Topological quantum computation may provide a robust approach for encoding and manipulating information utilizing the topological properties of anyonic quasiparticle excitations. We develop an efficient means to map between dense and sparse representations of quantum information (qubits) and a simple construction of multiqubit gates, for all anyon models from Chern-Simons-Witten SU(2)
k theory that support universal quantum computation by braiding ($k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$). In the process, we show how the constructions of topological quantum memory and gates for
k=2,4 connect naturally to those for $k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$, unifying these concepts in a simple framework. Furthermore, we illustrate potential extensions of these ideas to other anyon models outside of Chern-Simons-Witten field theory.
Abstract We investigate quantum circuits for graph representation learning, and propose equivariant quantum graph circuits (EQGCs), as a class of parameterized quantum circuits with strong relational inductive bias for learning over graph-structured data. Conceptually, EQGCs serve as a unifying framework for quantum graph representation learning, allowing us to define several interesting subclasses subsuming existing proposals. In terms of the representation power, we prove that the subclasses of interest are universal approximators for functions over the bounded graph domain. This theoretical perspective on quantum graph machine learning methods opens many directions for further work, and could lead to models with capabilities beyond those of classical approaches. We also provide experimental evidence, and observe that the performance of EQGCs scales well with the depth of the model.
Borrowing ideas from the relation between simply laced Lie algebras and Dynkin diagrams, a weighted graph theory representation of quantum information is addressed. In this way, the density matrix of a quantum state can be interpreted as a signless Laplacian matrix of an associated graph. Using similarities with root systems of simply laced Lie algebras, one-qubit theory is analyzed in some details and is found to be linked to a non-oriented weighted graph having two vertices. Moreover, this one-qubit theory is generalized to n-qubits. In this representation, quantum gates correspond to graph weight operations preserving the probability condition. A speculation from string theory, via D-brane quivers, is also given.