25 research outputs found
Constructing cell data for diagram algebras
We show how the treatment of cellularity in families of algebras arising from
diagram calculi, such as Jones' Temperley--Lieb wreaths, variants on Brauer's
centralizer algebras, and the contour algebras of Cox et al (of which many
algebras are special cases), may be unified using the theory of tabular
algebras. This improves an earlier result of the first author (whose hypotheses
covered only the Brauer algebra from among these families).Comment: Approximately 38 pages, AMSTeX. Revised in light of referee comments.
To appear in the Journal of Pure and Applied Algebr
From monoids to hyperstructures: in search of an absolute arithmetic
We show that the trace formula interpretation of the explicit formulas
expresses the counting function N(q) of the hypothetical curve C associated to
the Riemann zeta function, as an intersection number involving the scaling
action on the adele class space. Then, we discuss the algebraic structure of
the adele class space both as a monoid and as a hyperring. We construct an
extension R^{convex} of the hyperfield S of signs, which is the hyperfield
analogue of the semifield R_+^{max} of tropical geometry, admitting a one
parameter group of automorphisms fixing S. Finally, we develop function theory
over Spec(S) and we show how to recover the field of real numbers from a purely
algebraic construction, as the function theory over Spec(S).Comment: 43 pages, 1 figur
Normalizer Circuits and Quantum Computation
(Abridged abstract.) In this thesis we introduce new models of quantum
computation to study the emergence of quantum speed-up in quantum computer
algorithms.
Our first contribution is a formalism of restricted quantum operations, named
normalizer circuit formalism, based on algebraic extensions of the qubit
Clifford gates (CNOT, Hadamard and -phase gates): a normalizer circuit
consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic
phase gates associated to a set , which is either an abelian group or
abelian hypergroup. Though Clifford circuits are efficiently classically
simulable, we show that normalizer circuit models encompass Shor's celebrated
factoring algorithm and the quantum algorithms for abelian Hidden Subgroup
Problems. We develop classical-simulation techniques to characterize under
which scenarios normalizer circuits provide quantum speed-ups. Finally, we
devise new quantum algorithms for finding hidden hyperstructures. The results
offer new insights into the source of quantum speed-ups for several algebraic
problems.
Our second contribution is an algebraic (group- and hypergroup-theoretic)
framework for describing quantum many-body states and classically simulating
quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism
(PSF), wherein quantum states are written as joint eigenspaces of stabilizer
groups of commuting Pauli operators: while the PSF is valid for qubit/qudit
systems, our formalism can be applied to discrete- and continuous-variable
systems, hybrid settings, and anyonic systems. These results enlarge the known
families of quantum processes that can be efficiently classically simulated.
This thesis also establishes a precise connection between Shor's quantum
algorithm and the stabilizer formalism, revealing a common mathematical
structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite
original papers if possible. Appendix E contains unpublished work on Gaussian
unitaries. If you spot typos/omissions please email me at JLastNames at
posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk:
https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism
Posted on my birthda
Subfactors and Applications
The theory of subfactors connects diverse topics in mathematics
and mathematical physics such as tensor categories, vertex operator
algebras, quantum groups, quantum topology, free probability,
quantum field theory, conformal field theory,
statistical mechanics, condensed matter
physics and, of course, operator algebras.
We invited an international group of researchers from these areas
and many fruitful interactions took place during the workshop
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group