量子多体问题的变分嵌入
Variational embedding for quantum many-body problems
摘要：
量子嵌入理论是近似求解大规模强关联量子多体问题的有力工具。量子嵌入的主要思想是将局部尺度上高度精确的量子理论和全局尺度上不太精确的量子理论粘合在一起。我们介绍的第一个量子嵌入理论，也是变分的，因为它是保证提供一个片面的确切的基态能量的限制。我们的方法，我们称之为变分嵌入法，提供了这个数量的下限。该方法放宽了量子边际的可表示性条件，一组线性和半定的约束，在局部和全局尺度上操作，从而在一个半定的程序（SDP）进行数值求解。该方法的准确性可以系统地提高。该方法是通用的，可以应用于，特别是，量子自旋系统和费米子系统，如电子结构计算所产生的量子多体问题。我们描述了如何适当的概念量子边际，足够普遍，以适应这两种设置，应措辞在某些代数的运营商。我们还研究了我们的SDPs的对偶理论，这为我们的方法作为嵌入理论提供了有价值的观点。作为这项调查的副产品，我们描述了一个配方，有效地实现变分嵌入方法通过部分对偶过程和量子模拟的Kantorovich问题的最佳传输理论的解决方案。
英文摘要：
Quantum embedding theories are powerful tools for approximately solving large-scale strongly correlated quantum many-body problems. The main idea of quantum embedding is to glue together a highly accurate quantum theory at the local scale and a less accurate quantum theory at the global scale. We introduce the first quantum embedding theory that is also variational, in that it is guaranteed to provide a one-sided bound for the exact ground-state energy. Our method, which we call the variational embedding method, provides a lower bound for this quantity. The method relaxes the representability conditions for quantum marginals to a set of linear and semidefinite constraints that operate at both local and global scales, resulting in a semidefinite program (SDP) to be solved numerically. The accuracy of the method can be systematically improved. The method is versatile and can be applied, in particular, to quantum many-body problems for both quantum spin systems and fermionic systems, such as those arising from electronic structure calculations. We describe how the proper notion of quantum marginal, sufficiently general to accommodate both of these settings, should be phrased in terms of certain algebras of operators. We also investigate the duality theory for our SDPs, which offers valuable perspective on our method as an embedding theory. As a byproduct of this investigation, we describe a formulation for efficiently implementing the variational embedding method via a partial dualization procedure and the solution of quantum analogs of the Kantorovich problem from optimal transport theory.
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