This work presents a graphical language for dagger compact closed categories, and sketches a proof of its completeness for equational reasoning, and gives a general construction, the CPM construction, which associates to each Dagger compact closed category its ''category of completely positive maps'', and shows that the resulting category is again dagger compactclosed.Expand

This paper describes the syntax and semantics of a simple quantum programming language with high-level features such as loops, recursive procedures, and structured data types, and has an interesting denotational semantics in terms of complete partial orders of superoperators.Expand

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also… Expand

Quipper, a scalable, expressive, functional, higher-order quantum programming language, which is geared towards a model of computation that uses a classical computer to control a quantum device, but is not dependent on any particular model of quantum hardware.Expand

It is proved, via a categorical structure theorem, that the categorical semantics is equivalent to a CPS semantics in the style of Hofmann and Streicher, and that the call-by-name λμ-calculus forms an internal language for the dual co-control categories.Expand

Black-on-white images can be represented either as a bitmap or as a vector outline. A bitmap represents an image as a grid of black or white pixels. A vector outline describes an image via an… Expand

We prove that a unitary matrix has an exact representation over the Clifford+$T$ gate set with local ancillas if and only if its entries are in the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$. Moreover,… Expand

The search for a semantics for higher-order quantum computation leads naturally to the study of categories of normed cones. In the first part of this paper, we develop the theory of continuous normed… Expand

A class of circuits whose T- depth can be reduced to 1 by using sufficiently many ancillas is described, and it is shown that the cost of adding an additional control to any controlled gate is at most 8 additional T-gates, and T-depth 2.Expand