80 research outputs found
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
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
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
Abstract structure of unitary oracles for quantum algorithms
We show that a pair of complementary dagger-Frobenius algebras, equipped with
a self-conjugate comonoid homomorphism onto one of the algebras, produce a
nontrivial unitary morphism on the product of the algebras. This gives an
abstract understanding of the structure of an oracle in a quantum computation,
and we apply this understanding to develop a new algorithm for the
deterministic identification of group homomorphisms into abelian groups. We
also discuss an application to the categorical theory of signal-flow networks.Comment: In Proceedings QPL 2014, arXiv:1412.810
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
Normalization for planar string diagrams and a quadratic equivalence algorithm
In the graphical calculus of planar string diagrams, equality is generated by
exchange moves, which swap the heights of adjacent vertices. We show that left-
and right-handed exchanges each give strongly normalizing rewrite strategies
for connected string diagrams. We use this result to give a linear-time
solution to the equivalence problem in the connected case, and a quadratic
solution in the general case. We also give a stronger proof of the Joyal-Street
coherence theorem, settling Selinger's conjecture on recumbent isotopy
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
Orthogonality for Quantum Latin Isometry Squares
Goyeneche et al recently proposed a notion of orthogonality for quantum Latin
squares, and showed that orthogonal quantum Latin squares yield quantum codes.
We give a simplified characterization of orthogonality for quantum Latin
squares, which we show is equivalent to the existing notion. We use this
simplified characterization to give an upper bound for the number of mutually
orthogonal quantum Latin squares of a given size, and to give the first
examples of orthogonal quantum Latin squares that do not arise from ordinary
Latin squares. We then discuss quantum Latin isometry squares, generalizations
of quantum Latin squares recently introduced by Benoist and Nechita, and define
a new orthogonality property for these objects, showing that it also allows the
construction of quantum codes. We give a new characterization of unitary error
bases using these structures.Comment: In Proceedings QPL 2018, arXiv:1901.0947
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