论文标题
关于快速单量子移相门的量子控制景观的详细结构
On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation
论文作者
论文摘要
在这项工作中，我们研究了量子控制景观的详细结构，以在快速时间尺度上产生单量相移门的问题。在以前的工作中，在各种时间尺度上证明了此问题的陷阱的缺失。根据控制系统的参数，已知在量子控制景观中存在的一个特殊关键点被证明是鞍形或全球极值。但是，在鞍座的情况下，目前尚未研究Hessian的负和正理值的数量及其大小。同时，这些数字和幅度决定了在临界点附近实用优化的相对易度或难度。在这项工作中，我们计算出Hessian在此鞍点上的负和正征值的数量，此外，对这些特征值的幅度进行了估计。我们还显着简化了有关Hessian的鞍点的先前定理证明[b.o. 〜Volkov，O.V.〜Morzhin，A.N.理论。 {\ bf 54}，215303（2021）]。
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem~3 in B.O.~Volkov, O.V.~Morzhin, A.N.~Pechen, J.~Phys.~A: Math. Theor. {\bf 54}, 215303 (2021)].
