In this study we present a method that adapts the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. Making use of the the corresponding Pauli error models for amplitude and phase-damping models, we focus on codes that correct one or two amplitude-damping errors. As a result, we are able to exhaustively search for all possible codes up to length 9 by applying local Clifford operations on graph states. With a similar method, we also look at codes for the Pauli error models that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including non-additive codes and degenerate codes.
We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block lengthn≥ 8 exceptn=2r-1. We generalize this construction to employ classical codes overGF(3) with which we numerically obtain better performing codes up to length 14. Because the resulting codes are of the codeword stabilized (CWS) type, conceptually simple (though potentially computationally expensive) encoding and decoding circuits are available.
No quantum system can be perfectly isolated from the environment and, as a result, no physical implementation of quantum information processing tasks can be completely free from noise. The best tool for combating such noise is the use of quantum error correcting codes (QECCs). The general requirements for QECCs have been documented for a while and yet construction of good codes and understanding their effect remains a difficult and active area of study. In this thesis I outline the work I have completed looking into both of these problems. My general technique throughout this work is to first reduce the problem space as much as possible through use of group theory. Then to use numerical methods and bring computational power to bear on the problem. In chapter 2, I investigate the construction of good codes for the amplitude damping channel, using the codeword stabilized quantum code(CWS) framework. Through an exhaustive search method many new codes with better parameters than previously known are found. In chapter 3, I continue constructing good codes for the amplitude damping channel, this time using code concatenation techniques to find results that would be unfeasible to find via an exhaustive search. Finally, chapter 4 broaches the difficult problem of determining if a quantum channel has capacity; the ultimate use of QECCs in a sense. Expanding on the key works done on the problem, I develop the theory surrounding effective noise channels obtained from applying a QECC on multiple uses of a channel in order to determine if capacity exists. Using this framework and making use of computational power available today, these techniques allow us to find many very noisy non-Pauli channels that have positive capacity which previously had not been shown to have capacity.
The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal
n-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with
n of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find $((9,97\ensuremath{\le}K\ensuremath{\le}100,2))$ and $((11,387\ensuremath{\le}K\ensuremath{\le}416,2))$ codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that
n-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a crossover operation based on spectral bisection that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found ((13, 18, 4)) and ((13, 20, 4)) codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for $n\ensuremath{\le}9$, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the
3n possible equivalent error sets. By combining this error-set selection with the genetic algorithm approach, we have found ((11, 68)) and ((11, 80)) codes capable of correcting a single amplitude damping error and ((11, 4)), ((12, 4)), ((13, 8)), and ((14, 16)) codes capable of correcting two amplitude damping errors.
We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or nonadditive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.