Taming quantum systems: A tutorial for using shortcuts-to-adiabaticity, quantum optimal control, and reinforcement learning

TL;DR

This tutorial provides a structured, pedagogical tour of three central quantum-control paradigms—shortcuts to adiabaticity (STA) with counterdiabatic driving and adiabatic gauge potentials, quantum optimal control (QOC) via gradient-based methods, and reinforcement learning (RL) for autonomous protocol discovery. It blends rigorous foundational derivations with concrete model examples (e.g., Landau-Zener, Ising and Lipkin-Meshkov-Glick) and discusses the practicalities of experimental implementations and the hybridization of approaches. Key contributions include variational constructions of approximate AGPs that respect locality, explicit QOC strategies (GRAPE) with analytic gradients, and RL frameworks that can generalize to unseen control tasks and operate with partial quantum information. Together, these sections illuminate how STA, QOC, and RL can be combined to harness controllability limits, push toward faster protocols, and inform hardware-aware strategies for scalable quantum technologies. The work underscores the importance of hybrid algorithms that blend analytic insight with data-driven optimization to tackle open-system dynamics, complex many-body problems, and real-world imperfections in quantum devices.

Abstract

Precise manipulation of quantum effects at the atomic and nanoscale has become an essential task in ongoing scientific and technological endeavours. Quantum control methods are thus routinely exploited for research in areas such as quantum materials, quantum chemistry, and atomic and molecular physics, as well as in the development of quantum technologies like computing, simulation, and sensing. Here, we present a pedagogical introduction to the basics of quantum control methods in tutorial form, with the aim of providing newcomers to the field with the core concepts and practical tools to use these methods in their research. We focus on three areas: shortcuts to adiabaticity, quantum optimal control, and machine-learning-based control. We lay out the basic theoretical elements of each area in a pedagogical way and describe their application to a series of example cases. For these, we include detailed analytical derivations as well as extensive numerical results. As an outlook, we discuss quantum control methods in the broader context of quantum technologies development and complex quantum systems research, outlining potential connections and synergies between them.

Figures (16)

• Figure 1: Schematic of the main three approaches to quantum control that are discussed in this tutorial article: shortcuts to adiabaticity (Sec. \ref{['sec:STA']}, optimal control (Sec. \ref{['sec:QOC']}), and reinforcement learning (Sec. \ref{['sec:RL_theory']}).
• Figure 2: Time dependence of the energy eigenvalues for the Landau-Zener model (solid, black) and for the controlled evolution, Eq. \ref{['HCD_LZ']} (dashed) for two different protocol durations assuming a linear ramp $v(t)=-5+10\tfrac{t}{T}$ and $\Delta\!=\!1$.
• Figure 3: Ising model for $N=4$ with $g(t)=0.1+1.9 t/T$ fixing $T \!=\!1$ and starting in the ground state. We show the fidelity with the instantaneous ground state for the full counterdiabtic term (black) involving all two and three body interactions i.e. Eq. \ref{['eq:HCDIsingFull']}, implementing only the two-body terms i.e. Eq. \ref{['eq:HCDIsing2body']} (dashed, red), implementing only the three-body terms i.e. Eq. \ref{['eq:HCDIsing3body']} (dot-dashed, orange). We also show the evolution for the bare Hamiltonian (blue).
• Figure 4: (a) Ramping within the symmetry broken phase of the all-to-all model, with $g(t)=2-0.9 t$ where the aim is to connect the ground state of $H$ at $t=0$ with the ground state at $t=1$. We show the final target state fidelity at $t=1$ as a function of system size for the exact, numerically evaluated CD (black), bare evolution (blue), and using the QHO approximation, Eq. \ref{['eq:LMGCDterm']}, (red). (b) We fix $N=50$ and use the guess pulse Eq. \ref{['eq:GuessPulseLMG']} for the control given by the QHO approximation and sweep over the achievable fidelities.
• Figure 5: Implementation of local CD in the Ising spin model. The case of no control term(s) being implemented (blue) is compared to implementing one-body local CD (red) and two-body local CD (black). (a-c) The achieved fidelity ($1-\mathcal{F}$) for different total protocol times for (a) $N=2$, (b) $N=3$, and (c) $N=4$ spins. Note, for the case of (a) $N=2$ the two-body local CD gives unit fidelity to below single precision. (d) The achieved final fidelity for a fixed total time, $T=0.01 J^{-1}$, for various system sizes showing the saturation of the fidelity for the one and two-body implementations.