Stochastic subspace correction methods and fault tolerance

Quantum Error Correction and Fault-Tolerance

I give an overview of the basic concepts behind quantum error correction and quantum fault tolerance. This includes the quantum error correction conditions, stabilizer codes, CSS codes, transversal gates, fault-tolerant error correction, and the threshold theorem. 1 Quantum Error Correction Building a quantum computer or a quantum communications device in the real world means having to deal wit...

Abstract Fault Tolerance: A Model-Theoretic Approach to Fault Tolerance and Fault Compensation without Error Correction

This paper explores the concept of “pure fault tolerance”--when a system can fulfill its computational goal, even though some of its components do not fulfill theirs. This leads to a natural statement of the composition problem for a very general concept of architecture: to what extent are the properties of a system based on that architecture a function of the properties of the components? The ...

Subspace Correction Methods for Eigenvalue Problems 313

Subspace Correction Methods for Singular Systems

We investigate the application of minimal residual and orthogonal residualsubspace correction methods to singular linear systems Ax = b. Special em-phasis is put on the special case of Krylov subspace methods. If A has index 1(i.e., if all Jordan blocks associated with the eigenvalue λ = 0 of A are 1× 1)the behaviour of these iterative methods is well understood (see, e.g., ...

Subspace Correction Methods for Convex Optimization