# Glassy Phase of Optimal Quantum Control

## TL;DR

This paper reveals a spin-glass-like transition in quantum control landscapes, showing a complex, glassy phase with many near-optimal protocols and using machine learning to visualize the high-dimensional landscape structure.

## Contribution

It uncovers a universal glassy phase in quantum control landscapes and introduces ML techniques to visualize and analyze their complex geometry.

## Key findings

- Identification of a spin-glass-like transition in control landscapes
- Visualization of landscape clusters using t-SNE
- Mapping of control landscape to a frustrated Ising model

## Abstract

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.