Having a good understanding of the behavior of quantum systems allows us to design better strategies for their optimal use in quantum algorithms. For example, tensor network notation is a well-known diagrammatic formalism that has helped researchers compute and reason about quantum systems over the years.
A prominent diagram notation based on tensor networks is the ZX-calculus, which has been used extensively in quantum circuit optimization and compilation in recent years. This notation is based on a set of graphical generators representing linear maps in Hilbert spaces and a set of rewrite rules that can be used to simplify the representation of complex systems.
In this work, EQUALITY partners from Leiden University extend the language of the ZX-calculus to better accommodate behaviors relevant to near-term devices, such as noisy operations. This allows for example to reason about Quantum Error Mitigation (QEM) techniques using diagrams. To do this, the authors use the well-established framework of category theory for reasoning about quantum systems and their graphical representations. The authors achieve this extension of the ZX-calculus by considering monoidal categories that are enriched in the category of Eilenberg-Moore algebras for a monad.
Under the condition that this monad is monoidal and affine, they construct an adjunction between symmetric monoidal categories and symmetric monoidal categories enriched over algebras for the monad. This allows the group to devise an extension, and its semantics, of the ZX-calculus with probabilistic choices by freely enriching over convex algebras, which are the algebras of the finite distribution monad. Finally, they show how this construction can be used for diagrammatic reasoning of noise in quantum systems and QEM techniques.
Read the paper by clicking on the link below.
