Harnessing subspace controllability: Time-optimal Dicke-state generation in Heisenberg-coupled qubit arrays with a single local control

TL;DR

The paper investigates time-optimal generation of Dicke states $|D^{N}_{a}\rangle$ in linear qubit arrays with isotropic Heisenberg coupling under a single local $Z$ control. By leveraging subspace controllability on each fixed-$a$ excitation subspace and employing the dCRAB quantum optimal-control algorithm, it achieves high-fidelity Dicke-state preparation for $N\le 9$ qubits, with the minimal time $T_{\min}$ scaling approximately as $N^2$. The method uses smooth control pulses expanded in a truncated random Fourier basis and demonstrates robustness to control-field deviations, while providing explicit scaling trends (e.g., $T_{\min}(N) \approx 0.14 N^{2.17}$ for $|D^{N}_2\rangle$ and $0.12 N^{2.02}$ for $|W_N\rangle$). These results highlight the practicality of minimal-resource, analog control for generating highly entangled states in realistic quantum devices and motivate experimental exploration.

Abstract

We explore the feasibility of realizing Dicke states in qubit arrays with always-on isotropic Heisenberg coupling between adjacent qubits, assuming a single Zeeman-type control acting in the $z$ direction on an actuator qubit. The Lie-algebraic criteria of controllability imply that such an array is not completely controllable, but satisfies the conditions for subspace controllability on any subspace with a fixed number of excitations. Therefore, a qubit array described by the model under consideration is state-to-state controllable for an arbitrary choice of initial and final states that have the same Hamming weight. This limited controllability is exploited here for the time-optimal dynamical generation of an $a$-excitation Dicke state $|D^{N}_{a}\rangle$ ($a=1,2,\ldots, N-1$) in a linear array with $N$ qubits starting from a generic Hamming-weight-$a$ product state. To dynamically generate the desired Dicke states -- including $W$ states $|W_{N}\rangle$ as their special ($a=1$) case -- in the shortest possible time with a single local $Z$ control, we employ an optimal-control scheme based on the dressed Chopped RAndom Basis (dCRAB) algorithm. We optimize the target-state fidelity over the expansion coefficients of smoothly-varying control fields in a truncated random Fourier basis; this is done by combining Nelder-Mead-type local optimizations with the multistart-based clustering algorithm that facilitates searches for global extrema. In this manner, we obtain the optimal control fields for Dicke-state preparation in arrays with up to $N=9$ qubits. Based on our numerical results, we find that the shortest possible state-preparation times scale quadratically with $N$. Finally, we demonstrate the robustness of our control scheme against small control-field deviations from the optimal values.

Figures (12)

• Figure 1: Pictorial illustration of the concept of local control. System $S$ is described by the drift Hamiltonian $H_S$. The system is subject to external control fields $f_j^{C}(t)$, which couple only to its degrees of freedom that belong to the subsystem $C$; these degrees of freedom are described by the local Hamiltonians $H_j^{C}$. The total Hamiltonian of the system is then the one given by Eq. \ref{['HamiltLocal']}.
• Figure 2: Schematic illustration of possible complete-controllability scenarios in Heisenberg-coupled $N$-qubit arrays with a local control: (a) In an array with the fully anisotropic Heisenberg interaction a local $Z$ control is applied to qubit $2$ and local $X$ control to qubit $N$, (b) In an array with the isotropic Heisenberg interaction local $X$- and $Y$ controls are both applied to qubit $1$.
• Figure 3: Schematic illustration of possible subspace-controllability scenarios in Heisenberg-coupled $N$-qubit arrays with a local control effected through a unidirectional time-dependent magnetic field of magnitude $B(t)$: (a) In an array with the $XXZ$-type Heisenberg interaction a local $Z$ control is applied to qubit $1$, (b) In an array with the isotropic Heisenberg interaction a local $X$ control is applied to qubit $2$.
• Figure 4: Schematic illustration of an $N$-qubit array with nearest-neighbor isotropic Heisenberg interaction and a local $Z$ control acting on the first qubit in the array.
• Figure 5: Pictorial overview of the results obtained using the optimal-control approach based on the dCRAB formalism (with $M=15$ harmonics in the truncated random Fourier basis) for an $N$-qubit array with isotropic, nearest-neighbor Heisenberg coupling and a $Z$ control on the first qubit: (a) The shortest times $T_{\textrm{min}}$ required for the realization of Dicke states ($|D^{N}_a\rangle$) and $W$ states ($|W_N\rangle$); (b) The highest fidelities achieved numerically for each of the investigated Dicke- and $W$ states with an evolution of duration $T_{\rm min}$.