  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-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 machine learning (ML) inspired framework based on the manifold
learning algorithm t-SNE, 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.
