The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.

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Quantum optimal control theory

For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e., piecewise constant control amplitudes, iteratively into an optimized shape. Here, we present a comparative study of optimal-control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and Krotov-type methods which do so sequentially. Guidelines for their use are given and open research questions are pointed out. Moreover, we introduce a unifying algorithmic framework, DYNAMO (dynamic optimization platform), designed to provide the quantum-technology community with a convenient matlab-based tool set for optimal control. In addition, it gives researchers in optimal-control techniques a framework for benchmarking and comparing newly proposed algorithms with the state of the art. It allows a mix-and-match approach with various types of gradients, update and step-size methods as well as subspace choices. Open-source code including examples is made available at http://qlib.info.

Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework

Manipulation of quantum interference requires that the system under control remains coherent, avoiding (or at least postponing) the phase randomization that can ensue from coupling to an uncontrolled environment. We show that closed-loop coherent control can be used to mitigate the rate of quantum dephasing in a gas-phase ensemble of potassium dimers (K2), which acts as a model system for testing the general concepts of controlling decoherence. Specifically, we adaptively shaped the light pulse used to prepare a vibrational wave packet in electronically excited K2, with the amplitude of quantum beats in the fluorescence signal used as an easily measured surrogate for the purpose of optimizing coherence. The optimal pulse increased the beat amplitude from below the noise level to well above it, and thereby increased the coherence life time as compared with the beats produced by a transform-limited pulse. Closed-loop methods can thus effectively identify states that are robust against dephasing without any previous information about the system-environment interaction.

https://science.sciencemag.org/content/sci/320/5876/638.full.pdf

Coherent Control of Decoherence

Optimal control methods for implementing quantum modules with least amount of relaxative loss are devised to give best approximations to unitary gates under relaxation. The potential gain by optimal control fully exploiting known relaxation parameters against time-optimal control (the alternative for unknown relaxation parameters) is explored and exemplified in numerical and in algebraic terms: for instance, relaxation-based optimal control is the method of choice to govern quantum systems within subspaces of weak relaxation whenever the drift Hamiltonian would otherwise drive the system through fast decaying modes. In a standard model system generalizing ideal decoherence-free subspaces to more realistic scenarios, opengrape-derived controls realize a CNOT with fidelities beyond 95% instead of at most 15% for a standard Trotter expansion. As additional benefit their control fields are orders of magnitude lower in power than bang-bang decouplings.

https://iopscience.iop.org/article/10.1088/0953-4075/44/15/154013/pdf

Optimal control for generating quantum gates in open dissipative systems

This paper presents a constructive proof of complete kinematic state controllability of finite-dimensional open quantum systems whose dynamics are represented by Kraus maps. For any pair of states (pure or mixed) on the Hilbert space of the system, we explicitly show how to construct a Kraus map that transforms one state into another. Moreover, we prove by construction the existence of a Kraus map that transforms all initial states into a predefined target state (such a process may be used, for example, in quantum information dilution). Thus, in sharp contrast to unitary control, Kraus-map dynamics allows for the design of controls which are robust to variations in the initial state of the system. The capabilities of non-unitary control for population transfer between pure states illustrated for an example of a two-level system by constructing a family of non-unitary Kraus maps to transform one pure state into another. The problem of dynamic state controllability of open quantum systems (i.e., controllability of state-to-state transformations, given a set of available dynamical resources such as coherent controls, incoherent interactions with the environment, and measurements) is also discussed.

Controllability of open quantum systems with Kraus-map dynamics

A Markovian master equation describing the evolution of open quantum systems in the presence of a time-dependent external field is derived within the Bloch-Redfield formalism. It leads to a system-bath interaction which depends on the control field. Optimal control theory is used to select control fields which allow accelerated or decelerated system relaxation, or suppression of relaxation (dissipation) altogether, depending on the dynamics we impose on the quantum system. The control-dissipation correlation and the nonperturbative treatment of the control field are essential for reaching this goal. The optimal control problem is formulated within Pontryagin's minimum principle and the resulting optimal differential system is solved numerically. As an application, we study the dynamics of a spin-boson model in the strong coupling regime under the influence of an external control field. We show how trapping the system in unstable quantum states and transfer of population can be achieved by optimized control of the dissipative quantum system. We also used optimal control theory to find the driving field that generates the quantum $Z$ gate. In several cases studied, we find that the selected optimal field which reduces the purity loss significantly is a multicomponent low-frequency field including higher harmonics, all of which lie below the phonon cutoff frequency. Finally, in the undriven case we present an analytic result for the Lamb shift at zero temperature.

Quantum optimal control theory and dynamic coupling in the spin-boson model

General optimal coherent control of dissipative $N$-level systems in the Markovian time regime is formulated within Pointryagin's principle and the Lindblad equation In the present paper, we study feasibility and limitations of steering of dissipative two-, three-, and four-level systems from a given initial pure or mixed state into a desired final state under the influence of an external electric field The time evolution of the system is computed within the Lindblad equation and a conjugate gradient method is used to identify optimal control fields The influence of both field-independent population and polarization decay on achieving the objective is investigated in systematic fashion It is shown that, for realistic dephasing times, optimum control fields can be identified which drive the system into the target state with very high success rate and in economical fashion, even when starting from a poor initial guess Furthermore, the optimal fields obtained give insight into the system dynamics However, if decay rates of the system cannot be subjected to electromagnetic control, the dissipative system cannot be maintained in a specific pure or mixed state, in general

W. Pötz

Papers

Quantum optimal control theory and dynamic coupling in the spin-boson model

Optimal coherent control of dissipative N -level systems

State-independent control theory for weakly dissipative quantum systems