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Lobachevskii Journal of Mathematics

Numerical Analysis of Quantum Control Landscapes for Single-Qubit Gate Generation in Three-Level Fluxonium Systems

I. M. Korolev, Alexander Nikolaevich Pechen

Published Jun 1, 2025• 2 citations
Publisher

Abstract

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, $$\mathrm{T}$$ , and $$\mathrm{S}$$ 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 $$\Lambda$$ -type systems, that may indicate either the absence or low influence of trap at the null control for a fluxonium.

DOI
10.1134/s1995080225607830

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Lobachevskii Journal of Mathematics•Jun 1, 2025•7 citations
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.
Abstract In this work, we explore quantum control landscapes for the single qubit optimal gate generation problem for fluxonium models of superconducting systems in the vicinity of the null control. Ultimately, the fluxonium qubit is described by an infinite-dimensional Hamiltonian, for which using the two lowest levels provides only a minimal approximation. A more detailed analysis requires considering more energy levels for a single qubit. We consider two-, three-, four-, and five-level approximations for a single qubit, and as an example, the generation of a Hadamard gate. We investigate how taking into account different numbers of energy levels for a single qubit affects the efficiency of optimization and control landscape properties. As a result, we find an unexpected sharp difference in the behavior for various values of the parameters and for various numbers of energy levels used for the approximation, including non-monotonic behavior of the optimization efficiency in some cases.
In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a V−VV-VV−V system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.
In this work, we consider a general method for numerically comparing the efficiency of optimization over control landscapes with different trapping features, both for unconstrained (using GRAPE) and constrained (using gradient projection method) optimization. Traps are controls which locally appear optimal while not being globally optimal. Such controls can significantly affect the efficiency of finding controls close to optimal with local search, e.g. gradient-based, algorithms. As an explicit example, we apply the general method to study unconstrained control landscapes for two three-level quantum Ξ\Xi Ξ systems which are widely used in quantum optics and known to have different trapping features: a non-degenerate Ξ\Xi Ξ -system for which the null control is a third-order trap, and a degenerate Ξ\Xi Ξ -system for which the null control is a seventh-order trap. Using the general approach, we perform a comparative study of the impact of the null control on the efficiency of optimization and estimate the dependence of this impact on the maximal allowed number of iterations to check how the optimization is sensitive to this number. The results show that while quantitative behavior is different, the qualitative difference between the two control landscapes remains similar.
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.
An important problem in quantum computation is the generation of single-qubit quantum gates such as Hadamard (H) and π/8 (T) gates, which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to a decrease in the gate fidelity, in some cases, it can be used as a resource. Here, we consider the problem of the optimal generation of H and T gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we studied the quantum control landscape, which represents the behavior of the infidelity as a functional of the controls. We considered three landscapes, with infidelities defined by steering between two, three (via Goerz–Reich–Koch approach), and four matrices in the qubit Hilbert space. We observed that, for the H gate, which is a Clifford gate, for all three infidelities, the distributions of minimal values obtained with a gradient search have a simple form with just one peak. However, for the T gate, which is a non-Clifford gate, the situation is surprisingly different—this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, which might indicate the possible existence of two isolated minima in the control landscape. It is important that, among these three infidelities, only those defined with three and four matrices guarantee the closeness of the generated gate to a target and can be used as a good measure of closeness. We studied sets of optimized solutions for the most general and previously unexplored case of coherent and incoherent controls acting together and discovered that they form sub-manifolds in the control space, and unexpectedly, in some cases, two isolated sub-manifolds.
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.