Design and Dynamics of High-Fidelity Two-Qubit Gates with Electrons on Helium

TL;DR

This work addresses the challenge of implementing high-fidelity two-qubit gates using electrons trapped on liquid helium by employing time-dependent shaping of a confining double-well potential and TD-FCI–style dynamics to simulate gate operation. It demonstrates two non-Clifford operations, √iSWAP and CZ, achieving fidelities of up to 0.999 and 0.996 at nanosecond-scale durations, using two electrode-voltage strategies (V^β and V^ζ) and optimizing ramp/hold timings. The analysis highlights the critical roles of phase control, ZZ-coupling, and leakage to higher excited states, revealing design principles to suppress unwanted interactions and stabilize gate performance under realistic timing deviations. These results support the experimental feasibility of high-fidelity two-qubit gates with electrons on helium and provide concrete guidance for electrode geometry and control protocols to realize scalable quantum operations.

Abstract

Systems of individual electrons electrostatically trapped on condensed noble gas surfaces have recently attracted considerable interest as potential platforms for quantum computing. The electrons form the qubits of the system, and the purity of the noble gas surface protects the relevant quantum properties of each electron. Previous work has indicated that manipulation of a confining double-well potential for electrons on superfluid helium can generate entanglement suitable for two-qubit gate operations. In this work, we incorporate a time-dependent tuning of the potential shape to further explore operation of two-qubit gates with the superfluid helium system. Through numerical time evolution, we show that fast, high-fidelity two-qubit gates can be achieved. In particular, we simulate operation of the $\sqrt{i\mathrm{SWAP}}$ and CZ gates and obtain fidelities of 0.999 and 0.996 with execution times of 2.9 ns and 9.4 ns, respectively. Furthermore, we examine the stability of these gate fidelities under non-ideal execution conditions, which reveals new properties to consider in the device design. With the insights gained from this work, we believe that an experimental realization of two-qubit gates using electrons on helium is feasible.

Figures (12)

• Figure 1: Schematic microdevice, in which two electrons are trapped on the surface of a liquid helium basin in an electrostatic double-well potential created by electrodes 1–7. The two-qubit gates are driven by the Coulomb interaction $\kappa$ between the two electrons, while single qubit gates and readout are performed through the electron--cavity interaction $g$.
• Figure 2: The blue line shows the shape of $\lambda(t)$ with $t_\mathrm{ramp} = 0.3\times4\sqrt{2}ns\approx 1.7ns$ and $t_\mathrm{hold} = 3.0ns$. The orange and green lines illustrate how calculations with longer hold times, in this example 4.0ns and 5.0ns respectively, can be warm-started using previous calculations, thus reducing the computational cost.
• Figure 3: Grid search for a $\sqrt{i\mathrm{SWAP}}$ gate with step sizes $\Delta t_\mathrm{ramp} = 0.01\times4\sqrt{2}ns \approx 0.06ns$ and $\Delta t_\mathrm{hold} = 0.1ns$. The plots in the top row show the swap error $\epsilon_\mathrm{swap}$ for $\vb*{V}^\mathrm{\beta}$ in (a) and $\vb*{V}^\mathrm{\zeta}$ in (b). The corresponding gate fidelities of the two voltage functions are displayed in plots (c) and (d), respectively. The maximum fidelity achieved for $\vb*{V}^\mathrm{\beta}$ is $F=0.971$, obtained at $t_\mathrm{ramp}=0.51ns$ and $t_\mathrm{hold}=0.6ns$, and for $\vb*{V}^\mathrm{\zeta}$ the maximum fidelity is $F=0.999$, realized with $t_\mathrm{ramp}=1.41ns$ and $t_\mathrm{hold}=0.1ns$.
• Figure 4: Grid search for a CZ gate with step sizes $\Delta t_\mathrm{ramp} = 0.01\times4\sqrt{2}ns\approx0.06ns$ and $\Delta t_\mathrm{hold} = 0.1ns$. Note that the ranges of the axes are different between the two voltage functions. The plots in the top row show the leakage error $\epsilon_\mathrm{leak}$ for $\vb*{V}^\mathrm{\beta}$ in (a) and $\vb*{V}^\mathrm{\zeta}$ in (b). The corresponding gate fidelities are displayed in plot (c) and (d), respectively. The maximum achieved fidelity is $F=0.996$ for both voltage functions. With $\vb*{V}^\mathrm{\beta}$ this value is obtained at $t_\mathrm{ramp}=3.11ns$ and $t_\mathrm{hold}=3.2ns$, whereas it is given by $t_\mathrm{ramp}=1.30ns$ and $t_\mathrm{hold}=8.3ns$ with $\vb*{V}^\mathrm{\zeta}$.
• Figure 5: Gate fidelity for small deviations $\delta t_\mathrm{ramp}$ and $\delta t_\mathrm{hold}$ from the optimal ramp and hold time. The change in fidelity is presented for the $\sqrt{i\mathrm{SWAP}}$ gate in plot (a) using $\vb*{V}^\mathrm{\beta}$, and in plot (b) using $\vb*{V}^\mathrm{\zeta}$. For the CZ gate, the same results are shown in plot (c) with $\vb*{V}^\mathrm{\beta}$ and in plot (d) with $\vb*{V}^\mathrm{\zeta}$. Corresponding one-dimensional cross-sections are presented in plots (e)--(h). Note that the ramp time is crucial for the $\sqrt{i\mathrm{SWAP}}$ gate's performance.