Optimal Quantum Control of Multimode Couplings between Trapped Ion Qubits for Scalable Entanglement

We demonstrate entangling quantum gates within a chain of five trapped ion qubits by optimally shaping optical fields that couple to multiple collective modes of motion. We individually address qubits with segmented optical pulses to construct multipartite entangled states in a programmable way. This approach enables high-fidelity gates that can be scaled to larger qubit registers for quantum computation and simulation.

Figure 1

Improvement of entangled state creation using pulse shaping on $N=2$ trapped ion qubits. (a) Comparison of Bell state entanglement fidelity for a constant pulse (black) versus a five-segment pulse (red) over a range of detuning $\mu$, showing significant improvement with the segmented gate. (b) The segmented pulse pattern, parameterized by the Rabi frequency ${\mathrm{\Omega }}_i(t)$ with the particular detuning $\mu$ near the 2nd (“tilt”) motional mode [arrow in (a)] and measured state fidelity $\ge 94(2)\%$. (c) Phase space trajectories (arbitrary units) subject to pulse sequence in (b) for both CM and tilt modes of the two ions. The thickness of the curves from each segmented pulse is alternated to guide the trajectories. The five-segment pulse pattern brings the two trajectories back to their origins, simultaneously disentangling both modes of motion from the qubits.

Figure 2

Entanglement of qubit pairs within a chain of $N=5$ trapped ions. (a) Applied pulse pattern and measured parity oscillations for a constant pulse used to entangle ions 1 and 2. The gate time is ${\tau }_g=190\mu \mathrm{s}$ and the detuning is set to $\mu ={\omega }_x+2\pi /{\tau }_g$, which should exhibit the highest fidelity; we measure $F=82(3)\%$. (b) Same as (a), but with a nine-segment pulse. The gate time is again ${\tau }_g=190\mu \mathrm{s}$, and the detuning is set close to the tilt mode; we measure $F=95(2)\%$. (c) Different nine-segment pulse pattern used to entangle ions 2 and 3 with same gate time, and the detuning is set close to the 5th mode; we measure $F=95(2)\%$. (d) Phase space trajectories (arbitrary units but all on same scale) for the pulse pattern applied to the ion pair 1 and 2 at the detuning used in (b).

Figure 3

(a) Theoretical entanglement gate fidelity for the first two ions as a function of total number of ions in a chain using a constant (black), a five-segment (blue), and a nine-segment (red) pulse. The gate time is fixed to $\sim 90\mu \mathrm{s}$ and the minimum (central) ion spacing is $\sim 5\mu \mathrm{m}$. The advantages of the pulse segmentation are clearly seen for larger numbers of ions. (b) Theoretical entanglement gate fidelity for the first two ions as a function of detuning error $\mathrm{\Delta }\mu$ in a chain of 5 ions. The solid black (red) line corresponds to a constant (nine-segment) pulse on the two ions, where the pulse power is optimized at each value of the detuning offset. The dashed black (red) line corresponds to a constant (nine-segment) pulse optimized for $\mathrm{\Delta }\mu =0$. We see that the segmented pulse not only allows high-fidelity solutions at any detuning (solid line), but even when the detuning drifts, the segmented pulses mitigate this error (dashed line).

Figure 4

Programmable quantum operations to create tripartite entanglement. (a) Circuit for concatenated $XX$ gates between ions 1&2 and 2&3 and $\pi /2$ analysis rotations of ions 1&2 with phase $\varphi$. (b) Measured population after $XX$ gates on ions 1&2 and 2&3, where $P_N$ denotes the probability of finding $N$ ions in the $|1⟩$ state. (c) Parity oscillations of ions 1&2 with the phase $\varphi$ of the $\pi /2$ analysis rotations, after postselecting the state of the third ion, with periods $\pi$ (left) and $2\pi$ (right) for the two states $|0{⟩}_3$ and $|1{⟩}_3$, respectively [see Eq. (4)]. (d) Schematic for creating a GHZ “cat” state using two $XX$ gates on ions 1&2 and 2&3 as before, with additional individual qubit rotations, followed by $\pi /2$ analysis rotations of all three ions with phase $\varphi$. (e) Three-ion parity oscillation with phase $\varphi$ of the $\pi /2$ analysis rotations. The red solid line is fit to the data with period $2\pi /3$, while the blue dashed line is the expected signal assuming a perfect cat state with known systematic measurement errors.