Use Of Binary Representation Research Articles

Algebraic Methods for Quantum Codes on Lattices

This is a note from a series of lectures at Encuentro Colombiano de Computacion Cuantica, Universidad de los Andes, Bogota, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Clifford quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Clifford circuits.

Compact Mathematical Formulation for Graph Partitioning

The graph partitioning problem consists of dividing the vertices of a graph into clusters, such that the weight of the edges crossing between clusters is minimized. We present a new compact mathematical formulation of this problem, based on the use of binary representation for the index of clusters assigned to vertices. This new formulation is almost minimal in terms of the number of variables and constraints and of the density of the constraint matrix. Its linear relaxation brings a very fast computational resolution, compared with the standard one.