The Top Manifold Connectedness of Quantum Control Landscapes

TL;DR

The paper addresses whether the top manifold of quantum control landscapes for state transitions, observables, and unitary gates is path-connected when the control duration is fixed. It combines gradient-based sampling with the string method and D-MORPH to construct continuous paths between randomly chosen optimal controls, demonstrating empirical path-connectivity across multiple systems and objectives. The results show the top manifold is typically connected with near-straight connecting paths (ratio $R$ near unity) and only mild curvature, indicating features are navigable by local dynamics and enabling concurrent optimization of ancillary objectives. These findings provide practical algorithms for multi-criteria quantum control and suggest a general conjecture that path-connectedness holds under standard controllability and fixed-$T$ conditions, with implications for robust and multi-objective control design.

Abstract

The control of quantum systems has been proven to possess trap-free optimization landscapes under the satisfaction of proper assumptions. However, many details of the landscape geometry and their influence on search efficiency still need to be fully understood. This paper numerically explores the path-connectedness of globally optimal control solutions forming the top manifold of the landscape. We randomly sample a plurality of optimal controls in the top manifold to assess the existence of a continuous path at the top of the landscape that connects two arbitrary optimal solutions. It is shown that for different quantum control objectives including state-to-state transition probabilities, observable expectation values and unitary transformations, such a continuous path can be readily found, implying that these top manifolds are fundamentally path-connected. The significance of the latter conjecture lies in seeking locations in the top manifold where an ancillary objective can also be optimized while maintaining the full optimality of the original objective that defined the landscape.

Figures (10)

• Figure 1: Schematics of two examples of top manifolds for QCLs having differing connectedness properties. The landscape is illustrated as the objective function $J$ over two control variables (usually hundreds of variables in practice) (a) A connected top manifold formed by optimal controls. Two optimal control fields, $E_1(t)$ and $E_2(t)$ (depicted by the red dots), can be connected by a continuous path (black line) on the top. (b) A disconnected top manifold composed of two separate regions in which the flatter region of $A$ implies higher robustness against control noise than that of a local point at the top of $B$.
• Figure 2: Schematics of successful use of the string method. (a) A portion of the QCL specified in Fig. \ref{['fig:qcl_schematics']}(a). (b) The two-dimensional contour plot of the landscape. The string (white line) is first initialized as a straight line sampled by interpolated fields (interior yellow circles) between two given optimal controls $E_{\rm {start}}(t)$ and $E_{\rm {target}}(t)$ (interior red and green circles, respectively). Then, it gradually evolves following the gradient flows (gray arrows) and associated strings until reaching the top manifold.
• Figure 3: Schematics of the D-MORPH connecting algorithm. (a) A zoomed-in section of the top manifold specified in Fig. \ref{['fig:qcl_schematics']}(a), highlighting the presence of two possible impeding features. (b) The two-dimensional contour plot of the landscape. The D-MORPH connecting algorithm explores the high-level set $J_{\max}-\varepsilon$. Starting from one optimal control field $E_{\rm {start}}(t)$ (interior red circles), the differential change $\frac{\partial E(s,t)}{\partial s}$ is determined as the projection of $f_{\rm {dist}}(s,t)$ onto the subspace orthogonal to the local gradient $\frac{\delta J}{\delta E(s,t)}$, as depicted in (b). Note that the D-MORPH method for this purpose has been built so that the trajectory (black line), from $E_{\rm {start}}(t)$ towards $E_{\rm {target}}(t)$, remains on the top of the landscape. The numerical evidence shows that such trajectories almost always avoid getting trapped by a top manifold feature as schematically indicated, by moving on the edge in this figure.
• Figure 4: Distributions of the pairwise distances among optimal fields for STL (red), OCL (green), and UTL (blue). Each distribution, which records $\sim 5\times 10^5$ pairwise distances in total, is drawn from one thousand optimal controls in each landscape.
• Figure 5: (a) The convergence process of the string method to assess landscape connectedness illustrated for a particular STL. The vertical axis is the objective function $J$. Each dot in the figure characterizes the objective function $J$ of one interpolated field along the string. Initialized as a straight line (black line) between the two optimal fields $(E_{\rm start}(t), E_{\rm target}(t))$ in the top manifold, the intermediate interpolated fields initially produce $J$ values far from the top manifold. The set of intermediate fields of the string is each subjected to an iterative climb along a gradient flow of the landscape, and the set of fields forming the string finally reaches the top manifold at the iteration $18$ (red line) for this case. (b) A succession plot of some of the interpolated fields $\{\hat{E}_i(t):i=0,1,\cdots, N_{\rm st}\}$ along the final converged string in the top manifold for this case, where $N_{\rm st}=34$. The interpolated fields (distinguished by color scale) demonstrate that the starting field $E_{\rm start}(t)$ (orange on the bottom of (b)) is continuously transformed to the final control $E_{\rm target}(t)$ (red on the top) along the string.