Multi-ion entangling gates mediated by spectrally unresolved modes

Abstract

Entangling interactions between distant qubits can be mediated via an additional degree of freedom. In conventional trapped-ion schemes, realizing a well-defined, coherent gate typically requires spectrally addressing a specific bus mode. As the ion number increases, the coupling to each individual motional mode becomes weaker, so gates on large ion strings mediated by a single mode are necessarily slow. Moreover, addressing a large number of modes demands complex driving schemes, and the fundamentally perturbative character of these approaches imposes constraints on achievable gate speed and fidelity. Here, we introduce a scheme for entangling trapped-ion qubits using a time-dependent magnetic-field gradient, in which all axial motional modes participate in mediating the interaction and the gate construction is nonperturbative. The framework can be used to implement both multi-qubit gates and two-qubit gates between arbitrary pairs in a linear ion string. Through several explicit examples, we highlight the advantages over existing magnetic-gradient schemes and show how gates on multiple ion pairs can be carried out simultaneously.

Figures (8)

• Figure 1: Illustration of various interaction geometries, where arrows indicate interacting elements, and their color represent interaction strength (light for weak, and dark for strong). In (a), uniform coupling is shown across all ions, with equal interaction strength between each pair. In (b), a rainbow-style entanglement geometry is depicted, where symmetric pairs of ions are coupled with equal strength. In (c), distance-dependent coupling is represented, where the ion at site $j$ interacts with ions at sites $k>j$, with interaction strength decreasing as a function of the distance between them. This interaction pattern forms the basis of the controlled-phase gates used in implementing the quantum Fourier transform.
• Figure 2: (a) Required gate duration, expressed in units of the COM mode period, as a function of the Lamb-Dicke parameter for the COM mode (values for other modes follow accordingly), for a static magnetic field gradient (blue) a monochromatic oscillating gradient (orange), and the optimized drive $f_{\text{opt}}(t)$ from this work (green). (b) Corresponding gate fidelities for the three protocols as a function of the Lamb-Dicke parameter, highlighting the trade-off between speed and fidelity in each case. All results correspond to a four-ion chain and the generation of an effective long-range Ising interaction as in Eq. \ref{['eq:Hising']}, with target coupling strength $J=\pi/4$. Fidelity is computed as described in Sec. IV of the Supplemental Material supp.
• Figure 3: Global modulation $f(t)$ of the magnetic field gradient as a function of time $t$, in fractions of the total gate time $T$. Panels (a) and (b) show distinct modulation profiles designed to implement specific quantum operations. Panel (a) shows the pulse shape used to realize a uniform Ising interaction with effective coupling $J=\pi/4$ and gate time $T=2.321/2\pi\nu$; panel (b) shows the pulse for a rainbow coupling pattern (see Fig. \ref{['fig:geometries']}(b)) with coupling $J=\pi/4$ and a longer duration $T=8.125/2\pi\nu$. The solution in (a) assumes a magnetic field gradient of $\simeq 250$ T/m and a trap frequency of $\nu/2\pi=100$ kHz, leading to a maximum qubit-motion coupling of $\eta_{j1}\simeq 0.3$. In (b), the magnetic field gradient is set to $\simeq 125$ T/m, yielding $\eta_{j1}\simeq 0.15$.
• Figure 4: Phase-space trajectories of the four motional modes in a four-ion chain during an effective Ising gate with target strength $J=\pi/4$. Columns correspond to $l=1,\dots,4$ (left to right). In each panel, the real and imaginary parts of the displacement $g_l(t)$ are shown, with time indicated by color from dark blue (start) to yellow (end); the red cross marks the displacement at the gate time $T$. Row (a) shows the evolution for the initial qubit state $\ket{0010}$, and row (b) for $\ket{1010}$, highlighting the qubit-dependent nature of the dynamics.
• Figure 5: Dynamics of the von Neumann entropy $S$ for various qubit bipartitions in a four-ion system, shown as a function of the normalized gate time $t/T$. The results correspond to the application of an optimized global drive $f(t)$ designed to generate the gate in Eq. \ref{['eq:Urainbow']}.