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arXiv:2302.12483 · Physical Review A

Dicke-state preparation through global transverse control of Ising-coupled qubits

Vladimir M. Stojanović, Julian K. Nauth

Published Jul 7, 2023• 23 citations
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Abstract

We consider the problem of engineering the two-excitation Dicke state $|{D}_{2}^{3}\ensuremath{\rangle}$ in a three-qubit system with all-to-all Ising-type qubit-qubit interaction, which is also subject to global transverse (Zeeman-type) control fields. The theoretical underpinning for our envisioned state-preparation scheme, in which $|000\ensuremath{\rangle}$ is adopted as the initial state of the system, is provided by a Lie-algebraic result that guarantees state-to-state controllability of this system for an arbitrary choice of initial and final states that are invariant with respect to permutations of qubits. This scheme is envisaged in the form of a pulse sequence that involves three instantaneous control pulses, which are equivalent to global qubit rotations, and two Ising-interaction pulses of finite durations between consecutive control pulses. The design of this pulse sequence (whose total duration is $T\ensuremath{\approx}0.95\phantom{\rule{0.222222em}{0ex}}\ensuremath{\hbar}/J$, where $J$ is the Ising-coupling strength) leans heavily on the concept of the symmetric sector, a four-dimensional, permutationally invariant subspace of the three-qubit Hilbert space. We demonstrate the feasibility of the proposed state-preparation scheme by carrying out a detailed numerical analysis of its robustness to systematic errors, i.e., deviations from the optimal values of the eight parameters that characterize the underlying pulse sequence. Finally, we discuss how our proposed scheme can be generalized for engineering Dicke states in systems with $N\ensuremath{\ge}4$ qubits. For the sake of illustration, we describe the preparation of the two-excitation Dicke state $|{D}_{2}^{4}\ensuremath{\rangle}$ in a four-qubit system.

DOI
10.1103/physreva.108.012608

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Interconversions of WWW and Greenberger-Horne-Zeilinger states in various physical systems have recently been attracting considerable attention. We address this problem in the fairly general physical setting of qubit arrays with long-ranged (all-to-all) Ising-type qubit-qubit interaction, which are simultaneously acted upon by transverse Zeeman-type global control fields. Motivated in part by a recent Lie-algebraic result that implies state-to-state controllability of such a system for an arbitrary pair of states that are invariant with respect to qubit permutations, we present a detailed investigation of the state-interconversion problem in the three-qubit case. The envisioned interconversion protocol has the form of a pulse sequence that consists of two instantaneous ($\ensuremath{\delta}$-shaped) control pulses, each of them corresponding to a global qubit rotation, and an Ising-interaction pulse of finite duration between them. Its construction relies heavily on the use of the (four-dimensional) permutation-invariant subspace (symmetric sector) of the three-qubit Hilbert space. In order to demonstrate the viability of the proposed state-interconversion scheme, we provide a detailed analysis of the robustness of the underlying pulse sequence to systematic errors, i.e., deviations from the optimal values of its five characteristic parameters.
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 zzz 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-efficient dynamical generation of an aaa-excitation Dicke state ∣DaN⟩|D^{N}_{a}\rangle∣DaN​⟩ (a=1,2,…,N−1a=1,2,\ldots, N-1a=1,2,…,N−1) in a linear array with NNN qubits starting from a generic Hamming-weight-aaa product state. To dynamically generate the desired Dicke states -- including WWW states ∣WN⟩|W_{N}\rangle∣WN​⟩ as their special (a=1a=1a=1) case -- in the shortest possible time with a single local ZZZ control, we employ an optimal-control scheme based on the {\em 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 999 qubits. Based on our numerical results, we find that the shortest possible state-preparation times scale as O(N2.08)\mathcal{O}(N^{2.08})O(N2.08) for WWW states and O(N1.78)\mathcal{O}(N^{1.78})O(N1.78) for a=2a=2a=2 Dicke states.
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.
Global control provides a promising route to implementing multi-qubit gates without individual qubit addressing. This is especially appealing for permutation-invariant (PI) gates, whose symmetry is often broken when they are compiled into individually addressed one- and two-qubit gates. Important examples include SWAP, iSWAP\sqrt{i\text{SWAP}}iSWAP​, and the n-qubit controlled-Z gate, which is equivalent, up to two single-qubit Hadamard gates, to the multi-qubit Toffoli gate. Motivated by this global-control perspective, we show that all PI unitaries on an arbitrary number of qubits can be realized using the Tavis-Cummings (TC) interaction, the multi-qubit version of the Jaynes-Cummings interaction, together with global uniform z and x fields. Here, the nnn qubits are identically coupled to a single bosonic mode (oscillator), which is initialized in and returned to its vacuum state. A corollary is that all PI states, including GHZ and Dicke states, can be prepared using the same global control. For the case n=2 qubits, which is particularly important in quantum computing, we also find explicit pulse sequences for implementing all PI qubit unitaries that conserve angular momentum in the z direction, using only the TC interaction and global z fields. This includes controlled-Z, SWAP, and iSWAP\sqrt{i\text{SWAP}}iSWAP​.
Coherent manipulation of three-level systems is of critical significance to quantum information processing. Here, we propose a scheme to achieve quantum state engineering in arbitrary three-level systems, in which the controlling Hamiltonian can be inversely engineered based on Lewis-Riesenfeld invariants and a gauge transformation. With well-designed control pulses, the three-level system continually evolves from the initial state to a desired final state in one step. The proposed scheme allows diverse optimizations, which gives it a significant advantage in robustness and fidelity over the commonly used resonant-pulse multistep method. The scheme can also be extended to produce multiqubit entangled states provided with a similar state subspace. In a particular application, we demonstrate numerically that our protocol can reliably and robustly prepare qutrit states as well as three-qubit entangled WWW states within superconducting circuits.
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.