‌ irregularity on both the​ memory representation of basis​‌ elements and the sparsity​​ of the quantum transformations​​​‌ they undergo. Moreover, the​ high dimensionality of these​‌ problems often makes them​​ memory intensive, potentially requiring​​​‌ truncation methods during the​ simulation. One prototypical example​‌ of this would be​​ quantum causal graph dynamics​​​‌ (QCGD), which feature superpositions​ of colored graphs of​‌ different shapes and sizes,​​ driven by the application​​​‌ of local quantum transformations.​ Numerical observations show that​‌ their reversible counterparts typically​​ grow in size; understanding​​​‌ how this is affected​ in the quantum regime​‌ is an arduous computational​​ challenge requiring a particular​​​‌ HPC expertise. In this​ work, we address this​‌ challenge by developing a​​ computational framework for a​​​‌ scalable simulation of such​ general irregular quantum systems​‌ in distributed-memory parallel environments.​​ We lay out the​​​‌ computational challenges arising from​ the nature of such​‌ simulations and then propose​​ effective parallelization, load balancing,​​ memory management, and parallel​​​‌ sampling strategies to accelerate‌ them. We report parallel‌​‌ scalability and accuracy results​​ for up to 1548​​​‌ MPI processes on a‌ parallel cluster using our‌​‌ framework for the QCGD​​ simulation.

5.17 A Curry-Howard​​​‌ Correspondence for Linear, Reversible‌ Computation

Participant: Benoît Valiron‌​‌.

5 In this​​ paper, we present a​​​‌ linear and reversible programming‌ language with inductives types‌​‌ and recursion. The semantics​​ of the languages is​​​‌ based on pattern-matching; we‌ show how ensuring syntactical‌​‌ exhaustivity and non-overlapping of​​ clauses is enough to​​​‌ ensure reversibility. The language‌ allows to represent any‌​‌ Primitive Recursive Function. We​​ then give a Curry-Howard​​​‌ correspondence with the logic‌ μMALL: linear logic extended‌​‌ with least fixed points​​ allowing inductive statements. The​​​‌ critical part of our‌ work is to show‌​‌ how primitive recursion yields​​ circular proofs that satisfy​​​‌ μMALL validity criterion and‌ how the language simulates‌​‌ the cut-elimination procedure of​​ μMALL.

5.18 A Rewriting​​​‌ Theory for Quantum Lambda-Calculus‌

Participant: Benoît Valiron.‌​‌

9 Quantum lambda calculus​​ has been studied mainly​​​‌ as an idealized programming‌ language - the evaluation‌​‌ essentially corresponds to a​​ deterministic abstract machine. Very​​​‌ little work has been‌ done to develop a‌​‌ rewriting theory for quantum​​ lambda calculus. Recent advances​​​‌ in the theory of‌ probabilistic rewriting give us‌​‌ a way to tackle​​ this task with tools​​​‌ unavailable a decade ago.‌ Our primary focus are‌​‌ standardization and normalization results.​​

5.19 Some Results on​​​‌ Causal Modalities in General‌ Spacetimes

Participant: Marco Lewis‌​‌.

29 Causality is​​ one of the fundamental​​​‌ structures of spacetimes, it‌ determines the possible behaviour‌​‌ and propagation of physical​​ information through different relations.​​​‌ Causal structure can be‌ analysed through the various‌​‌ modal logics it induces.​​ The modal logics for​​​‌ the standard chronological and‌ causal relations of the‌​‌ archetypal Minkowski spacetime have​​ been classified. However only​​​‌ partial results have been‌ achieved for the irreflexive‌​‌ variant of the causal​​ relation, also known as​​​‌ the after relation. Our‌ work continues this analysis‌​‌ towards arbitrary spacetimes. By​​ utilizing the definition of​​​‌ the causal relations through‌ causal paths, we can‌​‌ lift known results about​​ the modal logics of​​​‌ Minkowski spacetime to general‌ spacetimes. In particular, for‌​‌ the after relation, we​​ show that a previously​​​‌ studied formula within the‌ logics of Minkowski spacetime‌​‌ holds in arbitrary spacetimes.​​ We introduce a related​​​‌ modal formula that demonstrates‌ that the logic of‌​‌ two-dimensional spacetimes are more​​ expressive than higher dimensional​​​‌ ones. Lastly, we study‌ the interrelation between the‌​‌ logical properties and physical​​ properties along the causal​​​‌ ladder, a classification of‌ causal structures according to‌​‌ a hierarchy of physically​​ relevant properties.

5.20 Canonical​​​‌ Quantization of the Complex‌ Scalar Field without Making‌​‌ Use of its Real​​ and Imaginary Parts

Participant:​​​‌ Pablo Arnault.

