Scalable Numerical Methods for Adiabatic Quantum Preparation

许多物理相关的模型包括时间相关的汉密尔顿运营商。当时间依赖性慢于光谱的能隙时，绝热定理意味着在初始哈密顿量的基态中制备的量子系统将完全遵循瞬时基态。这可以被利用来准备有趣的基态从简单的缓慢变形的控制参数的哈密顿量。例如，最终的基态可以是纠缠的，可以用于单向量子计算，甚至可以直接编码难题的解决方案。该方案具有很好的鲁棒性：当库温度显著小于基态以上的依赖于时间的能隙时，退相干相互作用可能会将系统拖向其基态，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，基态本身可能以非常突然的方式改变，这通常与几乎消失的能隙相关联。在无限大的极限下，这对应于量子相变，而对于有限尺寸的系统，通常具有有限的能隙。能隙的这种缩放行为对绝热实现（发散绝热运行时间）及其对热激励的鲁棒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尺寸极限下表现出量子相变，因此通常必须使用数值模拟。本建议建议的智能数值方法的发展依赖于时间的跟踪的特征值问题，并研究绝热算法的性能的情况下，开放和封闭的量子系统。更具体地说，这些新的算法必须有足够的精度，即使在关键参数区域。这将通过自适应匹配不仅离散化水平，步长的特征值跟踪器，而且计算的特征值的数量（因此也跟踪子空间的大小）通过误差和条件估计系统的行为。

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.