Jason Dominy

Singularities of quantum control landscapes

A quantum control landscape is defined as the objective to be optimized as a function of the control variables. Existing empirical and theoretical studies reveal that most realistic quantum control landscapes are generally devoid of false traps. However, the impact of singular controls has yet to be investigated, which can arise due to a singularity on the mapping from the control to the final quantum state. We provide an explicit characterization of such controls that are strongly Hamiltonian-dependent and investigate their associated landscape geometry. Although in principle the singularities may correspond to local traps, we did not find any in numerical simulations. Also, as they occupy a small portion of the entire set of possible critical controls, their influence is expected to be much smaller than controls corresponding to the commonly located regular extremals. This observation supports the established ease of optimal searches to find high-quality controls in simulations and experiments.

Environment-invariant measure of distance between evolutions of an open quantum system

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 environments 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

Exploring the top and bottom of the quantum control landscape

A controlled quantum system possesses a search landscape defined by the target physical objective as a function of the controls. This paper focuses on the landscape for the transition probability P(i → f) between the states of a finite level quantum system. Traditionally, the controls are applied fields; here, we extend the notion of control to also include the Hamiltonian structure, in the form of time independent matrix elements. Level sets of controls that produce the same transition probability value are shown to exist at the bottom P(i → f)=0.0 and top P(i → f)=1.0 of the landscape with the field and/or Hamiltonian structure as controls. We present an algorithm to continuously explore these level sets starting from an initial point residing at either extreme value of P(i → f). The technique can also identify control solutions that exhibit the desirable properties of (a) robustness at the top and (b) the ability to rapidly rise towards an optimal control from the bottom. Numerical simulations are presented to illustrate the varied control behavior at the top and bottom of the landscape for several simple model systems.

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.

Photonic reagent control of dynamically homologous quantum systems

The general objective of quantum control is the manipulation of atomic scale physical and chemical phenomena through the application of external control fields. These tailored fields, or photonic reagents, exhibit systematic properties analogous to those of ordinary laboratory reagents. This analogous behavior is explored further here by considering the controlled response of a family of homologous quantum systems to a single common photonic reagent. A level set of dynamically homologous quantum systems is defined as the family that produces the same value(s) for a target physical observable(s) when controlled by a common photonic reagent. This paper investigates the scope of homologous quantum system control using the level set exploration technique (L-SET). L-SET enables the identification of continuous families of dynamically homologous quantum systems. Each quantum system is specified by a point in a hypercube whose edges are labeled by Hamiltonian matrix elements. Numerical examples are presented with simple finite level systems to illustrate the L-SET concepts. Both connected and disconnected families of dynamically homologous systems are shown to exist.

Characterization of the Critical Sets of Quantum Unitary Control Landscapes

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrödinger equation. We examine the critical point structure of the kinematic landscapes JF (U) = ||(U - W)A||2 and JP (U) = ||A||4 - |Tr(AA†W†U)|2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A E (H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic (geodesic) distance on U(H) and the projective unitary group PU(H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.

Exploring families of quantum controls for generating unitary transformations

A numerical method is presented for exploring the intertwined roles of the control-field structure and the final time T in determining the unitary evolution operator U(T, 0) for finite-level quantum control systems. The algorithm can (a) identify controls achieving a target unitary operator W at time T up to machine precision and (b) identify a continuous family of controls producing the same operator W over a continuous interval of final times. The high degree of precision is obtained, in part, by exploiting the geometry of the unitary group. In particular, geodesics of the unitary group are followed, both for tracking to a target transformation and for error management.

Optimized pulses for the control of uncertain qubits.

Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for \pi/2- and \pi-pulses, and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from \pi/2- and \pi-pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter, to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, post facto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.

Dynamic homotopy and landscape dynamical set topology in quantum control

We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where “state” may mean a pure state |ψ⟩, an ensemble density matrix ρ, or a unitary propagator U(0, T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of “dynamical sets” realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.

Bounds on the curvature at the top and bottom of the transition probability landscape

The transition probability between the states of a controlled quantum system is a basic physical observable, and the associated control landscape is specified by the transition probability as a function of the applied field. An initial control likely will produce a transition probability near the bottom of the landscape, while the final goal is to find a field that results in a high transition probability value at the top. For controls producing either of the latter extreme landscape values, the Hessian of the transition probability with respect to the control field characterizes the curvature of the landscape and the ease of leaving either limit. Prior work showed that the Hessian spectrum possesses an upper bound on the number of non-zero eigenvalues as well as an infinite number of zero eigenvalues. The associated eigenfunctions accordingly specify the coordinated control field changes that either maximally or minimally influence the transition probability. We show in this paper that there exists a lower bound on the number of non-zero Hessian eigenvalues at either the top or bottom of the landscape. In particular, there is at least one non-zero eigenvalue at the top and generally one at the bottom. Under special circumstances, the Hessian may be identically zero at the bottom (i.e. it possesses no non-zero eigenvalues). These results dictate the curvature of the top and bottom of the landscape, which has important physical significance for seeking optimal control fields. At the top, a field that produces a single non-zero Hessian eigenvalue of small magnitude will generally exhibit a high degree of robustness to field noise. In contrast, at the bottom, working with a field producing the maximum number of non-zero eigenvalues will generally assure the most rapid climb towards a high transition probability.