On Quantum and Classical Error Control Codes: Constructions and Applications
arXiv:0812.5104
Abstract
It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this work, I study various aspects of quantum error control codes -- the key component of fault-tolerant quantum information processing. I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields, in addition to families of subsystem codes over symmetric and asymmetric channels. Particularly, many families of quantum BCH, RS, duadic, and convolutional codes are constructed over finite fields. Families of subsystem codes and a class of optimal MDS subsystem codes are derived over asymmetric and symmetric quantum channels. In addition, propagation rules and tables of upper bounds on subsystem code parameters are established. Classes of quantum and classical LDPC codes based on finite geometries and Latin squares are constructed.
Parts of PhD dissertation, Texas A&M University
References in corpus (45)
- Mixed State Entanglement and Quantum Error Correction
- Good Quantum Error-Correcting Codes Exist
- Decoherence Free Subspaces for Quantum Computation
- A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
- Reliable Quantum Computers
- Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
- Simple Quantum Error Correcting Codes
- Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
- Sparse Graph Codes for Quantum Error-Correction
- Quantum codes on a lattice with boundary
- Stabilizer Formalism for Operator Quantum Error Correction
- On optimal quantum codes
- A Unified and Generalized Approach to Quantum Error Correction
- Subsystem fault tolerance with the Bacon-Shor code
- Asymmetric quantum error correcting codes
- General entanglement-assisted quantum error-correcting codes
- Efficient Computations of Encodings for Quantum Error Correction
- Description of a quantum convolutional code
- Convolutional and tail-biting quantum error-correcting codes
- Theory of Initialization-Free Decoherence-Free Subspaces and Subsystems
- Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes
- Threshold Accuracy for Quantum Computation
- Graphs, Quadratic Forms, and Quantum Codes
- An Introduction to Quantum Error Correction
- On Protected Realizations of Quantum Information
- Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes
- A Proposed Quantum Low Density Parity Check Code
- Fault-Tolerant Postselected Quantum Computation: Threshold Analysis
- Error correction optimisation in the presence of X/Z asymmetry
- Constructions and performance of classes of quantum LDPC codes
- A simple family of nonadditive quantum codes
- Fault-Tolerant Logical Gate Networks for CSS Codes
- Quantum Stabilizer Codes and Classical Linear Codes
- Quantum Convolutional BCH Codes
- Quantum Block and Convolutional Codes from Self-orthogonal Product Codes
- Constructions of Quantum Convolutional Codes
- Distance bounds for convolutional codes and some optimal codes
- Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes
- Good Quantum Convolutional Error Correction Codes And Their Decoding Algorithm Exist
- Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error-Correcting Codes
- Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes
- Trellises for stabilizer codes: definition and uses
- Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets
- On doubly-cyclic convolutional codes
- A Note on Quantum Hamming Bound