Controlled-Z gates with giant atoms in structured waveguides

Abstract

Giant atoms are quantum emitters coupled to waveguides at multiple, spatially separated points, enabling interference effects that fundamentally change their light-matter interactions. A notable consequence of the interference is the emergence of decoherence-free interaction (DFI), which allows coherent excitation exchange between giant atoms via the waveguide without radiative loss. Leveraging DFI offers a promising route to implementing two-qubit quantum gates without the need for additional resources, positioning giant atoms as a versatile platform for scalable universal quantum simulators. However, existing work has focused primarily on continuous, Markovian waveguides; in structured waveguides, where non-Markovian effects become significant, only iSWAP gates have been explored. To address this gap, we introduce and analyze a protocol for implementing controlled-Z (CZ) gates with giant atoms in structured waveguides. We first show that while a minimal two-point coupling scheme supports DFI, it also exhibits strong non-Markovian effects that substantially degrade gate fidelity. To overcome this limitation, we propose an extended design featuring a third coupling point. This configuration suppresses non-Markovian effects and enables CZ gates with fidelities up to $97.7\%$ (assuming typical values for experimental imperfections). Our results broaden the accessible gate set for giant atoms in structured waveguides to include both iSWAP and CZ gates, advancing these systems as a pathway toward universal quantum simulators operating in non-Markovian environments.

Figures (4)

• Figure 1: Model and gate scheme. (a) Schematic of two giant atoms (GAs) coupled to a one-dimensional array of coupled cavities with nearest-neighbor hopping rate $J$. Each atom $m$ is connected with coupling strength $g_{mj}$ to the array at site $x_{mj}$. The atoms have distinct transition frequencies $\omega_{1(2)}$ and anharmonicities $\alpha_{1(2)}$. (b) The CZ gate in GAs can be implemented via the pathway CZ-02 (CZ-20), which coherently transfers the population of the $|11 \rangle$ state to the $|02 \rangle$ ($|20 \rangle$) state and then brings it back, accumulating a $\pi$ phase shift.
• Figure 2: Setup and results for performing a CZ gate between GAs with two coupling points each. (a) Schematic of the GA-based setup with two coupling points to perform a CZ gate. (b) Bath dispersion relation and the decoherence-free frequencies for $\Delta x = 4$ ($\omega_{\rm DF}/J=\pm\sqrt{2}$) and $\Delta x = 16$ ($\omega_{\rm DF}/J=\pm0.39$) shown with circles and triangles, respectively. (c) Energy-level diagram of the two GAs for settings enabling the CZ gate. (d) Dynamics in the decoherence-free regime, with parameters $N=100$, $g/J = 0.175$, $\omega_1/J = \sqrt{2}$, $\omega_2/J = -\sqrt{2}$, $\alpha_1/J = -2\sqrt{2}$, and $\alpha_2/J = -3$. (e) Same as in (d), but with increased separation $\Delta x = 16$ and parameters $\omega_1/J = 0.39$, $\omega_2/J = -0.39$, $\alpha_1/J = -0.78$, and $\alpha_2/J = -1$.
• Figure 3: Setup and results for performing a CZ gate between GAs with three coupling points each. (a) Schematic of the GA-based setup with three coupling points to perform a CZ gate, with $\Delta x = 2$ between adjacent coupling points. Compared to Fig. \ref{['fig2']}, a third, central coupling point with coupling strength $\zeta g$ is added. This configuration yields two decoherence-free (DF) frequencies for $0 \leq \zeta < 2$. (b) Decoherence-free frequencies as a function of $\zeta$, showing that the two frequencies merge at $\zeta = 2$ with the band of propagating frequencies between $[-2, 2]$ shown in light green. (c) Dynamics in the DFI regime for $N=100$, $\Delta x = 2$, $g/J = 0.1$, and $\zeta = 1$, yielding DF frequencies $\omega_\text{DF}/J = \pm 1$. The frequencies and anharmonicities of the GAs are chosen to fit in the CZ-gate configuration: $\omega_1/J = 1$, $\omega_2/J = -0.98$, $\alpha_1/J = -2$, and $\alpha_2/J = -1.52$. (d) Same as (c), but with $\zeta = 1.5$, giving DF frequencies $\omega_\text{DF}/J = \pm 0.71$. Here, we take parameters $\omega_1/J = 0.71$, $\omega_2/J = -0.69$, $\alpha_1/J = -1.42$, and $\alpha_2/J = -1.31$. (e) Same as (c) and (d), but with $\zeta = 1.97$, resulting in DF frequencies $\omega_\text{DF}/J = \pm 0.17$. Here, we take parameters $\omega_1/J = 0.17$, $\omega_2/J = -0.17$, $\alpha_1/J = -0.34$, and $\alpha_2/J = -0.67$.
• Figure 4: Process fidelity $\mathcal{F}_{\text{CZ}}$ of CZ gates performed with the setup in Fig. \ref{['fig3']}. (a) Process fidelity as a function of time $t$ for the same parameters as in Fig. \ref{['fig3']}. (b) Same as (a), but with a smaller $g=0.05J$. (c) The best process fidelity of CZ gates, $\mathcal{F}_{\text{CZ,max}}$, for different values of $g$, considering realistic qubit and cavity decay rates of $\Gamma_q=1.6\cdot10^{-5}J$ and $\Gamma_c=8\cdot10^{-5}J$. (d) The gate time $\tau$ required for the best CZ gate in (c).