Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.
Department of Physics, University of California, Berkeley, California 94720, USA.
Phys Rev Lett. 2019 Jan 18;122(2):020601. doi: 10.1103/PhysRevLett.122.020601.
We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glasslike transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm t-distributed stochastic neighbor embedding, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.
我们通过在 bang-bang 协议族中优化保真度来研究通过优化保真度将量子多体系统从初始状态制备到目标状态的问题。我们给出了令人信服的数值证据，证明了由协议持续时间控制的普遍自旋玻璃样转变。玻璃态临界点的特点是具有接近最优保真度的协议大量增殖，并且出现真正的最优值，这似乎很难通过指数级的方法找到。我们使用基于流形学习算法 t 分布随机邻居嵌入的机器学习 (ML) 启发式框架，能够在有效的低维表示中可视化高维控制景观的几何形状。在转变过程中，控制景观具有大量的簇，簇之间被广泛的势垒隔开，这与自旋玻璃和随机满足问题中的 replica 对称性破缺具有很强的相似性。我们进一步表明，量子控制景观映射到一个没有无序的自由能经典伊辛模型上，该模型具有受挫折的非局部多体相互作用。我们的工作突出了最优量子控制和自旋玻璃物理之间复杂但出人意料的联系，并展示了如何使用 ML 工具来可视化和理解玻璃态优化景观。
