Experimental preparation of Greenberger-Horne-Zeilinger states in an Ising spin model by partially suppressing the nonadiabatic transitions

The creation of multipartite entangled states is a key task of quantum information processing. Among the various implementations, shortcut to adiabaticity (STA) offers a fast and robust means for generating entanglement. The traditional counterdiabatic driving, as a conventional and simple method for STA, suppresses transitions with an auxiliary Hamiltonian, but its complex interactions in many-body systems may hamper the feasibility of experimental implementation. To avoid this drawback, a flexible and efficient way was proposed theoretically by Chen et al. [Phys. Rev. A 93, 052109 (2016)] by substituting the counterdiabatic terms. Inspired by this work, we devise a practical protocol for preparing the Greenberger-Horne-Zeilinger state on the Ising spin model via STA by partial suppression of the nonadiabatic transitions, which can obviously reduce the complexity in experiments compared with the original method. We also experimentally demonstrate the viability of our scheme with a nuclear magnetic resonance quantum simulator. This work provides an alternative method to realize fast coherent quantum control for a multiqubit system in experiments.

Figure 1

The time-dependent parameters ${\alpha }_k\left(t\right)$ deduced from the Eqs. (12) on the conditions (a). ${\alpha }_{1,3,4}=0$, (b). ${\alpha }_{1,3,6}=0$, (c). ${\alpha }_{2,4,6}=0$, (d). ${\alpha }_{1,3,5,6}=0$, respectively. The corresponding analytical solutions are listed in the Table 1.

Figure 2

Quantum state fidelity of the evolved state with different protocols: no additional Hamiltonian (black dashed line), auxiliary Hamiltonian $H_{add}^{{}^{\prime }}\left(t\right)$ (blue dashed line), additional Hamiltonian $H_{add}\left(t\right)$ (orange squares) and exact CD term $H_{cd}\left(t\right)$ (red solid line), respectively. Detailed explanation can be found in the main text.

Figure 3

Relevant properties of the natural spin chain consisting of ${}^{13}\mathrm{C}$, ${}^1\mathrm{H}$, ${}^{19}\mathrm{F}$ nuclear spins marked in the Diethyl-fluoromalonate. The table on the right side summarizes the relevant NMR parameters, i.e., the resonance frequencies ${\omega }_i$ (on the diagonal), the $J$-coupling constants $J_{ij}$ (above the diagonal), and the relaxation times $T_2$ in the last columns.

Figure 4

${}^{13}\mathrm{C}$ experimental spectra (red) after a ${[\pi /2]}_y$ readout pulse applied to carbon channel for equilibrium state (top), PPS (middle), and GHZ state (bottom). The blue spectra show the results by fitting. The four resonance lines are labeled by the corresponding states of the two other qubits.

Figure 5

Pulse sequence in experiments. The color rectangles represent the hard pulses applied to individual qubits and pulse phases is indicated inside them. Time durations ${\tau }_{12}=\frac{2J\left[t_l\right]\mathrm{\Delta }t}{\pi J_{12}cos\left({\phi }_2\right)}$, ${\tau }_{23}=\frac{2J\left[t_l\right]\mathrm{\Delta }t}{\pi J_{23}cos\left({\phi }_2\right)}$ and ${\tau }_{13}=\frac{2J\left[t_l\right]\mathrm{\Delta }t}{\pi |J_{13}|cos\left({\phi }_2\right)}$ represent free evolutions under the natural Hamiltonian.

Figure 6

The real parts (a), (b) and imaginary parts (c), (d) of the experimental density matrices for the initial state $|000\rangle$ (left part), and the GHZ state (right part). The color bar represents the value of density matrix ${\rho }_{\text{exp}}$.

Figure 7

Instantaneous state fidelity of each step in experiment with original Hamiltonian $H_0\left(t\right)$ (blue squares), with Hamiltonian $H_0\left(t\right)+H_{add}\left(t\right)$ (orange x symbols), respectively. The color dashed lines represent the theoretical expectations with the Trotter's formula for two case. Here, experimental pulse sequence for simulating $H_0\left(t\right)$ can be acquired from Ref. [64].