This research will investigate important issues of quantum computational universality in measurement-based quantum computation (MBQC), explore connections of MBQC to ideas in statistical mechanics and condensed matter physics, and other quantum computational models. Specifically, the Affleck-Kennedy-Lieb-Tasaki (AKLT) models supply a rich playground for exploring new universal resource states and for understanding the intricate relations of quantum computational universality to percolation, spatial connectivity, magnetic order, and phase transitions in computational power. The long standing open question of the existence of a finite spectral of any two-dimensional rotationally symmetric (including AKLT) Hamiltonians is important also for the stability of generating related resource states by cooling. This will be studied with both analytic and numerical means. The research also includes searching for new types of resource states and developing model Hamiltonians whose thermal states can be used for quantum computation without the need to switch off interactions. Furthermore, this program studies how topological order can be of use to quantum computation, and conversely, how MBQC offers an efficient means to create a large class of topologically ordered states. Intellectual Merit: MBQC is one of the several models for building quantum computers. Essentially, all that is needed is a suitable highly entangled resource state to begin with and the ability to perform local measurements. This approach of realizing a quantum computer is promising with several physical systems, such as ultracold atoms in optical lattices and photons, complementing other approaches of implementing quantum computation. MBQC also provides a conceptual framework for answering fundamental questions in quantum computation and for bridging to other areas of research. The questions that will be addressed include: (1) What entangled states can qualify as an universal resource and can they arise as unique ground states of physically reasonable Hamiltonians? A complete understanding may lead to novel characterization of states of matter in terms of computational capability. (2) Is there a generalized Haldane conjecture in higher dimensions and how to test it? Tackling the long standing open question of the spectral gaps of two-dimensional AKLT Hamiltonians will give insight to a possible generalized Haldane conjecture in 2D and pave the road for probing richer phases in isotropic spin Hamiltonians in higher dimensions. (3) Can topological order provide insight to the quest of new resource states? (4) Are there advantages over others that the MBQC model offers? The research findings of MBQC from this program will not only advance our knowledge on various aspects of quantum computation and its connection to ideas in condensed matter physics and statistical mechanics, but also have potential impact on future quantum computer technology. Broader Impacts : The PI is taking the initiative in organizing a forum for discussing scientific results in quantum information science and stimulating collaboration across disciplines at Stony Brook University. He will integrate his research on quantum computation in the courses that he is currently and will be developing for both undergraduate and graduate students. This project will also include training of a graduate student and mentoring of a postoctoral researcher.
