The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g. a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.
The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g. a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via the optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.
Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.
Quantum information technologies require careful control for generating and preserving a desired target quantum state. The biggest practical obstacle is of course decoherence. Therefore, the reachability analysis, which in our scenario aims to estimate the distance between the controlled state under decoherence and the target state, is of great importance to evaluate the realistic performance of those technologies. This paper presents a lower bound of the fidelity-based distance for a general open Markovian quantum system driven by the decoherence process and several types of control including feedback. The lower bound is straightforward to calculate and can be used as a guide for choosing the target state, as demonstrated in some examples. Moreover, the lower bound is applied to derive a theoretical limit in some quantum metrology problems based on a large-size atomic ensemble under control and decoherence.
The Hamiltonian and evolution time of a quantum unitary system completely determine a unitary operator, which turns an initial state ${\ensuremath{\rho}}_{0}$ into its time-evolved state ${\ensuremath{\rho}}_{t}$ and may also turn another state ${\ensuremath{\sigma}}_{0}$ into its time-evolved state ${\ensuremath{\sigma}}_{t}$. Although the two pairs of states $({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{t})$ and $({\ensuremath{\sigma}}_{0},{\ensuremath{\sigma}}_{t})$ are governed by the same Hamiltonian and experience equal evolution periods, their state distances are not necessarily equal to each other. Specifically, the evolution operator contains information about the evolution other than that reflected by two specific states connected through it. As a matter of fact, two pairs of states with different state distances may have equal path distances. Here we propose a geometric measure for the path distance contained in a quantum unitary evolution, which depends only on the unitary operator itself and is independent of any pair of states connected through it. This path distance meets the basic requirements for a good measure of distance and is valid during the whole evolution process, no matter how long the evolution time is. This path distance is bounded by the path length of the evolution from above and by the Bures angle between the two states connected through it from below. It has potential applications in quantum information processing, such as providing another way to study the optimal control between two quantum states. We finally present the form of Hamiltonian to realize a unitary operator on demand.
A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive trace preserving maps (i.e., Kraus maps), are investigated in details. The objective function, which is the expectation value of a target system operator, is defined on the Stiefel manifold representing the space of Kraus maps. Three practically important properties of the objective function are found: (a) the absence of local maxima or minima (i.e., false traps); (b) the existence of multi-dimensional sub-manifolds of optimal solutions corresponding to the global maximum and minimum; and (c) the connectivity of each level set. All of the critical values and their associated critical sub-manifolds are explicitly found for any initial system state. Away from the absolute extrema there are no local maxima or minima, and only saddles may exist, whose number and the explicit structure of the corresponding critical sub-manifolds are determined by the initial system state. There are no saddles for pure initial states, one saddle for a completely mixed initial state, and two saddles for other initial states. In general, the landscape analysis of critical points and optimal manifolds is relevant to the problem of explaining the relative ease of obtaining good optimal control outcomes in the laboratory, even in the presence of the environment.