2014/07/24 by Yuen-Lam Cheung, Dmitriy Drusvyatskiy, Cheung, Yuen-Lam +7
Computer Science · Engineering · Mathematics · #65F10 #81Q10 #90C22 #Advanced Optimization Algorithms Research #Algorithm #Combinatorics #Computer science #FOS: Mathematics #Geometry #Heuristics #Mathematical optimization #Mathematics #Numerical Analysis (math.NA) #Optimization and Variational Analysis #Physics #Point (geometry) #Projection (relational algebra) #Quantum #Quantum mechanics #Rank (graph theory) #Reflection (computer programming) #Set (abstract data type) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1407.6604
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2014/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states \ρ1, …, ρk\ to another such set \ρ1, …, ρk\. In other words, we must find a \em completely positive linear map, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the method of alternating projections and the Douglas-Rachford reflection method.