17‌ We proceed to the‌​‌ canonical quantization of the​​ complex scalar field without​​​‌ making use of its‌ real and imaginary parts.‌​‌ Our motivation is to​​ formally connect, as tightly​​​‌ as possible, the quantum-field‌ notions of particle and‌​‌ antiparticle—most prominently represented, formally,​​​‌ by creation and annihilation​ operators—to the initial classical​‌ field theory—whose main formal​​ object is the field​​​‌ amplitude at a given​ spacetime point. Our point​‌ of view is that​​ doing this via the​​​‌ use of the real​ and imaginary parts of​‌ the field is not​​ satisfying. The derivation demands​​​‌ to consider, just before​ quantization, the field and​‌ its complex conjugate as​​ independent fields, which yields​​​‌ a system of two​ copies of independent complex​‌ scalar fields. One then​​ proceeds to quantization with​​​‌ these two copies, which​ leads to the introduction​‌ of two families of​​ creation and annihilation operators,​​​‌ corresponding to particles on​ the one hand, and​‌ antiparticles on the other​​ hand. One realizes that​​​‌ having two such families​ is the only hope​‌ for being able to​​ "invert" the definitions of​​​‌ the creation and annihilation​ in terms of the​‌ Fourier quantized fields, so​​ as to obtain an​​​‌ expression of the direct-space​ fields in terms of​‌ these creation and annihilation​​ operators, because the real-field​​​‌ condition used in the​ case of a real​‌ scalar field does not​​ hold for a complex​​​‌ scalar field. This hope​ is then met by​‌ introducing the complex-conjugate constraint​​ at the quantum level,​​​‌ that is, that the​ second independent field copy​‌ is actually the complex​​ conjugate of the first.​​​‌ All standard results are​ then recovered in a​‌ rigorous and purely deductive​​ way. While we reckon​​​‌ our derivation exists in​ the literature, we have​‌ not found it.

6.1 Quandela​

Participants: Benoît Valiron,​‌ Pablo Arrighi, Nicolas​​ Heurtel.

In the​​​‌ context of a PhD​ funded by CIFRE, QuaCS​‌ and Quandela are building​​ a collaboration on the​​​‌ study of quantum linear​ optics. The approach is​‌ both theoretical –with the​​ development of a formal​​​‌ language for reasoning on​ optical circuits, and practical,​‌ targeted towards simulation.

6.2​​ Eviden

Participants: Marc Baboulin​​​‌, Océane Koska.​

The collaboration with Eviden​‌ (Cyril Allouche's group) is​​ related to Marc Baboulin's​​​‌ research on the convergence​ between Quantum Computing and​‌ HPC. It consists mainly​​ in designing hybrid quantum-classical​​​‌ algorithms that could accelerate​ existing HPC applications in​‌ scientific computing, data analytics​​ and optimization. For instance​​​‌ we are currently working​ on linear algebra solvers​‌ that leverage the capabilities​​ of quantum and classical​​​‌ processors to compute accurate​ solutions. We also work​‌ on hybrid programming models​​ that enable to concretely​​​‌ implement these algorithms in​ existing HPC codes.

7.1​​ International initiatives

7.2‌ International research visitors

7.3 European‌​‌ initiatives

7.4 National initiatives‌​‌

7.5​​ Regional initiatives

8.1​ Promoting scientific activities

8.2 Teaching​​ - Supervision - Juries​​​‌ - Educational and pedagogical‌ outreach

8.3 Popularization

Pablo Arrighi​‌ 's previous result on​​ the Arrow of Time​​​‌ (cf. last year's report)​ inspired two popular science​‌ papers:

• Pour La Science​​ in August
• New Scientist​​​‌ in July

9.1 Major publications

• 1​‌ articleP.Pablo Arrighi​​, G.Gilles Dowek​​​‌ and A.Amélia Durbec​. Time arrow without​‌ past hypothesis: a toy​​ model explanation.New​​​‌ Journal of Physics26​11November 2024,​‌ 113019HALDOI
• 2​​ articleM.Marc Baboulin​​​‌, O.Oguz Kaya​, T.Theo Mary​‌ and M.Matthieu Robeyns​​. Mixed precision iterative​​​‌ refinement for low-rank matrix​ and tensor approximations.​‌SIAM Journal on Scientific​​ Computing4752025​​​‌, A2906-A2935HALDOI​
• 3 inproceedingsA.Alexandre​‌ Clément, N.Nicolas​​ Heurtel, S.Shane​​​‌ Mansfield, S.Simon​ Perdrix and B.Benoît​‌ Valiron. A Complete​​ Equational Theory for Quantum​​​‌ Circuits.38th Annual​ ACM/IEEE Symposium on Logic​‌ in Computer Science (LICS)​​2023 38th Annual ACM/IEEE​​​‌ Symposium on Logic in​ Computer Science (LICS)Boston,​‌ United StatesIEEEJuly​​ 2023, 1-13HAL​​​‌DOI
• 4 articleN.​Nathanaël Eon, G.​‌ D.Giuseppe Di Molfetta​​, G.Giuseppe Magnifico​​​‌ and P.Pablo Arrighi​. A relativistic discrete​‌ spacetime formulation of 3+1​​ QED.Quantum7​​​‌November 2023, 1179​HALDOI

9.2 Publications​‌ of the year

9.3 Cited publications​​​‌

• 26 unpublishedP.Pablo​ Arrighi, M.Marios​‌ Christodoulou and A.Amélia​​ Durbec. On quantum​​​‌ superpositions of graphs, no-signalling​ and covariance.November​‌ 2020, working paper​​ or preprintHALback​​​‌ to text
• 27 article​M.Marc Baboulin,​‌ O.Oguz Kaya,​​ T.Theo Mary and​​​‌ M.Matthieu Robeyns.​ Mixed precision iterative refinement​‌ for low-rank matrix and​​ tensor approximations.SIAM​​​‌ Journal on Scientific Computing​4752025,​‌ A2906-A2935HALDOIback​​ to text
• 28 unpublished​​​‌J. S.Julien Saan​ Joachim, M.Marc​‌ de Visme, S.​​Stefan Haar and G.​​​‌Glynn Winskel. Quantum​ Petri Nets with Event​‌ Structure semantics.August​​ 2025, working paper​​​‌ or preprintHALDOI​back to text
• 29​‌ miscM.Marco Lewis​​ and N.Nesta van​​​‌ der Schaaf. Some​ Results on Causal Modalities​‌ in General Spacetimes.​​2026, URL: https://arxiv.org/abs/2601.14029​​​‌back to text