Quantum stabilizer codes and beyond

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.Comment: Ph.D. Dissertation, Texas A&M University, 200

Quantum error control codes

It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm is able to factor large integers in polynomial time on a quantum 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 dissertation, 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. This dissertation is organized into three parts: Quantum Block Codes. After introducing the theory of quantum block codes, I establish conditions when BCH codes are self-orthogonal (or dual-containing) with respect to Euclidean and Hermitian inner products. In particular, I derive two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum codes, as well as families of quantum codes derived from projective geometries. Subsystem Codes. Subsystem codes form a new class of quantum codes in which the underlying classical codes do not need to be self-orthogonal. I give an introduction to subsystem codes and present several methods for subsystem code constructions. I derive families of subsystem codes from classical BCH and RS codes and establish a family of optimal MDS subsystem codes. I establish propagation rules of subsystem codes and construct tables of upper and lower bounds on subsystem code parameters. Quantum Convolutional Codes. Quantum convolutional codes are particularly well-suited for communication applications. I develop the theory of quantum convolutional codes and give families of quantum convolutional codes based on RS codes. Furthermore, I establish a bound on the code parameters of quantum convolutional codes - the generalized Singleton bound. I develop a general framework for deriving convolutional codes from block codes and use it to derive families of non-catastrophic quantum convolutional codes from BCH codes. The dissertation concludes with a discussion of some open problems

Collective Spin-Cavity Ensembles and the Protection of Higher-dimensional Quantum Information

The development of devices leveraging their quantum nature to outperform classical analogues has been an ongoing effort for many decades now. In this thesis we investigate two facets related to this effort, albeit at different potential timescales of utility. First, we discuss our contributions in understanding collective spin ensembles interacting with a resonance cavity. This set up is common in superconducting qubit devices and electron spin-resonance experiments. The traditional model when considering this situation is the Tavis--Cummings model, although many of the methods could be adapted to other mesoscopic systems composed of ensembles of spins and cavities. In particular, we focus on characterizing the shifts in the energies due to dressing states, which are known as Lamb shifts. Before this line of work, most efforts focused on generating difference equations which could be solved iteratively to extract these shifts and dressed states. While this methodology works for systems involving hundreds of spins to thousands of spins, this iterative construction loses utility for larger ensembles due to the time needed to determine the parameters and prevents broad trends from being noted. Through these works we have stated how to determine the moments of the statistical distribution of the Lamb shifts, how to bound the largest of these shifts, and which of the subspaces are most important when finding these shifts. Beyond this, we have found that by including thermal effects we may use the moments of the Lamb shifts to greatly simplify a perturbative expansion to determine values of certain observables in optimal time (in spin ensemble size). These results provide greater insights into this model, provide faster simulation times, and can aide in experimental tests of these devices. Second, we discuss the contributions made in quantum error-correcting codes. The typical formalism used for quantum error-correcting codes is the stabilizer formalism. In our work we have extended this formalism to no longer directly depend on the local-dimension of the quantum computing device. For instance, most devices being currently designed and built run on qubits, which have local-dimension two, while qutrits have local-dimension of three. By removing this local-dimensional dependency we are able to generate many stabilizer codes, including codes with parameters previously unknown--amongst which are local-dimension-invariant forms, with the same distance parameter value, for the Steane code, the entire quantum analog of the classical Hamming family, and the Toric code. While these codes do not outperform the best known codes, this serves as an interesting pedagogical and extended framework and may provide for improved codes upon sufficient consideration, or aide in other work in fields closely related to stabilizers. Meanwhile, this extended framework permits for the importation of quantum error-correcting codes from lower local-dimension values to devices with higher local-dimension values, which at least provides some code options if a quantum computer is developed with easily tuned local-dimension value. These topics should be considered as disjoint, and all variable meanings are reset between the topics