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Peter Selinger

Publications and source records attributed to Peter Selinger.

49 records · Page 3Linked to original sources
arXiv2013

Applying quantitative semantics to higher-order quantum computing

Finding a denotational semantics for higher order quantum computation is a long-standing problem in the semantics of quantum programming languages. Most past approaches to this problem fell short in one way or another, either limiting the language to an unusably small finitary fragment, or giving up important features of quantum physics such as entanglement. In this paper, we propose a denotational semantics for a quantum lambda calculus with recursion and an infinite data type, using constructions from quantitative semantics of linear logic.

cs.LO↗
arXiv2013

An Introduction to Quantum Programming in Quipper

Quipper is a recently developed programming language for expressing quantum computations. This paper gives a brief tutorial introduction to the language, through a demonstration of how to make use of some of its key features. We illustrate many of Quipper's language features by developing a few well known examples of Quantum computation, including quantum teleportation, the quantum Fourier transform, and a quantum circuit for addition.

cs.PL↗
arXiv2013

Quipper: A Scalable Quantum Programming Language

The field of quantum algorithms is vibrant. Still, there is currently a lack of programming languages for describing quantum computation on a practical scale, i.e., not just at the level of toy problems. We address this issue by introducing Quipper, a scalable, expressive, functional, higher-order quantum programming language. Quipper has been used to program a diverse set of non-trivial quantum algorithms, and can generate quantum gate representations using trillions of gates. It 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. Quipper has proven effective and easy to use, and opens the door towards using formal methods to analyze quantum algorithms.

cs.PL↗
arXiv2013

Quantum circuits of T-depth one

We give a Clifford+T representation of the Toffoli gate of T-depth 1, using four ancillas. More generally, we describe a class of circuits whose T-depth can be reduced to 1 by using sufficiently many ancillas. We show that the cost of adding an additional control to any controlled gate is at most 8 additional T-gates, and T-depth 2. We also show that the circuit THT does not possess a T-depth 1 representation with an arbitrary number of ancillas initialized to 0.

quant-ph↗
arXiv2013

Exact synthesis of multiqubit Clifford+T circuits

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 Z[1/sqrt(2),i]. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+T circuit from any such n-qubit operator. We also characterize the Clifford+T operators that can be represented without ancillas.

quant-ph↗
arXiv2013

Presheaf models of quantum computation: an outline

This paper outlines the construction of categorical models of higher-order quantum computation. We construct a concrete denotational semantics of Selinger and Valiron's quantum lambda calculus, which was previously an open problem. We do this by considering presheaves over appropriate base categories arising from first-order quantum computation. The main technical ingredients are Day's convolution theory and Kelly and Freyd's notion of continuity of functors. We first give an abstract description of the properties required of the base categories for the model construction to work. We then exhibit a specific example of base categories satisfying these properties.

math.CT↗
arXiv2012

Proceedings 8th International Workshop on Quantum Physics and Logic

This volume contains the proceedings of the 8th International Workshop on Quantum Physics and Logic (QPL 2011), which was held October 27-29, 2011 at Radboud University Nijmegen. The goal of this workshop series is to bring together researchers working on mathematical foundations of quantum physics, quantum computing and spatio-temporal causal structures, and in particular those that use logical tools, ordered algebraic and category-theoretic structures, formal languages, semantic methods and other computer science methods for the study of physical behavior in general. Over the past few years, there has been growing activity in these foundational approaches, together with a renewed interest in the foundations of quantum theory, which complement the more mainstream research in quantum computation.

quant-ph↗
arXiv2012

Partially traced categories

This paper deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int-construction.

math.CT↗
arXiv2009

A survey of graphical languages for monoidal categories

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 to physicists, computer scientists, and others who use diagrammatic reasoning. We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs. Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study. Where possible, we provide pointers to more rigorous treatments in the literature. Where we include results that have only been proved in special cases, we indicate this in the form of caveats.

math.CT↗
arXiv2008

A linear-non-linear model for a computational call-by-value lambda calculus (extended abstract)

We give a categorical semantics for a call-by-value linear lambda calculus. Such a lambda calculus was used by Selinger and Valiron as the backbone of a functional programming language for quantum computation. One feature of this lambda calculus is its linear type system, which includes a duplicability operator "!" as in linear logic. Another main feature is its call-by-value reduction strategy, together with a side-effect to model probabilistic measurements. The "!" operator gives rise to a comonad, as in the linear logic models of Seely, Bierman, and Benton. The side-effects give rise to a monad, as in Moggi's computational lambda calculus. It is this combination of a monad and a comonad that makes the present paper interesting. We show that our categorical semantics is sound and complete.

cs.LO↗
arXiv2006

Simplicial cycles and the computation of simplicial trees

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted, and therefore Cohen-Macaulay.

math.AC↗
arXiv2004

A lambda calculus for quantum computation with classical control

The objective of this paper is to develop a functional programming language for quantum computers. We develop a lambda calculus for the classical control model, following the first author's work on quantum flow-charts. We define a call-by-value operational semantics, and we give a type system using affine intuitionistic linear logic. The main results of this paper are the safety properties of the language and the development of a type inference algorithm.

cs.LO↗