Graphical Structures for Design and Verification of Quantum Error Correction

We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the zx-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical [n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its zx-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at https://doi.org/10.15128/r1bn999672

Introduction to Forward-Error-Correcting Coding

This reference publication introduces forward error correcting (FEC) and stresses definitions and basic calculations for use by engineers. The seven chapters include 41 example problems, worked in detail to illustrate points. A glossary of terms is included, as well as an appendix on the Q function. Block and convolutional codes are covered

Error Correcting Codes for Distributed Control

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in applications of networked control systems. Although the work of Schulman and Sahai over the past two decades, and their development of the notions of "tree codes"\phantom{} and "anytime capacity", provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one needs real-time encoding and real-time decoding and a reliability which increases exponentially with decoding delay, which is what tree codes guarantee. We prove that linear tree codes occur with high probability and, for erasure channels, give an explicit construction with an expected decoding complexity that is constant per time instant. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight. This work takes an important step towards controlling plants over noisy channels, and we demonstrate the efficacy of the method through several examples.Comment: 39 page

Online error correcting output codes

a b s t r a c t This article proposes a general extension of the error correcting output codes framework to the online learning scenario. As a result, the final classifier handles the addition of new classes independently of the base classifier used. In particular, this extension supports the use of both online example incremental and batch classifiers as base learners. The extension of the traditional problem independent codings oneversus-all and one-versus-one is introduced. Furthermore, two new codings are proposed, unbalanced online ECOC and a problem dependent online ECOC. This last online coding technique takes advantage of the problem data for minimizing the number of dichotomizers used in the ECOC framework while preserving a high accuracy. These techniques are validated on an online setting of 11 data sets from UCI database and applied to two real machine vision applications: traffic sign recognition and face recognition. As a result, the online ECOC techniques proposed provide a feasible and robust way for handling new classes using any base classifier

Decoding Schemes for Foliated Sparse Quantum Error Correcting Codes

Foliated quantum codes are a resource for fault-tolerant measurement-based quantum error correction for quantum repeaters and for quantum computation. They represent a general approach to integrating a range of possible quantum error correcting codes into larger fault-tolerant networks. Here we present an efficient heuristic decoding scheme for foliated quantum codes, based on message passing between primal and dual code 'sheets'. We test this decoder on two different families of sparse quantum error correcting code: turbo codes and bicycle codes, and show reasonably high numerical performance thresholds. We also present a construction schedule for building such code states.Comment: 23 pages, 15 figures, accepted for publication in Phys. Rev.

The application of forward error correction techniques in wireless ATM

Bibliography: pages 116-121.The possibility of providing wireless access to an ATM network promises nomadic users a communication tool of unparalleled power and flexibility. Unfortunately, the physical realization of a wireless A TM system is fraught with technical difficulties, not the least of which is the problem of supporting a traditional ATM protocol over a non-benign wireless link. The objective of this thesis, titled "The Application of Forward Error Correction Techniques in Wireless ATM' is to examine the feasibility of using forward error correction techniques to improve the perceived channel characteristics to the extent that the channel becomes transparent to the higher layers and allows the use of an unmodified A TM protocol over the channel. In the course of the investigation that this dissertation describes, three possible error control strategies were suggested for implementation in a generic wireless channel. These schemes used a combination of forward error correction coding schemes, automatic repeat request schemes and interleavers to combat the impact of bit errors on the performance of the link. The following error control strategies were considered : 1. A stand alone fixed rate Reed-Solomon encoder/decoder with automatic repeat request. 2. A concatenated Reed-Solomon, convolution encoder/decoder with automatic request and convolution interleaving for the convolution codec. 3. A dynamic rate encoder/decoder using either a concatenated Reed-Solomon, convolution scheme or a Reed-Solomon only scheme with variable length Reed-Solomon words