21,998 research outputs found
Perfect forms over totally real number fields
A rational positive-definite quadratic form is perfect if it can be
reconstructed from the knowledge of its minimal nonzero value m and the finite
set of integral vectors v such that f(v) = m. This concept was introduced by
Voronoi and later generalized by Koecher to arbitrary number fields. One knows
that up to a natural "change of variables'' equivalence, there are only
finitely many perfect forms, and given an initial perfect form one knows how to
explicitly compute all perfect forms up to equivalence. In this paper we
investigate perfect forms over totally real number fields. Our main result
explains how to find an initial perfect form for any such field. We also
compute the inequivalent binary perfect forms over real quadratic fields
Q(\sqrt{d}) with d \leq 66.Comment: 11 pages, 2 figures, 1 tabl
Modular forms and elliptic curves over the cubic field of discriminant -23
Let F be the cubic field of discriminant -23 and let O be its ring of
integers. By explicitly computing cohomology of congruence subgroups of
GL(2,O), we computationally investigate modularity of elliptic curves over F.Comment: Incorporated referee's comment
Noise in One-Dimensional Measurement-Based Quantum Computing
Measurement-Based Quantum Computing (MBQC) is an alternative to the quantum
circuit model, whereby the computation proceeds via measurements on an
entangled resource state. Noise processes are a major experimental challenge to
the construction of a quantum computer. Here, we investigate how noise
processes affecting physical states affect the performed computation by
considering MBQC on a one-dimensional cluster state. This allows us to break
down the computation in a sequence of building blocks and map physical errors
to logical errors. Next, we extend the Matrix Product State construction to
mixed states (which is known as Matrix Product Operators) and once again map
the effect of physical noise to logical noise acting within the correlation
space. This approach allows us to consider more general errors than the
conventional Pauli errors, and could be used in order to simulate noisy quantum
computation.Comment: 16 page
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