Maity Krishnendu, Lakshminarayan Arul
Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Sep;74(3 Pt 2):035203. doi: 10.1103/PhysRevE.74.035203. Epub 2006 Sep 15.
We provide compelling evidence for the presence of quantum chaos in the unitary part of the operator usually employed in Shor's factoring algorithm. In particular we analyze the spectrum of this part after proper desymmetrization and show that the fluctuations of the eigenangles as well as the distribution of the eigenvector components follow the circular unitary ensemble of random matrices, of relevance to quantized chaotic systems that violate time-reversal symmetry. However, as the algorithm tracks the evolution of a single state, it is possible to employ other operators; in particular, it is possible that the generic quantum chaos found above becomes of a nongeneric kind such as is found in the quantum cat maps and in toy models of the quantum baker's map.
我们为通常用于Shor因式分解算法的算符酉部分中量子混沌的存在提供了令人信服的证据。特别是，我们在进行适当的去对称化后分析了这一部分的谱，并表明本征角的涨落以及本征向量分量的分布遵循随机矩阵的圆酉系综，这与违反时间反演对称性的量子化混沌系统相关。然而，由于该算法跟踪单个态的演化，因此有可能使用其他算符；特别是，有可能上述发现的一般量子混沌会变成非一般类型，例如在量子猫映射和量子面包师映射的玩具模型中发现的那样。
