2,896 research outputs found
Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations
We study how much noise can be tolerated by a universal gate set before it
loses its quantum-computational power. Specifically we look at circuits with
perfect stabilizer operations in addition to imperfect non-stabilizer gates. We
prove that for all unitary single-qubit gates there exists a tight depolarizing
noise threshold that determines whether the gate enables universal quantum
computation or if the gate can be simulated by a mixture of Clifford gates.
This exact threshold is determined by the Clifford polytope spanned by the 24
single-qubit Clifford gates. The result is in contrast to the situation wherein
non-stabilizer qubit states are used; the thresholds in that case are not
currently known to be tight.Comment: 4 pages, 2 figure
Some hyperbolic 4-manifolds with low volume and number of cusps
We construct here two new examples of non-orientable, non-compact, hyperbolic
4-manifolds. The first has minimal volume and two cusps.
This example has the lowest number of cusps among known minimal volume
hyperbolic 4-manifolds. The second has volume and one cusp. It has
lowest volume among known one-cusped hyperbolic 4-manifolds.Comment: 12 pages, 11 figure
A geometrically bounding hyperbolic link complement
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the
geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here
an example of non-compact, finite-volume hyperbolic 3-manifold that
geometrically bounds. The 3-manifold is the complement of a link with eight
components, and its volume is roughly equal to 29.311.Comment: 23 pages, 19 figure
Classification of integrable discrete equations of octahedron type
We use the consistency approach to classify discrete integrable 3D equations
of the octahedron type. They are naturally treated on the root lattice
and are consistent on the multidimensional lattice . Our list includes
the most prominent representatives of this class, the discrete KP equation and
its Schwarzian (multi-ratio) version, as well as three further equations. The
combinatorics and geometry of the octahedron type equations are explained. In
particular, the consistency on the 4-dimensional Delaunay cells has its origin
in the classical Desargues theorem of projective geometry. The main technical
tool used for the classification is the so called tripodal form of the
octahedron type equations.Comment: 53 pp., pdfLaTe
Cohen-Macaulay Circulant Graphs
Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2
\rfloor}, and let I(G) denote its the edge ideal in the ring R =
k[x_1,...,x_n]. We consider the problem of determining when G is
Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay
graph G must be well-covered, we focus on known families of well-covered
circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic
circulant graphs are Cohen-Macaulay. We end with the observation that even
though the well-covered property is preserved under lexicographical products of
graphs, this is not true of the Cohen-Macaulay property.Comment: 14 page
On the variational interpretation of the discrete KP equation
We study the variational structure of the discrete Kadomtsev-Petviashvili
(dKP) equation by means of its pluri-Lagrangian formulation. We consider the
dKP equation and its variational formulation on the cubic lattice as well as on the root lattice . We prove that, on a lattice
of dimension at least four, the corresponding Euler-Lagrange equations are
equivalent to the dKP equation.Comment: 24 page
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