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arXiv:2104.12699 · Journal of Physics A Mathematical and Theoretical

Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates

Boris O Volkov, Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen

Published Apr 29, 2021• 31 citations
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Abstract

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.

DOI
10.1088/1751-8121/abf45d

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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.
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]).
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.
Abstract In this work we study extrema of objective functionals for ultrafast manipulation by a qubit. Traps are extrema of the objective functionals which are optimal for manipulation by quantum systems only locally, not globally. Prior work has devoted a large amount of effort to the analysis of traps for quantum systems controlled by laser pulses which are long enough, and, for example, manipulation by a qubit with long control pulses was shown to be trap-free. Ultrafast femtosecond and attosecond control has now become widely applicable, which makes the analysis of traps on the ultrafast time scale a necessity. We complete such analysis for a qubit and show that ultrafast state transfer in a qubit remains trap-free for a wide range of the initial and final states of the qubit. We prove that for this range the probability of transition between the initial and the final states has a saddle but no traps.
In this work, we consider the problem of the analysis of quantum control landscapes in both situations when the controls during optimization are unconstrained and when they should satisfy a given constraint, for example their amplitude should be bounded by some constant value. For the analysis of unconstrained quantum control landscapes, a special approach was developed in [Pechen and Tannor, Israel J. of Chemistry 52 (5), 467–472 (2012)] based on a fixed step gradient ascent pulse engineering (GRAPE) algorithm. Here we also adopt a GRAPE approach based on the Pontryagin maximum principle (PMP). For the analysis of the constrained quantum control landscapes, we adopt the gradient projection method which was developed for quantum systems in [A. Oza et al., J. Phys. A: Math. Theor. 42, 205305 (2009)] (GPM; for our purpose the most suitable is a one-step GPM with a fixed step), both for closed and open quantum systems. As an explicit example, we apply these approaches to study quantum control landscape for optimization of an observable mean value for a three-level degenerate Ξ\XiΞ -system system having a seventh-order trap.
In this work, we consider a formulation of the optimal gate generation problem for a fluxonium model of superconducting systems and apply it to the numerical analysis of the corresponding quantum control landscape in a vicinity of the null control. The model includes Schrödinger equation with control and maximization of the objective functionals for gate generation and observable mean value. For a qutrit fluxonium model, the third level can be considered as one of the three computational states. We exploit gradient of the control objective and GRAPE algorithm to perform numerical investigation of the quantum control landscapes for generation of the Hadamard, T\mathrm{T}T , and S\mathrm{S}S gates and for maximizing mean value of a quantum observable in a vicinity of the null control. The obtained results show high efficiency of gradient-based optimization when it starts from the null control and decrease of the efficiency with increase of the distance of the initial control from the null control for the gate generation problems, and an opposite behaviour for the mean value maximization. For gate generation, the obtained behavior differs (is opposite) from the behavior found previously for maximizing mean value of an observable for Λ\LambdaΛ -type systems, that may indicate either the absence or low influence of trap at the null control for a fluxonium.
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.