As technological advances allow us to peer into and beyond microscopic phenomena, new theoretical developments are necessary to facilitate this exploration. In particular, the potential afforded by utilizing quantum resources promises to dramatically affect scientific research, communications, computation, and material science. \n \nControl theory is the field dedicated to the manipulation of systems, and quantum control theory pertains to the manoeuvring of quantum systems. Due to the inherent sensitivity of quantum ensembles to their environment, time-optimal solutions should be found in order to minimize exposure to external sources. Furthermore, the complexity of the Schr\\"odinger equation in describing the evolution of a unitary operator makes the analytical discovery of time-optimal solutions rare, motivating the development of numerical algorithms. \n \nThe seminal result of classical control is the Pontryagin Maximum Principle, which implies that a restriction to bounded control amplitudes admits two classifications of trajectories: bang-bang and singular. Extensions of this result to generalized control problems yield the same classification and hence arise in the study of quantum dynamics. While singular trajectories are often avoided in both classical and quantum literature, a full optimal synthesis requires that we analyze the possibility of their existence. \n \nWith this in mind, this treatise will examine the issue of time-optimal quantum control. In particular, we examine the results of existing numerical algorithms, including Gradient Ascent Pulse Engineering and the Kaya-Huneault method. We elaborate on the ideas of the Kaya-Huneault algorithm and present an alternative algorithm that we title the Real-Embedding algorithm. These methods are then compared when used to simulate unitary evolution. \n \nThis is followed by a brief examination on the conditions for the existence of singular controls, as well possible ideas and developments in creating geometry based numerical algorithms.
Recently there has been a lot of interest in the potential applications of performing computations on systems whose governing physical laws are quantum, rather than classical in nature. These quantum computers would have the ability to perform some calculations which would not be feasible for their classical counterparts. To date, however, a quantum computer large enough to perform useful calculations has yet to be built. Before this can be accomplished, a method must be developed to control the underlying quantum systems. This is a problem which can naturally be formulated in the language of control theory. This report outlines the basic control-theoretic approach to time-optimally controlling quantum systems evolving under the dynamics of the Schr\\"{o}dinger operator equation. It is found that under the assumption of non-singularity, the controls which produce time-optimal trajectories are bang-bang. With this in mind, a switching time algorithm is implemented to find optimal bang-bang controls.
Optimization is one of the key applications of quantum computing where a quantum speedup has been an eagerly anticipated outcome. A promising approach to optimization using quantum dynamics is to consider a linear combination $s(t)B+[1\ensuremath{-}s(t)]C$ of two noncommuting Hamiltonians
B and
C, where
C encodes the solution to the optimization problem in its ground state,
B is a Hamiltonian whose ground state is easy to prepare, and $s(t)\ensuremath{\in}[0,1]$ is the bounded ``switching schedule'' or ``path,'' with $t\ensuremath{\in}[0,{t}_{f}]$. This approach encompasses two of the most widely studied quantum-optimization algorithms: quantum annealing [QA; continuous
s(t)] and the quantum approximate optimization algorithm [QAOA; piecewise constant
s(t)]. While it is notoriously difficult to prove a quantum advantage for either algorithm, it is possible to compare and contrast them by finding the optimal
s(t). Here we provide a rigorous analysis of this quantum optimal control problem, entirely within the geometric framework of Pontryagin's maximum principle of optimal control theory. We extend earlier results, derived in a purely closed-system setting, to open systems. This is the natural setting for experimental realizations of QA and QAOA. In the closed-system setting it was shown that the optimal solution is a ``bang-anneal-bang'' schedule, with the bangs characterized by
s(t)=0 and
s(t)=1 in finite subintervals of
[0,tf], in particular,
s(0)=0 and
s(tf)=1, in contrast to the standard prescription
s(0)=1 and
s(tf)=0 of QA. As an example, we prove that for a single spin-
1/2, the optimal solution in the closed-system setting is the bang-bang schedule, switching midway from $s\ensuremath{\equiv}0$ to $s\ensuremath{\equiv}1$. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of
s(0)=0 and
s(tf)=1. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which
s(tf)=1, and even this result is not recovered in the fully Markovian limit, suggesting that the pure anneal-type schedule is optimal. Our open-system results have implications for the use of experimental quantum-information processors, which are by necessity noisy, and suggest that in this practical sense the optimal schedules for quantum optimization are likely to be continuous.
The time optimal problem for a two level quantum system is studied. We compare two different control strategies of bang-bang control and the geometric control, respectively, especially in the case of minimizing the time of steering the state from North Pole to South Pole on the Bloch sphere with bounded control. The time performances are compared for different parameters by the individual numerical simulation experiments, and the experimental results are analyzed. The results show that the geometric control spends less time than the bang-bang control does.
We apply advanced methods of control theory to open quantum systems and we determine finite-time processes which are optimal with respect to thermodynamic performances. General properties and necessary conditions characterizing optimal drivings are derived, obtaining bang-bang-type solutions corresponding to control strategies switching between adiabatic and isothermal transformations. A direct application of these results is the maximization of the work produced by a generic quantum heat engine, where we show that the maximum power is directly linked to a particular conserved quantity naturally emerging from the control problem. Finally we apply our general approach to the specific case of a two-level system, which can be put in contact with two different baths at fixed temperatures, identifying the processes that minimize heat dissipation. Moreover, we explicitly solve the optimization problem for a cyclic two-level heat engine driven beyond the linear-response regime, determining the corresponding optimal cycle, the maximum power, and the efficiency at maximum power.
Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength
umax. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small
umax, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.