Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.

中文翻译：

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哈密​​顿变量量子系统的插值方法及绝热定理

可以通过改变系统哈密顿量来实现量子控制。根据绝热定理，如果初始状态是基态，则缓慢变化的哈密顿量可以在演化过程中大致将系统保持在基态。在本文中，我们将此过程视为初始汉密尔顿和最终哈密顿量之间的插值。我们使用单个算子的平均值来测量最终状态与理想基态之间的距离。该措施类似于激发能或在热力学中执行的多余功，可以视为绝热近似的误差。我们证明在某些条件下，可以为任意给定的插值函数估计此误差。该误差估计可以用作诱导绝热演变的准则。根据我们的计算，绝热近似误差在许多情况下与系统哈密顿量变化的平均速度和能隙的倒数不成线性比例。特别地，我们将此分析应用于绝热定理的适用性值得怀疑的例子。