8,161 research outputs found
Modeling Quantum Behavior in the Framework of Permutation Groups
Quantum-mechanical concepts can be formulated in constructive finite terms
without loss of their empirical content if we replace a general unitary group
by a unitary representation of a finite group. Any linear representation of a
finite group can be realized as a subrepresentation of a permutation
representation. Thus, quantum-mechanical problems can be expressed in terms of
permutation groups. This approach allows us to clarify the meaning of a number
of physical concepts. Combining methods of computational group theory with
Monte Carlo simulation we study a model based on representations of permutation
groups.Comment: 8 pages, based on plenary lecture at Mathematical Modeling and
Computational Physics 2017, Dubna, July 3--7, 201
Large Fourier transforms never exactly realized by braiding conformal blocks
Fourier transform is an essential ingredient in Shor's factoring algorithm.
In the standard quantum circuit model with the gate set \{\U(2),
\textrm{CNOT}\}, the discrete Fourier transforms , can be realized exactly by
quantum circuits of size , and so can the discrete
sine/cosine transforms. In topological quantum computing, the simplest
universal topological quantum computer is based on the Fibonacci
(2+1)-topological quantum field theory (TQFT), where the standard quantum
circuits are replaced by unitary transformations realized by braiding conformal
blocks. We report here that the large Fourier transforms and the discrete
sine/cosine transforms can never be realized exactly by braiding conformal
blocks for a fixed TQFT. It follows that approximation is unavoidable to
implement the Fourier transforms by braiding conformal blocks
Yang-Mills theory and Tamagawa numbers
Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills
functional to calculate the Betti numbers of moduli spaces of vector bundles
over a Riemann surface, rederiving inductive formulae obtained from an
arithmetic approach which involved the Tamagawa number of SL_n. This article
surveys this link between Yang-Mills theory and Tamagawa numbers, and explains
how methods used over the last three decades to study the singular cohomology
of moduli spaces of bundles on a smooth complex projective curve can be adapted
to the setting of A^1-homotopy theory to study the motivic cohomology of these
moduli spaces.Comment: Accepted for publication in the Bulletin of the London Mathematical
Societ
Ultrametric Logarithm Laws, II
We prove positive characteristic versions of the logarithm laws of Sullivan
and Kleinbock-Margulis and obtain related results in Metric Diophantine
Approximation.Comment: submitted to Montasefte Fur Mathemati
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