8,161 research outputs found

    Modeling Quantum Behavior in the Framework of Permutation Groups

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    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

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    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 FN=(ωij)N×N,i,j=0,1,...,N−1,ω=e2πiNF_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(n2),n=logNO(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 FNF_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

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    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
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