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arXiv 1004.0509quant-phcond-mat.othermath-phmath.MP
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量子绝热演化和量子相变的内禀几何

Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions

Ali T. Rezakhani, Damian F. Abasto, Daniel A. Lidar, Paolo Zanardi

摘要:

我们阐明了量子绝热演化的几何。通过最小化绝热性的偏差,我们发现一个黎曼度量张量的绝热演化。配备了这个张量,我们确定了一个统一的几何描述的量子绝热演化和量子相变,它概括了以前的治疗,允许简并。相同的结构与量子信息处理中的应用相关,包括绝热和完整量子计算,其中控制参数的流形上的测地线对应于使误差最小化的路径。我们说明了这种几何结构的例子,我们明确发现绝热测地线。通过求解量子临界点附近的测地线方程,我们确定了量子相变最佳绝热通道的普适特征。特别是,我们表明,在一个临界点附近描述的二阶量子相变,测地线表现出幂律标度指数的两倍的空间和标度维度的产品的逆。

英文摘要:

We elucidate the geometry of quantum adiabatic evolution. By minimizing the deviation from adiabaticity we find a Riemannian metric tensor underlying adiabatic evolution. Equipped with this tensor, we identify a unified geometric description of quantum adiabatic evolution and quantum phase transitions, which generalizes previous treatments to allow for degeneracy. The same structure is relevant for applications in quantum information processing, including adiabatic and holonomic quantum computing, where geodesics over the manifold of control parameters correspond to paths which minimize errors. We illustrate this geometric structure with examples, for which we explicitly find adiabatic geodesics. By solving the geodesic equations in the vicinity of a quantum critical point, we identify universal characteristics of optimal adiabatic passage through a quantum phase transition. In particular, we show that in the vicinity of a critical point describing a second order quantum phase transition, the geodesic exhibits power-law scaling with an exponent given by twice the inverse of the product of the spatial and scaling dimensions.

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