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Journal of Physics A Mathematical and Theoretical

Control landscapes for open system quantum operations

Re-Bing Wu, HERSCHEL A. RABITZ

Published Nov 19, 2012• 17 citations
Publisher

Abstract

The reliable realization of control operations is a key component of quantum information applications. In practice, meeting this goal is very demanding for open quantum systems. This paper investigates the landscape defined as the fidelity J between the desired and achieved quantum operations with an open system. The goal is to maximize J as a functional of the control variables. We specify the complete set of critical points of the landscape function in the so-called kinematic picture. An associated Hessian analysis of the landscape reveals that, upon the satisfaction of a particular controllability criterion, the?critical topology is dependent on the particular environment, but no false traps (i.e. suboptimal solutions) exist. Thus, a gradient-type search algorithm should not be hindered in searching for the ultimate optimal solution with such controllable systems. Moreover, the maximal fidelity is proven to coincide with Uhlmann?s fidelity between the environmental initial states associated with the achieved and desired quantum operations, which provides a generalization of Uhlmann?s theorem in terms of Kraus maps.

DOI
10.1088/1751-8113/45/48/485303

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The reliable realization of control operations is a key component of quantum information applications. In practice, meeting this goal is very demanding for open quantum systems. This paper investigates the landscape defined as the fidelity J between the desired and achieved quantum operations with an open system. The goal is to maximize J as a functional of the control variables. We specify the complete set of critical points of the landscape function in the so-called kinematic picture. An associated Hessian analysis of the landscape reveals that, upon the satisfaction of a particular controllability criterion, the?critical topology is dependent on the particular environment, but no false traps (i.e. suboptimal solutions) exist. Thus, a gradient-type search algorithm should not be hindered in searching for the ultimate optimal solution with such controllable systems. Moreover, the maximal fidelity is proven to coincide with Uhlmann?s fidelity between the environmental initial states associated with the achieved and desired quantum operations, which provides a generalization of Uhlmann?s theorem in terms of Kraus maps.
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.
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 ρ i 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 ρ i and the target observable Θ. It is found that if the number of nonzero eigenvalues in ρ i remains constant, the search effort does not exhibit any significant dependence on N . If ρ i 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.
Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.
A broad class of quantum control problems entails optimizing the expectation value of an observable operator through tailored unitary propagation of the system density matrix. Such optimization processes can be viewed as a directed search over a quantum control landscape. The attainment of the global extrema of this landscape is the goal of quantum control. Local optima will generally exist, and their enumeration is shown to scale factorially with the system's effective Hilbert space dimension. A Hessian analysis reveals that these local optima have saddlepoint topology and cannot behave as suboptimal extrema traps. The implications of the landscape topology for practical quantum control efforts are discussed, including in the context of nonideal operating conditions.
A quantum control landscape is defined as the observable as a function(al) of the system control variables. Such landscapes were introduced to provide a basis to understand the increasing number of successful experiments controlling quantum dynamics phenomena. This paper extends the concept to encompass the broader context of the environment having an influence. For the case that the open system dynamics are fully controllable, it is shown that the control landscape for open systems can be lifted to the analysis of an equivalent auxiliary landscape of a closed composite system that contains the environmental interactions. This inherent connection can be analyzed to provide relevant information about the topology of the original open system landscape. Application to the optimization of an observable expectation value reveals the same landscape simplicity observed in former studies on closed systems. In particular, no false suboptimal traps exist in the system control landscape when seeking to optimize an observable, even in the presence of complex environments. Moreover, a quantitative study of the control landscape of a system interacting with a thermal environment shows that the enhanced controllability attainable with open dynamics significantly broadens the range of the achievable observable values over the control landscape.
Seeking an effective quantum control entails searching over a landscape defined as the objective as a functional of the control field. This paper considers the problem of driving a state-to-state transition in a finite level quantum system, and analyzes the local topology of the landscape of the final transition probability in terms of the variables specifying the control field. Numerical calculation of the eigenvalues of the Hessian of the transition probability with respect to the control field variables reveals systematic structure in the spectra reflecting the existence of a generic and simple control landscape topology. An illustration shows that the number of nonzero Hessian eigenvalues is determined by the number of quantum states in the system. The Hessian eigenvectors associated with its nonzero eigenvalues are shown to give insight into the cooperative roles of the control variables. The practical consequences of these findings for quantum control are discussed.