6,714 research outputs found
A Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests
Scenario programs have established themselves as efficient tools towards
decision-making under uncertainty. To assess the quality of scenario-based
solutions a posteriori, validation tests based on Bernoulli trials have been
widely adopted in practice. However, to reach a theoretically reliable
judgement of risk, one typically needs to collect massive validation samples.
In this work, we propose new a posteriori bounds for convex scenario programs
with validation tests, which are dependent on both realizations of support
constraints and performance on out-of-sample validation data. The proposed
bounds enjoy wide generality in that many existing theoretical results can be
incorporated as particular cases. To facilitate practical use, a systematic
approach for parameterizing a posteriori probability bounds is also developed,
which is shown to possess a variety of desirable properties allowing for easy
implementations and clear interpretations. By synthesizing comprehensive
information about support constraints and validation tests, improved risk
evaluation can be achieved for randomized solutions in comparison with existing
a posteriori bounds. Case studies on controller design of aircraft lateral
motion are presented to validate the effectiveness of the proposed a posteriori
bounds
Synthetic Topological Degeneracy by Anyon Condensation
Topological degeneracy is the degeneracy of the ground states in a many-body
system in the large-system-size limit. Topological degeneracy cannot be lifted
by any local perturbation of the Hamiltonian. The topological degeneracies on
closed manifolds have been used to discover/define topological order in
many-body systems, which contain excitations with fractional statistics. In
this paper, we study a new type of topological degeneracy induced by condensing
anyons along a line in 2D topological ordered states. Such topological
degeneracy can be viewed as carried by each end of the line-defect, which is a
generalization of Majorana zero-modes. The topological degeneracy can be used
as a quantum memory. The ends of line-defects carry projective non-Abelian
statistics, and braiding them allow us to perform fault tolerant quantum
computations.Comment: 4 pages + references + 3 pages of supplementary material, 2 figures.
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