Scalable and Parallel Tweezer Gates for Quantum Computing with Long Ion Strings

Trapped-ion quantum computers have demonstrated high-performance gate operations in registers of about ten qubits. However, scaling up and parallelizing quantum computations with long one-dimensional (1D) ion strings is an outstanding challenge due to the global nature of the motional modes of the ions, which mediate qubit-qubit couplings. Here, we devise methods to implement scalable and parallel entangling gates by using engineered localized phonon modes. We propose to tailor such localized modes by tuning the local potential of individual ions with programmable optical tweezers. Localized modes of small subsets of qubits form the basis to perform entangling gates on these subsets in parallel. We demonstrate the inherent scalability of this approach by presenting analytical and numerical results for long 1D ion chains and even for infinite chains of uniformly spaced ions. Furthermore, we show that combining our methods with optimal coherent control techniques allows realization of maximally dense universal parallelized quantum circuits.

Popular Summary

Quantum computers hold the promise of accomplishing computational tasks that exceed the capabilities of conventional classical computers. Trapped atomic ions are a leading platform for quantum-information processing, and have demonstrated high-performance gate operations in registers of about ten qubits. While state-of-the-art ion traps can sustain quantum registers of more than a hundred qubits, scaling up and, in particular, parallelizing gate operations remains an outstanding challenge. This is because gate operations use collective modes of oscillation of the crystal structure, which is formed by the ions, to mediate the transfer of quantum information between qubits, and these phonon modes become increasingly complex in large ion crystals. In our work, we develop new methods to address this challenge.

We propose to use optical tweezers to pin individual ions and in this way shape the structure of the phonon modes. In particular, localized phonon modes, which correspond to coupled oscillations of pairs of neighboring ions, can be used to implement parallel entangling two-qubit gates on these pairs. We demonstrate the inherent scalability of this approach by presenting analytical and numerical results for long one-dimensional ion crystals and even for infinite crystals of uniformly spaced ions. Our methods enable the implementation of universal parallelized quantum circuits, with a wide range of applications in quantum computation and simulation.

An intriguing perspective opened up by our work is to utilize localized phonon modes for the design of Hamiltonians in analog quantum simulation. Further, the optical segmentation of long ion chains through optical tweezers can lead to the realization of one-dimensional quantum networks.

Figure 1

Schematic setup and circuits. (a) Specific ions in a long 1D chain are pinned by optical tweezers. This leads to the formation of localized phonon modes, e.g., for the subregister formed by the three ions that are marked by cyan shading, and for the pair of distant qubits that are marked by pink shading and are pinned by equally strong optical potentials. (b) The localized phonon modes can be utilized to implement multiqubit entangling gates. Buffer ions at the ends of the chain are not used as qubits. Addressed lasers, which are required to perform entangling gates, are indicated schematically in (a) as blue and red beams. (c) In this work, we focus on implementing parallel two-qubit entangling gates, which use the localized phonon modes of pairs of neighboring pinned ions. (d) After a first layer of the circuit, which corresponds to the product of gates $U_1{U}_2{U}_3$ that act on distinct qubits, the tweezers are focused on different ions to perform the second layer corresponding to the gate operation $U_4{U}_5$.

Figure 2

(a) $N=60$ ions in an array of optical tweezers, which subdivides the chain into groups of $p=6$ six ions. The first and last five buffer ions are not used as qubits. (b) Mode matrix $M_{x,i}^n$ for transverse $x$ modes. Orange and gray shading indicates pinned and buffer ions, respectively. Blue shading marks modes that are superpositions of localized COM and stretch modes of pairs of pinned ions. (c) Phonon-mode spectrum for ${\nu }_0=0.4$ and, for comparison, without tweezers, i.e., for ${\nu }_0=0$. Transverse $x$ and $y$ modes and longitudinal $z$ modes are shown, respectively, in blue, green, and orange. We assume that the ratio of transverse trapping frequencies is given by ${\omega }_y/{\omega }_x=0.8$.

Figure 3

(a) Infinite ion chain with optical tweezers forming a periodic array with unit cell size $p=6$. (b) Mode matrix $M_{x,l,i}^{k,n,\lambda }$ for $l=k=0$ and $\lambda =1$. COM and stretch modes of the pinned ions are clearly decoupled from the ions that are not pinned. (c) Phonon-mode spectra for an infinite chain without and with optical tweezers, for ${\nu }_0=0$ and ${\nu }_0=0.4$, respectively. Transverse $x$ and $y$ modes and longitudinal $z$ modes are shown, respectively, in blue, green, and orange. The spatial periodicity imposed by tweezers causes the dense bands of the $x$ and $z$ modes in the infinite system to split up into $p=6$ bands, where the highest two bands correspond to COM and stretch modes of pinned pairs.

