Scalable Numerical Methods for Adiabatic Quantum Preparation

许多物理上相关的模型包括依赖时间的哈密顿算符。当与谱的能隙相比，时间依赖性较慢时，绝热定理意味着，在初始哈密顿量的基态下制备的量子系统将完全遵循瞬时基态。通过慢慢改变哈密顿量中的控制参数，可以利用这一点从简单的基态制备有趣的基态。例如，最终的基态可以是纠缠的，可以用于单向量子计算，甚至可以直接编码困难问题的解决方案。该方案具有很好的稳健性：当水库温度显著小于基态上方随时间变化的能隙时，退相干相互作用可能会将系统拖向基态，因此甚至可能有助于解决问题的目标。然而，对于临界参数值，基态本身可能会以一种非常突然的方式改变，这通常与几乎为零的能隙有关。在无限大的极限中，这对应于一个量子相变，而对于有限大小的系统，人们通常有一个有限的能隙。能隙的这种标度行为对绝热实现(发散绝热运行时)和它对热激励的稳健性都提出了严重的问题。只有少数几个精确可解的模型在无限大的尺寸范围内呈现出量子相变，因此通常人们必须使用数值模拟。本文建议发展智能数值方法来跟踪本征值问题，并研究绝热算法在开放和封闭量子系统中的性能。更具体地说，这些新算法即使在关键参数区域也必须具有足够的精度。这不仅通过使离散化水平、特征值跟踪器的步长自适应地匹配，而且通过误差和条件估计器使计算出的特征值的数目(以及被跟踪子空间的大小)与系统的行为相匹配来实现。

Many physically relevant models include time-dependent Hamilton operators. When the time-dependence is slow in comparison to the energy gaps of the spectrum, the adiabatic theorem implies that a quantum system prepared in the ground state of an initial Hamiltonian will completely follow the instantaneous ground state. This can be exploited to prepare interesting ground states from simple ones by slowly deforming control parameters in the Hamiltonian. The final ground states can - for example - be entangled, be useful for one-way quantum computation or even directly encode the solution to a difficult problem. The scheme has nice robustness features: When the reservoir temperature is significantly smaller than the time-dependent energy gap above the ground state, the decoherent interactions may drag the system towards its ground state and may therefore be even helpful in the goal to solve the problem. However, for critical parameter values the ground state itself may change in a quite abrupt manner, which is typically associated with a nearly vanishing energy gap. In the infinite size limit this corresponds to a quantum phase transition, whereas for finite-size systems one usually has a finite energy gap. This scaling behavior of the energy gap poses a severe problem both for the adiabatic implementation (diverging adiabatic runtime) and its robustness against thermal excitations. There exist only a few exactly solvable models that exhibit a quantum phase transition in the infinite size limit, such that in general one has to use numerical simulations. The present proposal suggests the development of intelligent numerical methods for the time-dependent tracking of eigenvalue problems and to study the performance of adiabatic algorithms in the case of open and closed quantum systems. More specifically, these new algorithms must have an adequate accuracy even in critical parameter regions. This shall be achieved by adaptively matching not only the discretization level, the step size of the eigenvalue tracer but also the number of calculated eigenvalues (and therefore also the size of the traced subspace) via error and condition estimators to the behavior of the system.