Quantum information processing is one of the rising forces of the information era. Encoding information within quantum systems and manipulating them promises to lead to great advantages, with three main application domains: quantum cryptography, quantum simulation, and quantum algorithmics. To understand its strengths and limits, we take a transversal stance and seek to capture which resources are granted to us by nature, at the fundamental level, for the sake of computing (e.g. quantum and spatial parallelism). We do so by abstracting away physics’ ability to compute, into formal models of quantum computation (e.g. quantum automata and graph rewriting models). We then verbalize its main structures as quantum programming languages (e.g. quantum lambda-calculus, process algebra). Actually, the process goes both ways, when developments in quantum programming languages lead to the discovery of new structures which may or may not be compilable into formal models of quantum computation, raising the sometimes fascinating question of the physicality of these resources.

One usually distinguishes three main fields of applications of Quantum Computing: quantum cryptography (short-term), quantum simulation (mid-term), quantum algorithmic (long-term). Quantum simulation then divides into two subfields: 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. In particular, as 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.

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:

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.

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.

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.

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.

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.

We achieved the first quantum cellular automaton model for real-life interacting particles, namely electrons and photons, a.k.a 3+1 quantum electrodynamics. We motivated our construction by proposing a discrete version of a fundamental symmetry of Physics, namely gauge symmetry, which turns out to be reminiscent of fault-tolerance in Computer Science. This opens the way for natively discrete formulations of quantum field theories, otherwise renowned for their ill-definedness. (Preprint 14)

We provided a discrete-spacetime action functional for 1+1 quantum electrodynamics, based on a discrete-time quantum walk. Let us explain the relevance of such a result. Remember that there are two mathematical frameworks for constructing a given theory made of quantum fields (matter fields and interaction fields): the path-integral manner, or the operator approach. The previous result follows the operator approach. An advantage of the path-integral approach in continuum spacetime is that it makes certain symmetries more manifest, especially the Lorentz symmetry, which cannot be manifest in the operator approach. We expect our result will bring these benefits to the discrete. (1)

We exploit the ZX-calculus and its simplification heuristic, together with new decompositions of pieces of diagrams into simpler ones to perform classical simulation of quantum programs. On the circuits that were benchmarked, this approach is a significant improvement upon the competition, allowing to simulate within minutes on a consumer laptop circuits of a size that would require a supercalculator to simulate naively. (8)

We designed a graphical language, called "Many-World-calculus" (MWC) that accommodates for both a tensor product and a coproduct in a natural way, where other graphical languages either have only one of the two products, or have to use three-dimensional diagrams with unintuitive ways to compose them. While the tensor product is the usual way to pair quantum systems together, the coproduct allows the representation of "branching", hence quantum control flow. The MWC is given a denotational semantics, allowing us to understand the diagrams as the operator they perform, and is given an equational theory, which we show to be complete with respect to that semantics (i.e. all diagrams that represent the same operator can be turned into one another using the rules of the equational theory). (Preprint:

12)

We introduced a new language called the LOv-calculus which is used for reasoning about linear optical quantum circuits. The language is graphical in nature, and has a specific set of rules to follow. We show that the language is sound and complete, meaning that two LOv-circuits that represent the same quantum process can be transformed into each other with the rules of the LOv-calculus. We also recover several known canonical forms of circuits from the literature. As a follow-up, we derive the first complete equational theory for quantum circuits, the standard language for quantum computation. (

6,

13)

We designed a universal and complete graphical language for the stabilizer fragments for qudits when d is prime. (Published in MFCS2022)

They are different interpretation of quantum mechanics, most of them come with their own mathematical formalism. This was not the case of relational quantum mechanics that was usually only formulated is an additional interpretational layer on top of the usual mathematical framework of quantuml mechanics. In this paper, we propose a refoundation of quantum mechanics whose premises are relational from the start. (Published in Foundations of Physics)

Monads in category theory are algebraic structures that can be used to model
computational effects in programming languages. We show how the notion of
“centre", and more generally “centrality", may be formulated
for strong monads acting on symmetric monoidal categories. We identify three
equivalent conditions which characterise the existence of the centre of a
strong monad and we show that every strong monad on many well-known naturally
occurring categories does admit a centre, thereby showing that this new
notion is ubiquitous. More generally, we study central submonads,
which are necessarily commutative, just like the centre of a strong monad. We
provide a computational interpretation for our ideas by formulating
equational theories of lambda calculi equipped with central submonads, we
describe categorical models for these theories and we prove soundness,
completeness and internal language results for our categorical semantics.
(Preprint: 10)

Vladimir Zamdzhiev has other semantics results published in 2022 (e.g. publications in LICS and POPL), but they are not reported here, because these results were obtained while he was a member of the MOCQUA team.

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.

The internationl consortium on Quantum Information Structure of Spacetime (2020–2023, 2023–2026, qiss.fr) is made of the top researchers, worldwide, on the question of the interaction between quantum information and quantum gravity. The Pablo Arrighi is a member.

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Courses/Undergraduate level

Discounting own students, Pablo Arrighi was PhD jury member for

Pablo Arnault was PhD jury member for Nathanaël Eon (Aix-Marseille U.)