论文标题
扰动理论和正方形的总和
Perturbation Theory and the Sum of Squares
论文作者
论文摘要
平方之和层次层次结构是一种基于半准编程的强大技术，可用于经典和量子优化问题。该层次结构以几个名称为单位。特别是，在量子化学中，它称为减少密度矩阵（RDM）方法。我们考虑了该层次结构对三种不同类型的系统重现弱耦合扰动理论的能力：旋转（或Qubit）系统，玻感系统系统（Anharmonic振荡器）和具有四分之一相互作用的典型系统。对于此类费米子系统，我们表明$ 4 $ SOS（称为$ 2 $ -RDM的量子化学物质）不会再现二阶扰动理论，而是$ 6 $ SOS（$ 3 $ -RDM）（并且我们猜测它会重现第三阶攻击理论）。的确，我们确定了一个可以做到这一点的$ 6 $ SOS的片段，这对于实用的量子化学计算可能很有用，因为可以实现此片段的成本少于全度$ 6 $ SOS。值得注意的是，这个片段与Hastings和O'Donnell为Sachdev-Ye-Kitaev（Syk）模型所研究的片段非常相似。
The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscillator), and fermionic systems with quartic interactions. For such fermionic systems, we show that degree-$4$ SoS (called $2$-RDM in quantum chemsitry) does not reproduce second order perturbation theory but degree-$6$ SoS ($3$-RDM) does (and we conjecture that it reproduces third order perturbation theory). Indeed, we identify a fragment of degree-$6$ SoS which can do this, which may be useful for practical quantum chemical calculations as it may be possible to implement this fragment with less cost than the full degree-$6$ SoS. Remarkably, this fragment is very similar to one studied by Hastings and O'Donnell for the Sachdev-Ye-Kitaev (SYK) model.
