**Level: ** M1 - M2

**Contact: ** Augustin Vanrietvelde <augustin.vanrietvelde [at] inria [dot] fr>

**Location: ** Laboratoire Méthodes Formelles, Université Paris-Saclay

**Keywords: ** Quantum Information, indefinite causal order, causality, quantum circuits

Read the internship proposal in pdf.

### Background

Ten years ago, it was discovered that quantum theory is compatible with the existence of exotic causal structures: the order in which events happen can itself be in a quantum superposition [ 1, 2 ]. This novel possibility, called indefinite causal order, has attracted considerable attention, due to two things: its foundational significance as a genuinely new quantum structure, potentially related to new physics, for example quantum gravity; and its practical applications, as it has been proved to yield advantages over causally ordered protocols in various computational and communicational tasks.

In that context, a particular issue has been that of the mathematical framework with which to describe the indefinite causal structures. While standard causal structures enjoy a natural and intuitive diagrammatic representation in terms of acyclic circuits [ 3 ], the non-standard ones cannot be represented without feedback loops. The problem is that very few circuits with feedback loops are valid ones: there needs to be a structural way to determine which ones are acceptable. Overall, this makes the non-standard causal structures extremely difficult to manipulate and understand, which considerably hinders progress in their study.

### The project

Important progress has recently been made on this problem: a way has been found to infer the validity of causal structures from their compositional structure [ 4 ]. This method allows to turn the study of nonstandard causal structures into a (much simpler) study of decorated graphs. This recent work opens the way for many research directions. One is to classify the valid graphs, in order to better understand the conceivable causal structures and their properties. Another is to make the new framework more formal by framing it as a type structure. Finally, one can merge this framework with other recent and complementary proposals, for example many-worlds calculus [ 5 ]. More details can be found in the proposal available here.

The intern will use mathematical methods and will also touch on conceptual aspects of the subject. This internship is also an opportunity to get in touch with current research themes in the very active field of quantum information, so as to find good opportunities for a PhD in this field, including in the Quacs Group.