Related papers: Tensors, !-graphs, and non-commutative quantum str…
Recent years have witnessed rapid advances in graph representation learning, with the continuous embedding approach emerging as the dominant paradigm. However, such methods encounter issues regarding parameter efficiency, interpretability,…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…
Quantum networks are important for quantum communication, enabling tasks such as quantum teleportation, quantum key distribution, quantum sensing, and quantum error correction, often utilizing graph states, a specific class of multipartite…
The ZX-Calculus is a graphical language for quantum mechanics. An axiomatisation has recently been proven to be complete for an approximatively universal fragment of quantum mechanics, the so-called Clifford+T fragment. We focus here on the…
Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the…
Quantum operations are the most widely used tool in the theory of quantum information processing, representing elementary transformations of quantum states that are composed to form complex quantum circuits. The class of quantum…
The Calculus of Conjunctive Queries (CCQ) has foundational status in database theory. A celebrated theorem of Chandra and Merlin states that CCQ query inclusion is decidable. Its proof transforms logical formulas to graphs: each query has a…
Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
Clifford algebras are used for definition of spinors. Because of using spin-1/2 systems as an adequate model of quantum bit, a relation of the algebras with quantum information science has physical reasons. But there are simple mathematical…
We define a categorical framework in which we build a systematic construction that provides generic invariants for C*-algebras. The benefit is significant as we show that any invariant arising this way automatically enjoys nice properties…
Encoder-decoder architectures are prominent building blocks of state-of-the-art solutions for tasks across multiple fields where deep learning (DL) or foundation models play a key role. Although there is a growing community working on the…
Several recent papers have proposed increasing the expressive power of graph neural networks by exploiting subgraphs or other topological structures. In parallel, researchers have investigated higher order permutation equivariant networks.…
A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in $\mathbb{R}^d$ (typically $d=2,3$) with…
One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice,…
We define a new mathematical structure ( graph quantum group) which combines the tower of algebras associated with a graph ${\cal G}$ and the structure of a Hopf algebra {\cal A}. In this structure Ocneanu's string operators become Hopf…
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such…
There has been a great deal of attention recently to graphs whose vertex set is a group, defined using the group structure. (The commuting graph, where two elements are joined if they commute, is the oldest and most famous example.) The…
We review briefly the motivations for introducing additional group-theoretic data in tensor models, leading to the richer framework of group field theories, themselves a field theory formulation of loop quantum gravity. We discuss how these…
We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…