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Известия Российской академии наук Серия математическая

On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

Boris Olegovich Volkov, Alexander Nikolaevich Pechen

Published Jan 1, 2023• 2 citations
Publisher

Abstract

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of a saddle the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem 3 in B. O. Volkov, O. V. Morzhin, A. N. Pechen, J. Phys. A: Math. Theor. 54, 215303 (2021)]. Bibliography: 42 titles.

DOI
10.4213/im9364

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In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of a saddle the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem 3 in B. O. Volkov, O. V. Morzhin, A. N. Pechen, J. Phys. A: Math. Theor. 54, 215303 (2021)]. Bibliography: 42 titles.
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).
Abstract Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as gradient ascent pulse engineering, differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.
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.
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology and structure provide a foundation to understand why optimal quantum control can be readily achieved.
There has been great interest in recent years in quantum control landscapes. Given an objective JJJ that depends on a control field $\ensuremath{\epsilon}$ the dynamical landscape is defined by the properties of the Hessian ${\ensuremath{\delta}}^{2}J/\ensuremath{\delta}{\ensuremath{\epsilon}}^{2}$ at the critical points $\ensuremath{\delta}J/\ensuremath{\delta}\ensuremath{\epsilon}=0$. We show that contrary to recent claims in the literature the dynamical control landscape can exhibit trapping behavior due to the existence of special critical points and illustrate this finding with an example of a 3-level $\ensuremath{\Lambda}$ system. This observation can have profound implications for both theoretical and experimental quantum control studies.
Optimization is ubiquitous in the control of quantum dynamics in atomic, molecular, and optical systems. The ease or difficulty of finding control solutions, which is practically crucial for developing quantum technologies, is highly dependent on the geometry of the underlying optimization landscapes. In this review, we give an introduction to the basic concepts in the theory of quantum optimal control landscapes, and their trap-free critical topology under two fundamental assumptions. Furthermore, the effects of various factors on the search effort are discussed, including control constraints, singularities, saddles, noises, and non-topological features of the landscapes. Additionally, we review recent experimental advances in the control of molecular and spin systems. These results provide an overall understanding of the optimization complexity of quantum control dynamics, which may help to develop more efficient optimization algorithms for quantum control systems, and as a promising extension, the training processes in quantum machine learning.