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Physical Review A

Preparation of Greenberger-Horne-Zeilinger and W states on a one-dimensional Ising chain by global control

Yanxiang Gao, Hui Zhou, Dong Zou, Xinhua Peng, Jiangfeng Du

Published Mar 28, 2013• 22 citations
Publisher

Abstract

We study how to prepare two important types of entangled states (i.e., GHZ and $W$ states) via global controls in closed linear Ising spin chains with nearest-neighbor couplings. To achieve the goal, we use a numerical optimization algorithm to search an optimal set of control parameters, and then analytically obtain the time-optimal solutions for a three-spin chain by a geometrical explanation on the Bloch sphere. For experimental demonstration, we also consider the most robust sequence under the realistic imperfections such as the errors of global controls and the time evolutions as well as in the numerical optimization algorithm, and experimentally implement the numerically derived robust pulse sequences in a three-spin system where the closed linear Ising spin chain is simulated by nuclear-magnetic-resonance technique. This experiment demonstrates that our scheme is feasible in realistic settings under the experimental imperfections.

DOI
10.1103/physreva.87.032335

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We study how to prepare two important types of entangled states (i.e., GHZ and WWW states) via global controls in closed linear Ising spin chains with nearest-neighbor couplings. To achieve the goal, we use a numerical optimization algorithm to search an optimal set of control parameters, and then analytically obtain the time-optimal solutions for a three-spin chain by a geometrical explanation on the Bloch sphere. For experimental demonstration, we also consider the most robust sequence under the realistic imperfections such as the errors of global controls and the time evolutions as well as in the numerical optimization algorithm, and experimentally implement the numerically derived robust pulse sequences in a three-spin system where the closed linear Ising spin chain is simulated by nuclear-magnetic-resonance technique. This experiment demonstrates that our scheme is feasible in realistic settings under the experimental imperfections.
Entanglement, a unique quantum resource with no classical counterpart, remains at the heart of quantum information. The Greenberger-Horne-Zeilinger (GHZ) and WWW states are two inequivalent classes of multipartite entangled states which cannot be transformed into each other by means of local operations and classic communication. In this paper, we present the methods to prepare the GHZ and WWW states via global controls on a long-range Ising spin model. For the GHZ state, general solutions are analytically obtained for an arbitrary-size spin system, while for the WWW state, we find a standard way to prepare the WWW state that is analytically illustrated in three- and four-spin systems and numerically demonstrated for larger-size systems. The number of parameters required in the numerical search increases only linearly with the size of the system.
We discuss how to prepare an Ising chain in a GHZ state using a single global control field only. This model does not require the spins to be individually addressable and is applicable to quantum systems such as cold atoms in optical lattices, some liquid- or solid-state NMR experiments, and many nanoscale quantum structures. We show that GHZ states can always be reached asymptotically from certain easy-to-prepare initial states using adiabatic passage, and under certain conditions finite-time reachability can be ensured. To provide a reference useful for future experimental implementations, three different control strategies to achieve the objective---adiabatic passage, Lyapunov control, and optimal control---are compared, and their advantages and disadvantages discussed, in particular in the presence of realistic imperfections such as imperfect initial state preparation, system inhomogeneity, and dephasing.
We study multiple-spin coherence transfers in linear Ising spin chains with nearest-neighbor couplings. These constitute a model for efficient information transfers in future quantum computing devices and for many multidimensional experiments for the assignment of complex spectra in nuclear magnetic resonance spectroscopy. We complement prior analytic techniques for multiple-spin coherence transfers with a systematic numerical study where we obtain strong evidence that a certain analytically motivated family of restricted controls is sufficient for time optimality. In the case of a linear three-spin system, additional evidence suggests that prior analytic pulse sequences using this family of restricted controls are time optimal even for arbitrary local controls. In addition, we compare the pulse sequences for linear Ising spin chains to pulse sequences for more realistic spin systems with additional long-range couplings between nonadjacent spins. We experimentally implement the derived pulse sequences in three- and four-spin systems and demonstrate that they are applicable in realistic settings under relaxation and experimental imperfections---in particular---by deriving broadband pulse sequences which are robust with respect to frequency offsets.
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.
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 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 JJJ 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.