The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Yet, preparing states quickly and with high fidelity remains a formidable challenge. In this work we implement cutting-edge Reinforcement Learning (RL) techniques and show that their performance is comparable to optimal control methods in the task of finding short, high-fidelity driving protocol from an initial to a target state in non-integrable many-body quantum systems of interacting qubits. RL methods learn about the underlying physical system solely through a single scalar reward (the fidelity of the resulting state) calculated from numerical simulations of the physical system. We further show that quantum state manipulation, viewed as an optimization problem, exhibits a spinglass-like phase transition in the space of protocols as a function of the protocol duration. Our RL-aided approach helps identify variational protocols with nearly optimal fidelity, even in the glassy phase, where optimal state manipulation is exponentially hard. This study highlights the potential usefulness of RL for applications in out-of-equilibrium quantum physics. S = {s = (t, h x (t))}, A = {a = δh x }, R = {r ∈ [0, 1]}.

In these lecture notes, partly based on a course taught at the Karpacz Winter School in March 2014, we explore the close connections between non-adiabatic response of a system with respect to macroscopic parameters and the geometry of quantum and classical states. We center our discussion around adiabatic gauge potentials, which are the generators of unitary basis transformations in quantum systems and generators of special canonical transformations in classical systems. In quantum systems, eigenstate expectation values of these potentials are the Berry connections and the covariance matrix of these gauge potentials is the geometric tensor, whose antisymmetric part defines the Berry curvature and whose symmetric part is the Fubini-Study metric tensor. In classical systems one simply replaces the eigenstate expectation value by an average over the micro-canonical shell. For complicated interacting systems, we show that a variational principle may be used to derive approximate gauge potentials. We then express the non-adiabatic response of the physical observables of the system through these gauge potentials, specifically demonstrating the close connection of the geometric tensor to the notions of Lorentz force and renormalized mass. We highlight applications of this formalism to deriving counter-diabatic (dissipationless) driving protocols in various systems, as well as to finding equations of motion for slow macroscopic parameters coupled to fast microscopic degrees of freedom that go beyond macroscopic Hamiltonian dynamics. Finally, we illustrate these ideas with a number of simple examples and highlight a few more complicated ones drawn from recent literature.Comment: Based on lectures at the Karpacz Winter School, March 2014. Significant revisions have been made to include approximate gauge potentials for interacting models and update to published versio

Such driving in principle allows one to realize arbitrarily fast annealing protocols or implement fast dissipationless driving, circumventing standard adiabatic limitations requiring infinitesimally slow rates. These ideas were tested and used both experimentally and theoretically in small systems, but in larger chaotic systems, it is known that exact counterdiabatic protocols do not exist. In this work, we develop a simple variational approach allowing one to find the best possible counterdiabatic protocols given physical constraints, like locality. These protocols are easy to derive and implement both experimentally and numerically. We show that, using these approximate protocols, one can drastically suppress heating and increase fidelity of quantum annealing protocols in complex many-particle systems. In the fast limit, these protocols provide an effective dual description of adiabatic dynamics, where the coupling constant plays the role of time and the counterdiabatic term plays the role of the Hamiltonian.counterdiabatic driving | adiabatic gauge | transitionless driving | variational principle | complex systems D espite the time-reversal symmetry of the microscopic dynamics of isolated systems, losses are ubiquitous in any process that tries to manipulate them. Whether it is the heat produced in a car engine or the decoherence of a qubit, all losses arise from our lack of control on the microscopic degrees of freedom of the system. Since the early days of thermodynamics and actually, even before, the adiabatic process has emerged as a universal way to minimize losses, leading to the concept of Carnot efficiency-the cornerstone of modern thermodynamics. Despite its conceptual importance, practical implications of the Carnot efficiency are limited, because the maximal efficiency goes hand in hand with zero power. Nonetheless, by sacrificing some of the efficiency, one can run the same Carnot cycle at finite power (1). Heat engines might seem to be a problem of the past, but the understanding of finite-time thermodynamics in small (quantum) systems has become increasingly important because of developments in quantum information and nanoengineering.Developing and understanding methods to induce quasiadiabatic dynamics at finite times are paramount to the advancement of quantum information technologies. In general, one could distinguish between two (complementary) approaches. On one hand, one can, for a fixed setup, try to develop optimal driving protocols that result in minimal loss under certain constraints. Such protocols were recently suggested as appropriate geodesic paths in the parameter space in both the context of thermodynamics (2) and the context of adiabatic-state preparation (3, 4). The optimal protocols were also analyzed numerically using various optimum control ideas (5-9). On the other hand, one can try to engineer fast nonadiabatic protocols that lead to the same result as the fully adiabatic protocol. In particular, transitionless driving protocols were recently proposed and explored in small...