Categorical Quantum Circuits

TL;DR

This paper develops a categorical framework for quantum circuits that models finite-dimensional quantum systems of arbitrary dimensions, unifying traditional quantum circuit models with tensor network states using category theory.

Contribution

It introduces a dagger-compact category model that generalizes quantum circuits and tensor networks, enabling new algebraic and diagrammatic methods for quantum information science.

Findings

Framework models finite-dimensional quantum systems of arbitrary dimensions

Circuit diagrams are morphisms in a categorical setting

Applicable to tensor network states like matrix product states

Abstract

In this paper, we extend past work done on the application of the mathematics of category theory to quantum information science. Specifically, we present a realization of a dagger-compact category that can model finite-dimensional quantum systems and explicitly allows for the interaction of systems of arbitrary, possibly unequal, dimensions. Hence our framework can handle generic tensor network states, including matrix product states. Our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science. The circuit diagrams themselves now become morphisms in a category, making quantum circuits a special case of a much more general mathematical framework. We introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information.