Quantum computers will likely provide the ultimate cure for the explosive growth of information data and the increasingly pressing demand to solve complex and comprehensive problems (such as factoring and discrete problems). Of critical importance to quantum computation and communication is quantum error correction, the mathematical-physical mechanism that protests the fragile quantum states from unwanted evolutions, thus enabling robust implementation of quantum processing devices and reliable transmission over noisy quantum channels. Proof-of-concept and the first constructive quantum error correction code did not appear until 1995, and the research is still at its infancy.
This research entails serious theoretic and algorithmic study to advance the state of quantum error correction. New methods will be investigated to provide a refreshing angle to the systematic construction of better families of quantum stabilizer codes and especially unrestricted low-density parity-check (LDPC) stabilizer codes. New and efficient decoding algorithms will be designed to revive several existing stabilizer codes. New concepts and tools will be developed to analyze and evaluate quantum codes. The tools used in the research is highly mathematical-physical, and involve quantum mechanics, quantum information theory, linear algebra, Euclidean/projective geometry, matrix theory, graph theory and convex optimization.
This project aims at designing new quantum codes and decoding algorithms, developing new theoretical results on error correction in general and quantum error correction in particular, and cross-fertilizing the areas of quantum coding, digital coding and analog coding by leveraging the useful ideas and concepts of one another. The project has resulted in a rich array of technical outcomes, including (1) the invention of five new designs of quantum non-CSS (i.e. unrestricted) stabilizer error correction codes based on binary linear digital codes (rather than F-4 linear codes); (2) the development of three practical decoding algorithms for quantum LDPC codes, quantum convolutional codes and quantum LDPC-convolutional codes, respectively; (3) the implementation and the simulation of these quantum codes in digital computers, and the assessment of their simulated error correction capabilities over quantum channels; (4) the extension of three analytical performance bounds from CSS stabilizer codes to non-CSSS stabilizer codes; (5) the derivation of a new and more comprehensive classification scheme for quantum stabilizer codes ; (6) the construction of four new classes of analog codes based on chaos maps, and their respective maximum-likelihood and/or minimum-mean-square-error decoders; (7) a soft encoding and soft decoding strategy for signal-relay and multi-relay cooperative communication systems; and (8) the develop of a hybrid digital-analog code for transmitting real-/complex-valued data with low bandwidth and low power. These research results and findings were published in more than 50 technical papers in high-quality peer-reviewed international conferences (36) and journal (17), including IEEE Information Society flagship conference (ISIT), IEEE Communication Society flagship conferences (ICC & GLOBECOM), and IEEE Signal Processing Society flagship conference (ICASSP), and top IEEE journals such as IEEE Transaction on Information Theory, IEEE Transaction on Wireless Communications, and IEEE Communication Letters. This project has helped support (partially) six Ph.D. students over a course of five years. It helped the PI and her team to gain tremendous research experience in the theory and practice of quantum, analog, and digital coding. It also helped increase the PI’s visibility. Positive outcomes also include her election to an IEEE Distinguished Lecturer, and the public reports on her quantum error correction research by Lehigh REVOLVE Magazine and by "A-to-Z on Nanotechnology" web (azonano.com).
