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arXiv:2512.24406 · Physical Review Research

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

Vladimir M. Stojanovic, Tommaso Calarco, Andrea Muratori

quant-phSubmitted Dec 30, 2025• Announced Dec 31, 2025• 0 citations
v1 · Dec 30, 2025v2 · Mar 30, 2026v3 · Jun 22, 2026v4 · Jul 23, 2026
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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.

Journal reference
Phys. Rev. Research 8, 033095 (2026)
DOI
10.1103/t2zk-4nxh
Comments
16 pages, 12 figures

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The ability to engineer and control quantum states of many-body systems is a central challenge in quantum information science. For a register of NNN qubits, the full Hilbert space dimension grows exponentially as 2N2^N2N, rendering generic state preparation and control infeasible without exploiting structure or symmetry. A particularly important and physically motivated restriction is to the fully symmetric subspace, spanned by the Dicke states, which are simultaneous eigenstates of collective spin J=N/2J=N/2J=N/2. Ensembles of Rydberg atoms interacting via electric dipoles in two-dimensional tweezer arrays form a promising platform for achieving such control. However, the finite range of dipole-dipole interactions poses a challenge to generating and controlling the Dicke manifold because the Hamiltonian incurs leakage from the computational subspace. To counteract this leakage, we perform quantum optimal control algorithms on a truncated Hilbert space according to our newly developed method of ``irrep distillation'' (IRD), which captures the process by which the symmetric subspace couples to leakage error-spaces, using only linear-scaling Hilbert dimension. We implement gradient ascent pulse engineering (GrAPE) on control schemes with little or no local addressing, to generate resourceful states like Greenberger-Horne-Zeilinger, Dicke, and extremal quantum states. We benchmark each scheme of IRD-GrAPE for its quantum speed limit (QSL), as well as exactly testing pulse fidelities on small system sizes and predicting fidelities using higher-order IRD on larger systems.
We investigate the feasibility of single-shot Toffoli- and Fredkin-gate realizations in qubit arrays with Heisenberg-type exchange interactions between adjacent qubits. As follows from the Lie-algebraic criteria of controllability, such an array is rendered completely controllable---equivalent to allowing universal quantum computation---by a Zeeman-like control field with two orthogonal components acting on a single ``actuator'' qubit. Adopting this local-control setting, we start our analysis with piecewise-constant control fields and determine the global maxima of the relevant figure of merit (target-gate fidelity) by combining the multistart-based clustering algorithm and quasi-Newton type local optimization. We subsequently introduce important practical considerations, such as finite frequency bandwidth of realistic fields and their leakage away from the actuator. We find the shortest times required for high-fidelity Toffoli- and Fredkin-gate realizations and provide comparisons to their respective two-qubit counterparts---controlled-not and exponential-swap. In particular, the Toffoli-gate time compares much more favorably to that of controlled-not than in the standard decomposition-based approach. This study indicates that the use of the single-shot approach can alleviate the burden on control-generating hardware in future experimental realizations of multiqubit gates.
WWW- and Dicke-state engineering using optimal global control in nearest-neighbor coupled ring-shaped qubit arrays
Motivated by a compelling need for time-efficient and robust schemes for quantum-state engineering in systems of neutral atoms in optical tweezers, we consider a ring-shaped array of qubits with nearest-neighbor Ising-type (zzzzzz) coupling and transverse (xxx and yyy) global control fields. This system to a large extent mimics -- outside of the Rydberg-blockade regime -- a circular array of neutral atoms interacting through van-der-Waals type interaction. We investigate the preparation of WWW and Dicke states in this system starting from the default initial state ∣00…0⟩|00\ldots 0\rangle∣00…0⟩ using two different optimal-control approaches: (i) NMR-like pulse sequence, which consists of instantaneous (delta-shaped) control- and Ising-interaction pulses, and (ii) time-dependent control scheme, which entails shaped control pulses in the presence of always-on Ising interaction between adjacent qubits. By making use of the underlying dihedral symmetry of this system -- which allows one to use a symmetry-adapted computational basis with O(2N/N)\mathcal{O}(2^N / N)O(2N/N) states in an NNN-qubit system -- and utilizing advanced global-optimization methods, we find optimal sequences of pulses for realizing WWW and Dicke states within both approaches. In addition, we demonstrate robustness of these sequences against unavoidable control errors. Finally, using typical values of parameters in realistic Rydberg-atom systems, we show that our control schemes enable the preparation of the desired multiqubit states on time scales much shorter than the relevant coherence times of those systems.
We propose an efficient yet simple protocol to generate arbitrary symmetric entangled states with only global single-qubit rotations in a torn Hilbert space. The system is based on spin-1/2 qubits in a resonator such as atoms in an optical cavity or superconducting qubits coupled to a main bus. By sending light or microwave into the resonator, it induces ac Stark shifts on particular angular-momentum eigenstates (Dicke states) of qubits. Then we are able to generate barriers that hinder transitions between adjacent Dicke states and tear the original Hilbert space into pieces. Therefore, a simple global single-qubit rotation becomes highly nontrivial, and thus generates entanglement among the many-body system. By optimal control of energy shifts on Dicke states, we are able to generate arbitrary symmetric entangled states. We also exemplify that we can create varieties of useful states with near-unity fidelities in only one or very few steps, including W states, spin-squeezed states (SSSs), and Greenberger-Horne-Zeilinger states. Particularly, the SSS can be created by only one step with a squeezing parameter ξ R 2 ∼ 1 / N 0.843 approaching the Heisenberg limit. Our finding establishes a way for universal entanglement generations with only single-qubit drivings where all the multiple-qubit controls are integrated into simply switching on or off microwave. It has direct applications in the variational quantum optimizer, which is available with existing technology. Published by the American Physical Society 2024
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