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Some properties of the ferromagnetic XXZ spin chain and their applications to quantum computation.

机译：铁磁XXZ自旋链的某些性质及其在量子计算中的应用。

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摘要

One of the key requirements of quantum computation is the ability to encode qubits and construct unitary gates that are protected from the effects of environmental noise. The ferromagnetic XXZ chain is one of the best studied quantum spin models. In this thesis we extend the study of the low lying spectrum of the one dimensional ferromagnetic XXZ chain with special boundary conditions by proving the existence of isolated interface states called 'kinks'. We then capitalize on the isolated nature of these kink states to use them to encode a qubit and construct an implementation scheme for quantum gates.;In chapter 3 we investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. In this chapter we prove that for J ≥ 3/2 and sufficiently large Delta, the lowest excited eigenvalues of the XXZ chain determined by a fixed value of M are separated by a gap from the rest of the spectrum, uniformly in the length of the chain.;The existence of isolated eigenvalues in the low energy spectrum is very good from a quantum computing point of view since they provide natural protection from the noise effects. In chapter 4 we demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent DMRG method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin-3/2 XXZ chains, such as Not, Hadamard, Pi-8, Phase, and C-Not, with fidelity levels exceeding 99%.;Our methods could easily be adapted to a variety of other systems.
机译：量子计算的关键要求之一是能够对qubit进行编码并构造单一门的能力，这些门可免受环境噪声的影响。铁磁XXZ链是研究最深入的量子自旋模型之一。在本文中，我们通过证明存在被称为“扭结”的孤立界面态，扩展了对具有特殊边界条件的一维铁磁XXZ链低能谱的研究。然后，我们利用这些扭结态的孤立性质，将其用于编码量子比特并构建量子门的实现方案。在第3章中，研究具有Ising各向异性Delta的自旋J铁磁XXZ链的低激发态。和扭结边界条件。由于总磁化强度的第三个分量M是守恒的，因此研究M的每个固定值的频谱都是有意义的。在本章中，我们证明对于J≥3/2和足够大的Delta，M的最低激发本征值由M的固定值确定的XXZ链与谱的其余部分之间有一个间隙，并且在链的长度上均匀地分开。;从量子计算的角度来看，低能谱中孤立特征值的存在非常好因为它们提供了自然保护免受噪声影响。在第四章中，我们演示了使用一维自旋J铁磁XXZ链的unit态演化构造量子门的实现方案。我们使用基于时间的DMRG方法和最佳控制理论的技术，基于链的模拟给出了数值结果。仅使用几个控制参数，我们发现可以在自旋3/2 XXZ链系统（例如Not，Hadamard，Pi-8，Phase和C-Not）上实现一比特和二比特门，保真度超过99％。;我们的方法可以轻松地适用于多种其他系统。

著录项

  • 作者

    Mulherkar, Jaideep S.;

  • 作者单位

    University of California, Davis.;

  • 授予单位 University of California, Davis.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 81 p.
  • 总页数 81
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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