16,901 research outputs found
Quantum Speedup and Categorical Distributivity
This paper studies one of the best known quantum algorithms - Shor's
factorisation algorithm - via categorical distributivity. A key aim of the
paper is to provide a minimal set of categorical requirements for key parts of
the algorithm, in order to establish the most general setting in which the
required operations may be performed efficiently.
We demonstrate that Laplaza's theory of coherence for distributivity provides
a purely categorical proof of the operational equivalence of two quantum
circuits, with the notable property that one is exponentially more efficient
than the other. This equivalence also exists in a wide range of categories.
When applied to the category of finite dimensional Hilbert spaces, we recover
the usual efficient implementation of the quantum oracles at the heart of both
Shor's algorithm and quantum period-finding generally; however, it is also
applicable in a much wider range of settings.Comment: 17 pages, 11 Figure
A Categorical Critical-pair Completion Algorithm
AbstractWe introduce a general critical-pair/completion algorithm, formulated in the language of category theory. It encompasses the KnuthâBendix procedure for term rewriting systems (also modulo equivalence relations), the GrĂśbner basis algorithm for polynomial ideal theory, and the resolution procedure for automated theorem proving. We show how these three procedures fit in the general algorithm, and how our approach relates to other categorical modeling approaches to these algorithms, especially term rewriting
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
Geometry of abstraction in quantum computation
Quantum algorithms are sequences of abstract operations, performed on
non-existent computers. They are in obvious need of categorical semantics. We
present some steps in this direction, following earlier contributions of
Abramsky, Coecke and Selinger. In particular, we analyze function abstraction
in quantum computation, which turns out to characterize its classical
interfaces. Some quantum algorithms provide feasible solutions of important
hard problems, such as factoring and discrete log (which are the building
blocks of modern cryptography). It is of a great practical interest to
precisely characterize the computational resources needed to execute such
quantum algorithms. There are many ideas how to build a quantum computer. Can
we prove some necessary conditions? Categorical semantics help with such
questions. We show how to implement an important family of quantum algorithms
using just abelian groups and relations.Comment: 29 pages, 42 figures; Clifford Lectures 2008 (main speaker Samson
Abramsky); this version fixes a pstricks problem in a diagra
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