A method of an on-demand beamsplitter for trapped-ion quantum computers

TL;DR

This work tackles the challenge of achieving switchable entangling gates between local modes in CV-encoded trapped-ion quantum information processing by introducing an on-demand beamsplitter based on dynamic frequency control. The authors derive an analytic transformation for the time-dependent harmonic-oscillator system using Lewis–Riesenfeld theory, capturing phase shifts, squeezing, and a central beamsplitter operation, and validate the approach with TDVP-MPS simulations showing Hong–Ou–Mandel interference and SWAP of finite-energy GKP states. They further analyze and mitigate unwanted hopping via a sawtooth frequency configuration and dynamical decoupling (C3PO), enabling scalable coupling across multiple modes while preserving state preparation and readout integrity. The results indicate practical feasibility for implementing large-scale CV-encoded trapped-ion quantum computers and simulators, with potential extensions to inter-module connectivity and multi-mode entangling gates.

Abstract

Quantum information processing using local modes of trapped ions has been applied to implementing bosonic quantum error correction codes and conducting efficient quantum simulation of bosonic systems. However, control of entanglement among local modes remains difficult because entanglement among resonant local modes is governed by the Coulomb interaction, which is not switchable. We propose a method of a beamsplitter for a trapped-ion architecture, where the secular frequency of each mode is dynamically controllable. The neighboring modes are far detuned except when the beamsplitter needs to be applied to them. We derive the analytical formula of the proposed procedure and numerically confirm its validity.

Figures (5)

• Figure 1: Sawtooth configuration of frequencies for $N=4$. Ions are aligned along $Z$ axis and separated by $d$. Local modes along $X$ is used for computation. A gap between a memory frequency $\omega'_{\alpha}\,(\alpha=0,1,\dots,N-1)$ (solid horizontal bar) and a gate frequency $\omega_{\beta-1,\beta}\,(\beta=1,2,\dots,N)$ should be large enough to avoid unwanted phonon hopping. For the on-demand beamsplitter between $(j-1)$th and $j$th modes with $j\neq0\mod N$, the frequency of each mode is changed by $\texttt{FC}$ (purple arrow) to the gate frequency $\omega_{\alpha,\beta}\,(\alpha=j-1,\,\beta=j\mod N)$ (dashed horizontal bar). Cross-talk error occurs during the on-demand beamsplitter between $(j-1)$th and $j$th modes with $j=0\mod N$. The largest contribution to the cross-talk error comes from $(j-2)$th mode, which becomes resonant with the $(j-1)$th mode, and from $(j+1)$th mode, which becomes resonant with the $j$th mode during $\texttt{FC}$. Phonon hopping between resonant modes separated by $Nd$ can be mitigated by C3PO, e.g., phonon hopping between the 2nd and the 6th modes can be mitigated by applying $(-\pi)$-phase shift gates $\texttt{P}_{-\pi}$ (red arrow) to the 6th mode. $\texttt{P}_{-\pi}$ is implemented by $\texttt{FC}$ and $i\texttt{FC}$ but the amount of change in frequency can be smaller than that needed for the on-demand beamsplitter. Note that $\texttt{P}_{-\pi}$ is equivalent to $\texttt{P}_{\pi}$ up to global phase and we choose either of them depending on the situation.
• Figure 2: The auxiliary function $b(t)$ (dashed lines) for the upward (blue) and the downward (orange) frequency changing defined by equations \ref{['eq:fu_erf']} and \ref{['eq:fd_erf']}, respectively. The corresponding $\tilde{\omega}(t)$ (solid lines) is defined by equation \ref{['eqApp:omega(t)_by_b(t)']}.
• Figure 3: (a) The populations of three states relevant in the HOM effect, $\ket{1}_1\ket{1}_0$, $\ket{2}_1\ket{0}_0$, and $\ket{0}_1\ket{2}_0$ in the memory-frequency basis (solid lines) and the gate-frequency basis (dashed lines) (b) and (c) shows the expanded view during $\texttt{FC}$ and $i\texttt{FC}$, respectively.
• Figure 4: The simulation of the $\texttt{SWAP}$ gate between two GKP states using the on-demand beamsplitter. The marginal distributions in the position space of each mode are shown in (a)-(f). The schematic of the on-demand beamsplitter is shown in (g), where the gates are applied from the left to right. Colors of the harmonic well and the marginal distributions correspond to the frequencies $\omega_{h}$ (orange), $\omega_{m}$ (green), and $\omega_{l}$ (blue). The initial state is the product of GKP states $\ket{1_L}_1\ket{0_L}_0$ characterized by the parameters $\Delta=\epsilon=0.3.$ In (a) and (d) ((b) and (e)), the solid lines represent the input to $\texttt{FC}$ ($i\texttt{FC}$) while the dashed lines represent the output from $\texttt{FC}$ ($i\texttt{FC}$). (c) and (f) are the output of the phase-shift gate $\texttt{P}$, which completes the $\texttt{SWAP}$ gate.
• Figure 5: Time dependent frequencies of four modes involved in SWAP and C3PO. (a) Resonant hopping during $\tilde{\omega}_3=\tilde{\omega}_4$ implements the on-demand beamsplitter while resonant hopping during $\tilde{\omega}_2=\tilde{\omega}_6$ causes error. The crossing of $\tilde{\omega}_2$ and $\tilde{\omega}_3$ causes the NN cross-talk and the crossing of $\tilde{\omega}_3$ and $\tilde{\omega}_6$ causes the non-NN cross-talk. The tiny dips of $\tilde{\omega}_6$ at the middle and at the end correspond to the application of $\texttt{P}_{-\pi}$. (b) The expanded view of (a) around the end of the simulation.