The modular multiplication operator and the quantized bakers maps

Irreversible Quantum Baker Map

We propose a generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of a unitary Floquet operator. A corresponding quantum system is constructed as a completely positive map acting in the space of density matrices. We investigate spectral properties of this super-operator and their link with the increase of the entropy of initially pure states.

Walsh量子化面包师映射的量子方差与涨落

Quantum variance and fluctuations for Walsh-quantized baker's maps

该研究证明Walsh量子化面包师映射（除D=4外）的矩阵元涨落渐近高斯，方差由经典关联决定，并给出ETH版本，揭示本征态微观关联区别于Haar随机向量。

Quantum baker maps with controlled-NOT coupling

The characteristic stretching and squeezing of chaotic motion is linearized within the finite number of phase space domains which subdivide a classical baker map. Tensor products of such maps are also chaotic, but a more interesting generalized baker map arises if the stacking orders for the factor maps are allowed to interact. These maps are readily quantized, in such a way that the stacking interaction is entirely attributed to primary qubits in each map, if each subsystem has power-of-two Hilbert space dimension. We here study the particular example of two baker maps that interact via a con…

Baker-Campbell-Hausdorff relation for special unitary groups SU(N)

Multiplication of two elements of the special unitary group SU(N) determines uniquely a third group element. A BAker-Campbell-Hausdorff relation is derived which expresses the group parameters of the product (written as an exponential) in terms of the parameters of the exponential factors. This requires the eigen- values of three (N-by-N) matrices. Consequently, the relation can be stated analytically up to N=4, in principle. Similarity transformations encoding the time evolution of quantum mechanical observables, for example, can be worked out by the same means.

Shor算法中模乘动力系统的混沌根

Chaotic Roots of the Modular Multiplication Dynamical System in Shor's Algorithm

本文从量子动力系统视角研究Shor算法中模乘算子可积性与经典混沌的悖论，证明该算子可表为量子化A-baker映射的叠加，其可积性源于同族其他混沌量子化的干涉，为量子算法中研究可积性、遍历性与混沌的相互作用奠定基础。

Quantum chaos in the spectrum of operators used in Shor's algorithm

We provide compelling evidence for the presence of quantum chaos in the unitary part of 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 CUE 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 foun…