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arXiv quant-ph/0608013quant-phcond-mat.str-elhep-th
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纠缠、量子相变和量子算法

Entanglement, quantum phase transitions and quantum algorithms

Roman Orus

摘要:

本论文的工作是在量子信息科学、量子多体物理和量子场论的交叉点上进行的。我们使用这三个领域的工具来分析跨学科交叉中出现的问题。更具体地说,在第一章中,我们从量子信息的角度利用优控制理论和共形场论研究了重整化群流的不可逆性。在第二章中,我们利用共形场论和Toeplitz矩阵的技巧计算了各种模型下的单拷贝二体量子系统的纠缠。第三章计算了所谓的Lipkin-Meshkov-Glick模型的纠缠熵,并与(1+1)维量子系统的纠缠熵进行了类比。第四章将量子相变中的量子关联标度的思想应用到量子算法的研究中,重点研究了Shor的因子分解算法和绝热演化求解NP完全问题的量子算法以及搜索问题。此外,在第5章中,我们使用最初受凝聚态物理启发的技术,利用所谓的矩阵乘积态,发展绝热量子算法的经典模拟。最后,在第6章中,我们从优控制理论的角度考虑了一些量子算法族的行为。每一章的结构都是这样的,最后一节总是总结基本结果。第7章简要讨论了一些一般性结论和未来可能的方向。附录A、附录B和附录C分别介绍了优控理论、共形场论和经典复杂性理论的一些基本概念。

英文摘要:

The work that we present in this thesis tries to be at the crossover of quantum information science, quantum many-body physics, and quantum field theory. We use tools from these three fields to analyze problems that arise in the interdisciplinary intersection. More concretely, in Chapter 1 we consider the irreversibility of renormalization group flows from a quantum information perspective by using majorization theory and conformal field theory. In Chapter 2 we compute the entanglement of a single copy of a bipartite quantum system for a variety of models by using techniques from conformal field theory and Toeplitz matrices. The entanglement entropy of the so-called Lipkin-Meshkov-Glick model is computed in Chapter 3, showing analogies with that of (1+1)-dimensional quantum systems. In Chapter 4 we apply the ideas of scaling of quantum correlations in quantum phase transitions to the study of quantum algorithms, focusing on Shor's factorization algorithm and quantum algorithms by adiabatic evolution solving an NP-complete and the searching problems. Also, in Chapter 5 we use techniques originally inspired by condensed-matter physics to develop classical simulations, using the so-called matrix product states, of an adiabatic quantum algorithm. Finally, in Chapter 6 we consider the behavior of some families of quantum algorithms from the perspective of majorization theory. The structure within each Chapter is such that the last section always summarizes the basic results. Some general conclusions and possible future directions are briefly discussed in Chapter 7. Appendix A, Appendix B and Appendix C respectively deal with some basic notions on majorization theory, conformal field theory, and classical complexity theory.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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