Using graphs for the analysis and construction of permutation distance-preserving mappings

Abstract: A new way of looking at permutation distance-preserving mappings (DPMs) is presented by making use of a graph representation. The properties necessary to make such a graph distance-preserving, are also investigated. Further, this new knowledge is used to analyze previous constructions, as well as to construct a new general mapping algorithm for a previous multilevel construction

Analysis of permutation distance-preserving mappings using graphs

Abstract A new way of analyzing permutation distance preserving mappings is presented by making use of a graph representation. The properties necessary to make such graphs distance-preserving and how this relates to the total sum of distances that exist for such mappings, are investigated. This new knowledge is used to analyze previous constructions, as well as showing the existence or non-existence of simple algorithms for mappings attaining the upper bound on the sum of distances. Finally, two applications for such graphs are considered

New distance concept and graph theory approach for certain coding techniques design and analysis

Abstract: A New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from the k-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in the k-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze

Graph limits and exchangeable random graphs

We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.Comment: 26 page

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape