Educação matemática pela arte
Gusmão, Lucimar Donizete
2013-08-28
Search results
14 records were found.
Comment: 18 pages, 12 figures; clarified intermidiate steps in the proof
Comment: amsart 39 pages; (v2) minor change, ref. added. (v3) stronger
conclusion about topological charges
We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have stringlike logical operators, which make the codes non-self-correcting. We introduce a notion of “logical string segments” to avoid difficulties in defining one-dimensional objects in discrete lattices. We prove that every stringlike logical operator of our code can be deformed to a disjoint union of short segments, each of which is in the stabilizer group. The code has surfacelike logical operators whose partial implementation has unsatisfied stabilizers along its boundary.
We study unfrustrated spin Hamiltonians that consist of commuting tensor
products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians
that belong to the same phase of matter is described by a map between modules over
the translation-group algebra, so homological methods are applicable. In any dimension
every point-like charge appears as a vertex of a fractal operator, and can be isolated with
energy barrier at most logarithmic in the separation distance. For a topologically ordered
system in three dimensions, there must exist a point-like nontrivial charge. A connection
between the ground state degeneracy and the number of points on an algebraic set is
discussed. Tools to handle local Clifford unitary transformations are given.
We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model H_A in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of H_A on a lattice of lattice spacing ɑ to the tensor product of ground spaces of two independent Hamiltonians H_A and H_B on lattices of lattice spacing 2ɑ. We further find a disentangling unitary for the ground space of H_B ...
MIT Department of Physics Pappalardo Program
Comment: 13 pages, 1 figure. (v3): Weaker conclusions about self correction in
two dimensions
Comment: 10 pages, 2 figures
Comment: 11 pages, 3 figures
