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 $F_N=(\omega^{ij})_{N\times N},i,j=0,1,..., N-1, \omega=e^{\frac{2\pi i}{N}}$, can be realized exactly by quantum circuits of size $O(n^2), n=\textrm{log}N$, 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 $F_N$ 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