Robust trapped-ion quantum logic gates by continuous dynamical decoupling

We introduce a scheme that combines phonon-mediated quantum logic gates in trapped ions with the benefits of continuous dynamical decoupling. We demonstrate theoretically that a strong driving of the qubit decouples it from external magnetic-field noise, enhancing the fidelity of two-qubit quantum gates. Moreover, the scheme does not require ground-state cooling, and is inherently robust to undesired ac Stark shifts. The underlying mechanism can be extended to a variety of other systems where a strong driving protects the quantum coherence of the qubits without compromising the two-qubit couplings.

Figure 1

(a) Hyperfine structure of ${}^{25}{\mathrm{Mg}}^{+}$. The states $|0\rangle =|F=3,m_F=3\rangle$ and $|1\rangle =|2,2\rangle$ form our qubit. Two laser beams ${\Omega }_1,{\Omega }_2$ drive the red sideband via an off-resonant excited state, and a microwave ${\Omega }_{\mathrm{d}}$ directly couples to the transition. (b) Spin-dependent ${\sigma }^x$ force acting on a single trapped ion. The phonons associated with the states $|{+}_x\rangle$,$|{-}_x\rangle$ are displaced in phase space according to $D_{|{\pm }_x\rangle }$, and form different closed paths that lead to the geometric phases ${\phi }_{\pm }$. (c) Trotterization of the combined ${\sigma }^x$ and ${\sigma }^y$ forces. The ${\sigma }^x$ displacement $D_{|{+}_x\rangle }$ shall be followed by the two possible ${\sigma }^y$ displacements $D_{|{\pm }_y\rangle }$ since $|{+}_x\rangle \propto \left(\right|{+}_y\rangle +i|{-}_y\rangle$. Hence, the phase-space trajectory is not generally closed. (d) Schematic spin-echo refocusing of the ${\sigma }^y$ force. By applying a $\pi$-pulse $X_i^{\pi }={\sigma }_i^x$ (box) at half the ${\sigma }^y$ displacements, one obtains $|{\pm }_y\rangle \to |{\mp }_y\rangle$, such that the displacements are reversed (dotted arrows), and the trajectory is refocused, yielding a well-defined geometric phase.

Figure 2

(a) Dynamics of the swap probabilities $P_{10}\left(t\right)$ (squares) and $P_{01}\left(t\right)$ (circles) for the effective gate (2), compared to the exact spin-phonon Hamiltonian for a two-ion crystal with $\overline{n}=\{0,0.1,1,2,4\}$ (solid lines). The phonon Hilbert space is truncated to $n_{\max }=20$ excitations per mode. (b) Error $\epsilon =1-\mathcal{F}$ for the generation of the Bell state $|{\Psi }^{-}\rangle$ as a function of the mean phonon number and $n_{\max }=14$. As the driving is increased, ${\Omega }_{\mathrm{d}}/{\omega }_z\in \{0,2,3.8,5.2\}$, the error lies within the fault-tolerance (FT) threshold.

Figure 3

The two upper rows represent the circuit model for the two qubits, and the lower rows represent the switching of the laser beams (LB) and microwaves (MW). The initialization consists of optical pumping $\mathcal{P}$, followed by a local $\pi$-pulse $X_i^{\pi }={\sigma }_i^x$. Global $\pi$-pulses $Z_i^{\pi }={\sigma }_i^z$ correct possible synchronization errors. The two-qubit coupling is applied by switching on the red sideband (LB), and the carrier (MW) yields the noise decoupling.

Figure 4

(a) Exponential decay of the coherences $\langle {\sigma }^x\left(t\right)\rangle$ for the initial states $|{\psi }_0\rangle =|{\pm }_x\rangle$. The circles represent the numerical statistical average of the $N=5\times {10}^3$ evolutions, and the solid line represents the exponential fit. (b) Error in the generation of the Bell state $|{\Phi }^{-}\rangle$ for the additional magnetic-field noise, together with a strong driving ${\Omega }_{\mathrm{d}}/2\pi =5.2$ MHz. The error is presented as a function of the $T_2$ time. In the inset, we represent the evolution of the fidelity for the typical noise dephasing time $T_2=5$$f$ ms.