20 research outputs found
A 2-Categorical Analysis of Complementary Families, Quantum Key Distribution and the Mean King Problem
This paper explores the use of 2-categorical technology for describing and
reasoning about complex quantum procedures. We give syntactic definitions of a
family of complementary measurements, and of quantum key distribution, and show
that they are equivalent. We then show abstractly that either structure gives a
solution to the Mean King problem, which we also formulate 2-categorically.Comment: In Proceedings QPL 2014, arXiv:1412.810
Bicategorical Semantics for Nondeterministic Computation
We outline a bicategorical syntax for the interaction between public and
private information in classical information theory. We use this to give
high-level graphical definitions of encrypted communication and secret sharing
protocols, including a characterization of their security properties.
Remarkably, this makes it clear that the protocols have an identical abstract
form to the quantum teleportation and dense coding procedures, yielding
evidence of a deep connection between classical and quantum information
processing. We also formulate public-key cryptography using our scheme.
Specific implementations of these protocols as nondeterministic classical
procedures are recovered by applying our formalism in a symmetric monoidal
bicategory of matrices of relations.Comment: 21 page
Groupoid Semantics for Thermal Computing
A groupoid semantics is presented for systems with both logical and thermal
degrees of freedom. We apply this to a syntactic model for encryption, and
obtain an algebraic characterization of the heat produced by the encryption
function, as predicted by Landauer's principle. Our model has a linear
representation theory that reveals an underlying quantum semantics, giving for
the first time a functorial classical model for quantum teleportation and other
quantum phenomena.Comment: We describe a groupoid model for thermodynamic computation, and a
quantization procedure that turns encrypted communication into quantum
teleportation. Everything is done using higher category theor
The Topology of Quantum Algorithms
We use a categorical topological semantics to examine the Deutsch-Jozsa,
hidden subgroup and single-shot Grover algorithms. This reveals important
structures hidden by conventional algebraic presentations, and allows novel
proofs of correctness via local topological operations, giving for the first
time a satisfying high-level explanation for why these procedures work. We also
investigate generalizations of these algorithms, providing improved analyses of
those already in the literature, and a new generalization of the single-shot
Grover algorithm.Comment: 33 pages. Updated to match the final published articl
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
A classical groupoid model for quantum networks
We give a mathematical analysis of a new type of classical computer network
architecture, intended as a model of a new technology that has recently been
proposed in industry. Our approach is based on groubits, generalizations of
classical bits based on groupoids. This network architecture allows the direct
execution of a number of protocols that are usually associated with quantum
networks, including teleportation, dense coding and secure key distribution
Mixed quantum states in higher categories
There are two ways to describe the interaction between classical and quantum
information categorically: one based on completely positive maps between
Frobenius algebras, the other using symmetric monoidal 2-categories. This paper
makes a first step towards combining the two. The integrated approach allows a
unified description of quantum teleportation and classical encryption in a
single 2-category, as well as a universal security proof applicable
simultaneously to both scenarios.Comment: In Proceedings QPL 2014, arXiv:1412.810
A Classical Groupoid Model for Quantum Networks
We give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution