量子场论中的符号问题:经典和量子方法

S. Lawrence
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引用次数: 13

摘要

在晶格场理论框架内的蒙特卡罗计算提供了对量子场平衡物理的非摄动访问。当应用于某些费米子系统或非平衡物理计算时,这些方法遇到了所谓的符号问题,并且计算资源需求变得不切实际。这些困难阻碍了量子色动力学状态方程第一原理的计算,以及量子场论中输运系数的计算等。本文详细介绍了减轻或避免符号问题的两种方法。首先,通过场变量的复化和柯西积分定理的应用,可以改变符号问题的难度。这需要寻找一个合适的积分轮廓。讨论了几种求等高线的方法,以及在等高线上进行积分的步骤。本文强调了两个值得注意的例子:在一种情况下,存在完全消除符号问题的轮廓,而在另一种情况下,可证明没有轮廓可用于改善符号问题,超过(参数化)少量。作为一种替代方案,物理模拟可以在量子计算机的帮助下进行。量子计算的形式要素——即希尔伯特空间、作用于其上的酉算子和待测量的厄米观测值——可以与量子场论的形式要素相匹配。通过这种方式,可以使纠错量子计算机成为一个控制良好的实验室。提出了这项任务的精确算法,特别是在量子色动力学的背景下。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Sign Problems in Quantum Field Theory: Classical and Quantum Approaches
Monte Carlo calculations in the framework of lattice field theory provide non-perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource requirements become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy's integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation - that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured - can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
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