Geometric quantum information processing in open systems

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The theoretical promise of quantum information processing (QIP) is widely regarded as one of the most exciting developments in computer science in many years. This is because QIP appears to be able to efficiently solve problems which are classicaly intractable (such factoring, Shors algorithm), and is provably capable of providing significant computational speedups in problems of wide interest (such as database search, Grover's algorithm). As a result QIP has spawned an avalance of activity across many disciplines, including also physics, electrical engineering, chemistry, and materials science. However,how to best implement QIP is still a wide open question. In particular, not only is it still unclear which physical system is best suited for QIP, it is also unresolved whether QIP should be implemented by means of dynamical or geometriclogic gates. This proposal is concerned with the geometric approach.Although the most widely studied version of QIP is based on dynamical evolution of quantum states, other approaches could very well turn out to be easier to implement and/or more robust against unwanted interactions and imperfections. In this proposal, the investigators intend to explore an alternative and promising version of QIP called holonomic quantum computation (HQC, introduced by the PI) that has garnered increasing interest. In HQC a quantum system that has a set of degenerate lowest energy (?ground?) states is driven slowly (adiabatically) around a loop in its control parameter space. In the process the system acquires a so-called geometric phase, meaning that its state changes in accordance with the geometrical properties of this loop. These state changes can then be combined in order to execute a complete quantum algorithm.The resulting geometrical transformations are not only of fundamental interest, but have the advantage over the standard dynamical ones that they are rather robust to certain errors. The reason is that the geometric phase depends only on the area the loop encloses in parameter space, but not on its shape, or on the speed the loop is traversed, provided it is slow. HQC has already attracted the attention of various groups attempting to build quantum computers, especially using trapped ions, because of the aforementioned robustness and because it is a more natural approach to the implementation of quantum logic gates than is the dynamical model.The potential of HQC is exciting, but there are crucial missing elements in HQC theory. Most importantly, the theory of HQC error correction is still primitive, even though error correction will undoubtedly be indispensable for a working holonomic quantum computer. It is essential to develop this general theory as well as detailed insight into specific physical systems that could be used to realize fault tolerant HQC. At the same time, it is vital to investigate HQC?s potential for new algorithms and new insight into physical processes. This proposal presents strategies for addressing these fundamentalopen problems.The success of HQC depends on the ability to keep the quantum system in its ground energy subspace, which in turn depends on maintaining an energy gap between the ground and next lowest energy states.Opening the HQC to interactions with its environment leads to processes that either shrink this gap or cause transitions across it, thus ruining the computation (a process known as decoherence). Equipped with a formulation of geometric phases for open systems recently introduced by the co-PI, the researchers intend to achieve a deep understanding of the effects of decoherence on HQC. The open system geometric phase provides a tool that can quantitatively capture the relationship between error correction and the gap.To thoroughly explore error correction in HQC, the authors plan to leverage their experience in circuit QC with decoherence-free subspaces, dynamical decoupling, and error correcting codes. They intend to use open system adiabatic theory (introduced by the co-PI) to provide thorough analysis of the validity and utility of their proposed error correcting techniques.