Classical and Quantum Error-Correction Coding in Genetics

The subject of this chapter is the use of classical/quantum information theory and coding in genetics and evolution. The chapter starts with the description of using the concepts from both classical and quantum information theories to describe the evolution of biological channel capacity through generations. In order to do so, several classical and quantum biological channel models are employed including the Markovian classical and Markovian-like quantum model, hybrid quantum-classical model, multilevel symmetric channel model, and Kimura model-based Markovian process. In order to describe the reliable long-time storage of genetic information in DNA, the use of unequal error protection (UEP) coding is studied. Several classes of error-correction codes suitable for UEP on a cellular level are described including nested coding, multilevel coding (MLC), rate-adaptive coding, and generalized LDPC coding. The use of concepts of constrained coding to describe the genetic information flow from DNA to proteins is also described as well as joint-constrained and error-correction coding. After that, the use of quantum error-correction concepts to deal with environmental errors including canonical quantum error-correction and stabilizer codes is briefly described. One particular class of stabilizer codes, known as topological codes, is then described that might be relevant to biological processes as they only involve the local qubits in encoding process. Another relevant class of codes, the subsystem codes, is then described. The key idea behind subsystem codes is to decompose the quantum code as the tensor product of two subsystems, exon subsystem A and intron subsystem B, and we are concerned with correcting errors only on the exon subsystem. Finally, we describe the use of nonbinary quantum stabilizer codes to deal with nucleobase substitution errors, both random and burst errors. We also briefly discuss the possible use of both classical and quantum error-correction concepts to improve tolerance to tumor and cancer introducing errors.