Optimal Control


Minuscule objects that follow the laws of quantum mechanics have the promise of carrying out delicate tasks fundamentally better than macroscopic objects that are bound by the laws of Newtonian classical mechanics. The superposition principle permits individual quantum mechanical objects to follow multiple trajectories in parallel, and pairs (or larger collections) of such objects can be entangled with each other, such that a measurement on one object affects the properties of the other objects even if they are far apart. The superposition principle and entanglement provide the basis for applications like precision metrology or quantum computation that are expected to revolutionise our technology, just like the steam engine or the advent of electricity has done in the past.
The explicit utilisation of these quantum mechanical effects for useful applications, however, requires extremely accurate control over quantum objects and their interaction with their surroundings. Trapped ions are one of the leading systems in this context. Selected energy levels of an ion define a qubit, which is the elementary unit of a quantum computer, just like a classical computer is comprised of many bits. Confined by electric and magnetic trapping fields, ions can be manipulated with laser beams, and the collective motion of strings of ions enables the exchange of information between several qubits. For this to work with high accuracy it is typically required to cool the ions to a temperature close to absolute zero. Once such a temperature has been reached, one makes use of the fact that any manipulation of the ions changes their motional state in order to implement logical operations that define the elementary building blocks of a quantum algorithm.
Since the ions' motion is easily heated by its room-temperature environment, the intentionally induced changes in the motional state can be accompanied by uncontrolled heating processes, and any deviation from the desired change in motional state results in reduced accuracy of the operations being implemented. The goal of the present project is the development and experimental implementation of laser control of trapped ions that achieves desired operations with high accuracy and robustness in the presence of undesired heating and other experimental imperfections. In a strong collaboration between theory and experiment, control sequences will be developed and tested in a novel ion trap whose parameters can be varied over a wide range. The ability to tune the strength of the interaction between qubits and motion (the Lamb-Dicke parameter) and the strength of thermal effects will allow us to identify the control strategies that deal with each type of imperfection in an ideal fashion. Most current experiments are conducted with a rather weak interaction between qubits and motion, but we aim at the realisation of logical operations between qubits that interact strongly with the motion. The increased manipulation speed that comes with the strong interaction increases the number of logical operations that can be implemented within the limits imposed by finite decoherence time, and as such will help us to move from proof-of-principle experiments to a practical application.
The immediate goal of our work is the improvement in the control of trapped ions for quantum computing, but the advanced control techniques we will develop directly apply to any type of coherent manipulation of trapped ions. Since strong interactions between qubits are beneficial for fast information transfer but challenging for the implementation of accurate manipulations in essentially any quantum system, the control techniques to be developed are expected to find application in a broad range of other systems in quantum optics and quantum electronics.
This application proposes an integrated theoretical and experimental investigation of the use of optimal control techniques for the coherent control of ions in an ion trap. Our hypothesis is that the use of optimal control techniques will provide significant improvements to the speed and
robustness of trapped ion quantum logic gates. We aim to devise control schemes for trapped ions that do not rely on the Lamb-Dicke approximation, and to test the efficacy of such schemes in a large-scale radiofrequency ion trap with an engineered environmental coupling.

List of objectives

Hence, our specific objectives are to:
* use our existing Penning trap apparatus to determine which parameters are most important to the fidelity of single qubit gates beyond the Lamb-Dicke regime and to test the validity of the approximations we expect to use in the theory part of the proposed work;
* investigate methods of engineering environmental noise, to produce a tuneable, artificial heating mechanism and test it in the Penning trap;
* design pulses that drive specific transitions (either a carrier transition or a sideband transition) and that actively suppress contributions of undesired transitions;
* implement improved sideband cooling through these optimised control pulses;
* design and build a large, strong, millimeter-scale linear radiofrequency ion trap with extremely low intrinsic heating rate;
* trap and cool calcium ions to the motional ground-state in this trap;
* experimentally test optimised single-ion quantum gates regarding the effects of unwanted off-resonant transitions and noise processes;
* implement a two-ion Molmer-Sorensen entangling gate in the new trap with bi-chromatic and optimised polychromatic driving, and test the gate performance in dependence of heating rate;
* design and implement optimised control pulses for two- or multi-qubit gates beyond the Lamb-Dicke regime, and demonstrate the robustness of these gates against heating;
* investigate the use of a second ion species to prepare coherent states of the motion beyond the Lamb-Dicke approximation with optical dipole forces, without the loss of quantum information;
* investigate the application of our optimal control techniques to other quantum systems which can be used as platforms for quantum information.