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Spectral graph theory with applications to quantum adiabatic optimization.

机译：光谱图理论及其在量子绝热优化中的应用。

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摘要

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as "stoquastic" or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above.;I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically.;Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing "Fundamental Gap Conjecture".
机译：在这篇论文中，我通过控制这些过程及其相关光谱的算子在量子绝热优化，光谱图论，热扩散和亚随机过程之间建立了联系。特别地，我们研究最近被称为“随机”或等效地称为亚随机过程生成器的哈密顿量。与这些哈密顿量相对应的算子在上面提到的所有设置中都感兴趣。我主要研究算子的光谱间隙或该算子的两个最低能量之差与某些平衡行为之间的联系。在绝热优化的情况下，这对应于解决感兴趣的优化问题的可能性。我将提供一个优化问题的实例，该经典问题很容易解决，但给绝热难题留下了可能。除此具体示例外，本文的工作主要是数学上的工作，并且我们着重于限制谱隙。我们执行此操作的主要工具是频谱图理论，它可以通过简单考虑宿主图子图的Dirichlet特征值来提供最自然的方法。我将为一维，超立方体和一般凸子图的间隙导出紧密边界。所使用的技术还将适应Andrews和Clutterbuck最近用于证明长期存在的“基本差距猜想”的方法。

著录项

  • 作者

    Baume, Michael Jarret.;

  • 作者单位

    University of Maryland, College Park.;

  • 授予单位 University of Maryland, College Park.;
  • 学科 Quantum physics.;Computer science.;Mathematics.
  • 学位 Ph.D.
  • 年度 2016
  • 页码 132 p.
  • 总页数 132
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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