Geometric quantum information processing in open systems

量子信息处理的理论前景被广泛认为是多年来计算机科学中最令人兴奋的发展之一。这是因为QIP似乎能够有效地解决经典的棘手问题(如因式分解、Shors算法)，并被证明能够在广泛感兴趣的问题(如数据库搜索、Grover算法)上提供显著的计算加速比。因此，QIP催生了许多学科的活动，包括物理、电气工程、化学和材料科学。然而，如何最好地实施QIP仍是一个悬而未决的问题。特别是，不仅仍然不清楚哪个物理系统最适合QIP，而且QIP应该通过动态门还是几何逻辑门来实现也是悬而未决的。这一提议与几何方法有关。尽管最广泛研究的QIP版本是基于量子态的动态演化，但其他方法很可能被证明更容易实现和/或对不必要的相互作用和缺陷更健壮。在这项提议中，研究人员打算探索一种替代的、有前途的QIP版本，即由PI引入的完整量子计算(HQC)，该版本已经引起了越来越多的兴趣。在HQC中，具有一组简并的最低能量(基态)的量子系统。状态被缓慢地(绝热地)围绕其控制参数空间中的回路驱动。在这个过程中，系统获得了一个所谓的几何阶段，这意味着它的状态根据这个回路的几何性质而变化。然后，这些状态变化可以组合在一起，以执行一个完整的量子算法。由此产生的几何变换不仅具有基本的意义，而且比标准的动力学变换具有优势，因为它们对某些误差相当稳健。原因是几何阶段仅取决于循环在参数空间中的封闭面积，而不取决于其形状或循环遍历的速度(如果循环速度较慢)。HQC已经引起了试图建立量子计算机的各种团体的注意，特别是使用囚禁离子，因为前面提到的健壮性，因为它是一种比动力学模型更自然的实现量子逻辑门的方法。HQC的潜力是令人兴奋的，但HQC理论中有关键的缺失元素。最重要的是，HQC纠错理论仍然是原始的，尽管纠错无疑将是工作的完整量子计算机不可缺少的。必须发展这一一般理论，以及对可用于实现容错HQC的特定物理系统的详细了解。同时，研究HQC S对新算法和对物理过程的新见解的潜力也是至关重要的。这一提议提出了解决这些基本开放问题的策略。HQC的成功取决于将量子系统保持在其地面能量子空间的能力，而这反过来又取决于维持地面和次低能态之间的能量差距。开放HQC与其环境的相互作用导致过程要么缩小这种差距，要么导致跨越它的跃迁，从而破坏计算(这个过程被称为退相干)。配备了最近由共同PI引入的开放系统的几何相位公式，研究人员打算实现对退相干对HQC的影响的深入理解。开放系统几何阶段提供了一种工具，可以定量地捕捉纠错和缺口之间的关系。为了深入探索HQC中的纠错，作者计划利用他们在电路QC中的经验，通过无去相干的子空间、动态解耦和纠错码来实现。他们打算使用开放系统绝热理论(由共同PI引入)来彻底分析他们提出的纠错技术的有效性和实用性。

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.