广义Harper模型的量子扩散

研究了广义Ｈａｒｐｅｒ模型的量子扩散,分析了自关联函数Ｃ（ｔ）在不同哈密顿量参数区域的长时特征。结果表明,在金属区Ｃ（ｔ）－Ｔ＾－１,而在绝缘体区Ｃ（ｔ）－ｔ＾０。     

周期驱动的Harper模型的量子计算鲁棒性与量子混沌

以周期驱动的量子Harper(quantum kicked Harper, QKH)模型为例，研究复杂量子动力系统的量子计算在各种干扰下的稳定性.通过对Floquet算子本征态的统计遍历性及其Husimi函数的分析，比较随机噪声干扰和静态干扰对量子计算不同程度的影响.进一步的保真度摄动分析表明，在随机噪声干扰下保真度随系统演化呈指数衰减，而静态干扰下的保真度为高斯衰减，并通过数值计算得到了干扰下的可信计算时间尺度.与经典混沌仿真中误差使状态产生指数分离不同，量子计算对状态干扰的稳定性和仿真模型的动力学特性无关. 关键词： 量子Harper模型 量子计算 量子混沌 保真度     

Approximating spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

     

量子Harper模型的量子计算鲁棒性与耗散退相干

运用量子轨迹和量子Monte Carlo仿真的方法，研究耗散退相干对周期驱动的量子Harper (quantum kicked Harper, QKH)模型量子计算的影响.数值仿真结果表明，一定强度的耗散干扰将破坏QKH特征状态的动态局域化以及相空间的随机网结构.以相位阻尼信道噪声模型为例分析了保真度的衰减规律以及可信计算时间尺度.与静态干扰相比，在干扰强度小于某一阈值时，耗散干扰下的可信计算时间尺度随量子比特的增加而快速下降；而在干扰强度大于该阈值时，静态干扰下的可信计算时间尺度下降更快.     

A one-to-two-dimensional mapping using a partial Fast Fourier Transform

It will be shown how to map a simple one-dimensional tight binding model with a cosine potential in one dimension exactly to a two-dimensional tight binding model with periodic boundary conditions in the presence of a single flux quantum spread evenly on the torus. The mapping is achieved by a partial sequence of “Fast Fourier Transform” (FFT) steps which if completed would be an exact Fourier transform of the original model. Each step of the FFT recursively maps a tight binding model into two decoupled sublattices of half the lattice length.     

Quantum Hall Effect on the Hyperbolic Plane in the Presence of Disorder

We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in Comm. Math. Phys. 190 (1998), 629–673, to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.