Figure 4

Phase-space trajectories for parallel entangling gates in an infinite chain. (a) Contribution to the displacement of the phonon mode $k,n,\lambda$ due to its coupling, Eq. (5), to the qubit at position $l=i=1$, which is assumed to be in the $+1$ eigenstate of ${\sigma }_{l,i}^x$. For the chosen detuning close to ${\mu }_{\mathrm{stretch}}$ in Eq. (19), the stretch mode with $n=\lambda =2$ and $k\approx 1.55$ such that ${\omega }_{k,2}={\omega }_2$ equals the mean frequency of the stretch band experiences the strongest displacement. The phase-space trajectory of the COM mode with the same values of $k$ and $\lambda$ forms a smaller circle, which is traversed twice. Trajectories of modes with $n\ge 3$ are not visible on the scale of the figure. (b) Closer inspection reveals that the phase-space curves for different values of $k$ are not perfectly closed. The area under the squared modulus of the qubit-phonon coupling at the end of the gate operation, shown here for $n=\lambda =2$, determines the infidelity (11). Parameters of the ion chain are $p=6$, ${\nu }_0=0.4$, and $ϵ=0.07$.

Figure 5

Gate performance for an infinite chain. The main panel shows the dependence of the infidelity on the parameters ${\nu }_0$ and $ϵ$. $\delta F={10}^{-3}$ on the black line across the diagonal. In the white region, the optical potential is insufficient to decouple the pinned ions from the other ions in the chain. The insets show the gate duration $\tau$ and the crosstalk $C$ for fixed infidelity $\delta F={10}^{-3}$.

Figure 6

Tweezer gates in finite chains. We show individual detunings and Rabi frequencies for $17$ maximally entangling tweezer gates targeting the stretch mode along a chain of $130$ ions with $15$ buffer ions on each side ($p=6$, $ϵ\approx 0.07$). Optical tweezers generate a trapping frequency that is uniform ${\nu }_0=0.4$ and alternating between ${\nu }_{01}=0.4$ and ${\nu }_{02}=0.36$ for (a) and (b), respectively. For comparison, we also show the detuning ${\mu }_{\mathrm{\infty }}$ and Rabi frequency ${\mathrm{\Omega }}_{\mathrm{\infty }}$ for an infinite chain with $ϵ=0.07$ and ${\nu }_0=0.4$.

Figure 7

Tweezer misadjustments. We show the infidelity Eq. (11) (blue) and the over- and under-rotation error Eq. (13) (orange) as a function of the strength of fluctuations $\sigma$ for three types of misadjustments and for their combination. For each type of misadjustment, the infidelity and the over- and under-rotation error are averaged over $40$ realizations. We consider here a chain of $130$ ions with $15$ buffer ions on each side, and parameters $p=6$, $ϵ\approx 0.07$, and ${\nu }_0=0.4$.

Figure 8

Schematic setup for dense circuits. (a) In a setup in which pairs of ions that are pinned and not pinned alternate, gates can be performed on all ions in parallel. This leads to the realization of dense circuits as illustrated in (b). Maintaining high fidelity and low crosstalk in such circuits requires optimal coherent control techniques.

Figure 9

Pulse optimization in an infinite chain. (a) Phonon-mode spectrum for $p=4$, ${\nu }_0=0.4$, and $ϵ=0.07$. (b) The minimization of the cost function $L$ is performed independently for pinned and not pinned ions. The upper and lower panels show the cost function for optimal sequences of Rabi frequencies for a range of detunings around the COM and stretch bands of pinned and not pinned ions, respectively. (c) For the optimal detunings we show the corresponding $G=4$ independent sequences of $S=8$ Rabi frequencies both in the time domain and in discrete Fourier space. Purple and green correspond to positive and negative Rabi frequencies, respectively. The Fourier representations are dominated by few modes, which belong to subspaces of Fourier space that alternate along the chain: a pulse sequence that contains only odd Fourier components is followed by a sequence that is composed of even Fourier components and vice versa.

Figure 10

Pulse optimization in a finite chain. We show the chosen detunings (see main text) to implement a dense circuit of $50$ parallel, maximally entangling tweezer gates in a chain of $130$ ions with $15$ buffer ions on each side ($p=4$, ${\nu }_0=0.4$, $ϵ\approx 0.07$). The horizontal lines mark the mode frequencies. We allow $S=8$ segments and set the gate duration to ${\omega }_x\tau =1500$, which requires a maximal Rabi frequency of ${\eta }_0\mathrm{\Omega }/{\omega }_x=0.007$. We get average infidelity $\delta F={10}^{-5}$ and crosstalk $C=2.77\times {10}^{-4}$.

Figure 11

Infidelity due to spontaneous scattering for $ϵ=0.07$, ${\nu }_0=0.4$, and ${\omega }_x=2\pi \times 3$ MHz for different atomic species. The steep slope at short wavelengths is caused by the excessive scattering close to the resonance of the $S\iff P$ transition. At long wavelengths, the infidelity increases due to an enlarged spot size for a diffraction limited spot for the assumed numerical aperture.

Figure 12

Spontaneous scattering infidelity for parallel gates performed on ${}^{24}{\mathrm{Mg}}^{+}$ ground-state qubits with 400-nm tweezer wavelength for varying ion distance $d$ and radial trap frequency ${\omega }_x$. By choosing $d$ and ${\omega }_x$, $ϵ$ is fixed, and ${\nu }_0$ is determined in order to reach the strong pinning regime. The gray region marks parameter combinations that lead to a zigzag configuration of the ion string.