High-fidelity method for a single-step $N$-bit Toffoli gate in trapped ions

Conditional multiqubit gates are a key component for elaborate quantum algorithms. In a recent work, Rasmussen et al. [Phys. Rev. A 101, 022308 (2020)] proposed an efficient single-step method for a prototypical multiqubit gate, a Toffoli gate, based on a combination of Ising interactions between control qubits and an appropriate driving field on a target qubit. Trapped ions are a natural platform to implement this method, since Ising interactions mediated by phonons have been demonstrated in increasingly large ion crystals. However, the simultaneous application of these interactions and the driving field required for the gate results in undesired entanglement between the qubits and the motion of the ions, reducing the gate fidelity. In this work, we propose a solution based on adiabatic switching of these phonon-mediated Ising interactions. We study the effects of imperfect ground-state cooling and use spin-echo techniques to undo unwanted phase accumulation in the achievable fidelities. For gates coupling to all axial modes of a linear crystal, we calculate high-fidelity $(>99\%)$ $N$-qubit rotations with $N=3–7$ ions cooled to their ground state of motion and a gate time below 1 ms. Finally, we study the effect of laser intensity fluctuations and find that the proposed gate requires intensity stabilization with subpercentage noise levels. The high fidelities obtained also for large crystals could make the gate competitive with gate-decomposed, multistep variants of the $N$-qubit Toffoli gate, at the expense of requiring ground-state cooling of the ion crystal.

Figure 1

Energies of noninteracting eigenstates ($J=0$) and interacting (dressed) states ($J>0$). The two target states $|111\rangle ,|011\rangle$ are highlighted. Because their energy gap is unique, an appropriate drive field can couple the states resonantly.

Figure 2

Time evolution of states under the action of ${\tilde{H}}_{\text{T,sm}}$ (a) Phase-space trajectories (zoomed in, dimensionless) of motional wave function during evolution with ${\widehat{U}}_{\text{T}}$. Note that the adiabatic ramp ensures that dynamics take place along the momentum axis in this frame, as explained more in detail in Appendix pp1. (b) Real and (c) imaginary part of process unitary matrix for the motional ground state $\left(\right|n=0\rangle )$ subspace. (d) Evolution in the Bloch sphere of the two resonant states, and (e) the projections along $x$ ($-·$), $y$ ($--$), and $z$ ($-$) of the trajectory of initial state $|111\rangle$. Time is indicated with the color intensity from light ($t=0$) to dark ($t={\tau }_g$) in panel (d). The gate parameters are ${\delta }_{\text{CM}}/2\pi =20\mathrm{kHz}$, $J/2\pi =2\mathrm{kHz}$ ($\mathrm{\Omega }/2\pi =126.491\mathrm{kHz}$), $g/2\pi =1\mathrm{kHz}$ for a gate time of ${\tau }_g=\pi /g=500\mu \mathrm{s}$.

Figure 3

Multimode unitaries and spectrum of multiple beatnotes for “echo” step for phases cancellation. (a) $i$-Toffoli unitary for a three-ion crystal considering all-mode couplings without and (inset) with echo step. Frequency and amplitude of beatnotes for (b) three and (c) seven-ion gate with detunings ${\delta }_{\text{CM}}/2\pi =-20\mathrm{kHz}$ and ${\delta }_{\text{CM}}/2\pi =-50\mathrm{kHz}$, respectively. The phonon mode frequencies are indicated in dashed red lines. The parameters of panel (a) are ${\omega }_{\text{CM}}/2\pi =1\mathrm{MHz}$, ${\delta }_{\text{CM}}/2\pi =-20\mathrm{kHz}$, $g/2\pi =1\mathrm{kHz}$ and for panels (b) and (c) the interaction time is $t_{\text{mb}}=5\mu \mathrm{s}$.

Figure 4

Process error in function of Ising strength and gate time assuming single-mode coupling. The results are for a $i$-Toffoli gate of 3, 5, 7, and 9 ions for detunings of (a) 50, and (b) 200 kHz. We also show the result ($7^{*}$) for a seven-ion crystal including an “echo” step. In this case, the total process time corresponds to $2t_{\text{T}}$.

Figure 5

Effect of average phonon number in process fidelity for gate with multimode coupling. The Ising and drive strengths ($J/4\pi =g/2\pi$) are (a) 4.762 kHz and (b) 1 kHz. The detuning, ${\delta }_{\text{CM}}/2\pi$, and the center-of-mass frequency, ${\omega }_{\text{CM}}/2\pi$, are $-200\mathrm{kHz}$ and 1 MHz, respectively.

Figure 6

Gate error for fluctuating Rabi frequency $\tilde{\mathrm{\Omega }}$ at frequency $f_{\tilde{\mathrm{\Omega }}}$ for two different variances ${\sigma }_{\mathrm{\Omega }}$. The gate parameters are ${\omega }_{\text{CM}}/2\pi =1\mathrm{MHz}$, ${\delta }_{\text{CM}}/2\pi =-200\mathrm{kHz}$, $g/2\pi =1\mathrm{kHz}$ for a gate time of $1500\mu \mathrm{s}$. The results are the average and deviation from 50 realizations.

Figure 7

Strength of Hamiltonian terms during length $i$-Toffoli gate. The Ising interaction (blue, lower row) is increased before acting with the drive field (purple, middle row) and then lowered again. A “echo” step (red, upper row) can be applied at the end of the gate to correct for residual entanglement or dynamical phases.

Figure 8

Phase-space trajectories (dimensionless) of target states under the application of ${\widehat{H}}_{\text{q-ph}}$. Trajectories of $|111\rangle$ (red, outer trajectory) and $|011\rangle$ (blue, inner trajectory) wave packages and evolution of momentum expectation value (blue, solid line) of $|011\rangle$ due to (a), (c) a quench activation of $500\mu \mathrm{s}$ and (b), (d) an adiabatic modulation of ${\widehat{H}}_{\text{q-ph}}$ (orange, dashed line). $\langle x\rangle =0$ at the times indicated in panels (a) and (b).