Katharine W. Moore

On the relationship between quantum control landscape structure and optimization complexity

It has been widely observed in optimal control simulations and experiments that state preparation is surprisingly easy to achieve, regardless of the dimension N of the system Hilbert space. In contrast, simulations for the generation of targeted unitary transformations indicate that the effort increases exponentially with N. In order to understand such behavior, the concept of quantum control landscapes was recently introduced, where the landscape is defined as the physical objective, as a function of the control variables. The present work explores how the local structure of the control landscape influences the effectiveness and efficiency of quantum optimal control search efforts. Optimizations of state and unitary transformation preparation using kinematic control variables (i.e., the elements of the action matrix) are performed with gradient, genetic, and simplex algorithms. The results indicate that the search effort scales weakly, or possibly independently, with N for state preparation, while the search effort for the unitary transformation objective increases exponentially with N. Analysis of the mean path length traversed during a search trajectory through the space of action matrices and the local structure along this trajectory provides a basis to explain the difference in the scaling of the search effort with N for these control objectives. Much more favorable scaling for unitary transformation preparation arises upon specifying an initial action matrix based on state preparation results. The consequences of choosing a reduced number of control variables for state preparation is also investigated, showing a significant reduction in performance for using fewer than 2N-2 variables, which is consistent with the topological analysis of the associated landscape.

Search complexity and resource scaling for the quantum optimal control of unitary transformations

The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.

Universal characteristics of chemical synthesis and property optimization

A common goal in chemistry is to optimize a synthesis yield or the properties of a synthesis product by searching over a suitable set of variables (e.g., reagents, solvents, reaction temperature, etc.). Synthesis and property optimizations are regularly performed, yet simple reasoning implies that meeting these goals should be exceedingly difficult due to the large numbers of possible variable combinations that may be tested. This paper resolves this conundrum by showing that the explanation lies in the inherent attractive topology of the fitness landscape specifying the synthesis yield or property value as a function of the variables. Under simple physical assumptions, the landscape is shown to contain no suboptimal local extrema that could act as traps on the way to the optimal outcome. The literature contains broad evidence supporting this “OptiChem” theory. OptiChem theory implies that increasing the number of variables employed should result in more efficient and effective optimization, contrary to intuition.

NMR landscapes for chemical shift prediction.

The ability to reliably predict NMR chemical shifts plays an important role in elucidating the structure of organic molecules. Additionally, an intriguing question is how the multitude of variable factors (structural, electronic, and environmental) correlate with the actual electromagnetic shielding effect that determines the chemical shift value. This work presents NMRscape as a new tool for understanding these correlations by constructing the landscape that describes the relationship between the chemical shift value and the moieties bonded to a molecular scaffold. The scaffold may be as small as a single atom probed by NMR or a larger molecular framework containing the probed atom. NMRscape operates with only a list of the chemical moieties bonded to the scaffold, without utilizing any potentially biasing chemometric descriptors. The corresponding chemical shift landscape is constructed based on fundamental physical principles, which makes NMRscape a credible chemical shift prediction and analysis tool. As an illustration, we demonstrate that NMRscape can predict (13)C chemical shifts with an accuracy exceeding the substituent chemical shift (SCS) increment, hierarchical organization of spherical environments (HOSE), and neural networks (NN), methods for three distinct families of molecules sharing a common scaffold structure with moieties placed at two variable sites. The constructed NMR landscapes confirmed known empirical rules relating chemical shift values to the variation of chemical moieties on a scaffold, as well as uncovered hitherto hidden relationships. The practical importance of NMRscape is discussed.

Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution

A quantum control landscape is defined as the expectation value of a target observable Θ as a function of the control variables. In this work, control landscapes for open quantum systems governed by Kraus map evolution are analyzed. Kraus maps are used as the controls transforming an initial density matrix ρi into a final density matrix to maximize the expectation value of the observable Θ. The absence of suboptimal local maxima for the relevant control landscapes is numerically illustrated. The dependence of the optimization search effort is analyzed in terms of the dimension of the system N, the initial state ρi and the target observable Θ. It is found that if the number of nonzero eigenvalues in ρi remains constant, the search effort does not exhibit any significant dependence on N. If ρi has no zero eigenvalues, then the computational complexity and the required search effort rise with N. The dimension of the top manifold (i.e., the set of Kraus operators that maximizes the objective) is found to positively correlate with the optimization search efficiency. Under the assumption of full controllability, incoherent control modeled by Kraus maps is found to be more efficient in reaching the same value of the objective than coherent control modeled by unitary maps. Numerical simulations are also performed for control landscapes with linear constraints on the available Kraus maps, and suboptimal maxima are not revealed for these landscapes.