Quantum Error-Correcting Hybrid Codes

Abstract

Remarkable contributions made in the field of quantum algorithms and theory since 1994 have paved the way for quantum information and quantum computing. Their substantial speed-up over classical algorithms encouraged further developments in quantum information theory that enable information transmission in a reliable and fault-tolerant manner. A huge family of error-correcting codes have been developed since then with improved parameters and code-generating methods to process quantum information in the presence of noise and imperfect quantum gates. Stabilizer codes are one of the important classes of quantum error correcting codes. Their simple structure makes these codes easier to implement in a fault-tolerant manner. Promising work in the domain of hybrid quantum error-correcting codes has further shown their advantages over general quantum error correction.

In this thesis, we show various techniques for constructing error-correcting quantum codes, especially hybrid codes that transmit quantum-classical information over a single channel. A hybrid code can simultaneously transmit m bits of classical information and k bits of quantum information by building a collection of m quantum codes where each quantum message is associated with a classical message. Such codes have been shown to have better code parameters than the best known quantum codes using the same number of physical qubits. The first model is based on the use of codeword stabilized codes and union stabilizer codes while the second model uses subsystem codes by encoding the classical information in the gauge subsystem of the code. We also discuss various examples of good hybrid code constructions using these models and introduce the notion of using the framework of graph codes to encode and transmit both quantum and classical information since they allow for simpler fault-tolerant procedures. We finally propose various future directions to continue the work.