Glassy and Correlated Phases of Optimal Quantum Control

Abstract

Modern Machine Learning (ML) relies on cost function optimization to train model parameters. The non-convexity of cost function landscapes results in the emergence of local minima in which state-of-the-art gradient descent optimizers get stuck. Similarly, in modern Quantum Control (QC), a key to understanding the difficulty of multiqubit state preparation holds the control landscape -- the mapping assigning to every control protocol its cost function value. Reinforcement Learning (RL) and QC strive to find a better local minimum of the control landscape; the global minimum corresponds to the optimal protocol. Analyzing a decrease in the learning capability of our RL agent as we vary the protocol duration, we found rapid changes in the search for optimal protocols, reminiscent of phase transitions. These "control phase transitions" can be interpreted within Statistical Mechanics by viewing the cost function as "energy" and control protocols – as "spin configurations". I will show that optimal qubit control exhibits continuous and discontinuous phase transitions familiar from macroscopic systems: correlated/glassy phases and spontaneous symmetry breaking. I will then present numerical evidence for a universal spin-glass-like 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 ML inspired framework based on the manifold learning algorithm t-SNE, we 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 random satisfiability problems.