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Scalable Numerical Methods for Adiabatic Quantum Preparation

用于绝热量子制备的可扩展数值方法

基本信息

批准号：
169213306
负责人：
Professor Dr. Tobias Brandes (†)
金额：
--
依托单位：
Fachgebiet Numerische Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2010
资助国家：
德国
起止时间：
2009-12-31 至 2013-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/169213306?language=en
关键词：
Scalable Numerical Methods Adiabatic Quantum

项目摘要

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.

许多物理上相关的模型包括时间相关的汉密尔顿算子。当与光谱的能量间隙相比，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。这可以通过在哈密顿量中缓慢变形控制参数来从简单的基态制备有趣的基态。例如，最终基态可以纠缠在一起，可以用于单向量子计算，甚至可以直接编码一个难题的解决方案。该方案具有良好的鲁棒性：当储层温度明显小于基态以上的时变能隙时，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实现解决问题的目标。然而，对于关键参数值，基态本身可能以一种非常突然的方式变化，这通常与几乎消失的能隙有关。在无限大小的极限中，这对应于量子相变，而对于有限大小的系统，通常具有有限的能隙。这种能量间隙的标度行为对绝热实现（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。只有少数可精确解的模型能在无限大小的极限下表现出量子相变，因此通常必须使用数值模拟。本建议建议发展特征值问题随时间跟踪的智能数值方法，并研究在开放和封闭量子系统情况下绝热算法的性能。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。这将通过自适应匹配来实现，不仅是离散化水平，特征值跟踪器的步长，还有计算的特征值的数量（因此也是跟踪子空间的大小），通过误差和条件估计器来匹配系统的行为。