High-fidelity regimes of resonator-mediated controlled-Z gates between quantum-dot qubits

Abstract

Semiconductor double quantum dot (DQD) qubits coupled via superconducting microwave resonators provide a powerful means of long-range manipulation of the qubits' spin and charge degrees of freedom. Quantum gates can be implemented by parametrically driving the qubits while their transition frequencies are detuned from the resonator frequency. Long-range two-qubit controlled-Z (CZ) gates have been proposed for the DQD spin qubit within the rotating-wave approximation (RWA). Rapid gates demand strong coupling, but RWA breaks down when coupling strengths become significant relative to system frequencies. Therefore, understanding the errors arising from approximations used is critical for high-fidelity operation. Here, we go beyond RWA to study CZ gate fidelity for both DQD spin and charge qubits. We propose a novel parametric drive on the charge qubit that produces smaller errors and show that the fidelity of the CZ gate outperforms its spin counterpart, resulting in a much smaller fidelity loss of $0.05\%$ compared to $0.80\%$ for the spin qubit, and greater robustness against qubit dephasing and photon loss. We find that drive amplitude -- a parameter dropped in RWA -- is critical for optimizing fidelity, with the charge qubit exhibiting better tolerance to drive amplitude variations. Our results demonstrate the necessity of going beyond RWA in understanding how long-range gates can be realized in DQD qubits, with charge qubits offering considerable advantages in high-fidelity operation.

Figures (5)

• Figure 1: (a) Schematic of two spatially separated DQD qubits in a circuit QED system. (b) Single-electron DQD charge qubit and energy scales. Tunnel coupling $t_c$ and orbital detuning $\epsilon$ control the charge qubit frequency $\omega_c$. Logical states (blue) are eigenstates of $\hat{H}_\text{DQD}$. Simultaneous $t_c (t)$ and $\epsilon(t)$ drives with amplitude $2\Omega = \omega_c$ and drive frequency $\omega = \Delta$ enables red-sideband transitions. (c) Single-electron DQD spin qubit. The Zeeman $B_z$ field spin-splits each orbital; a spin-orbit field $B_x$ admixes spin and orbital states, enabling electrical control of the logical states (red). A resonant $\epsilon(t)$ drive with amplitude $2\Omega = \Delta$ enables red-sideband transitions.
• Figure 2: Gate fidelity of spin (top row) and charge (bottom row) qubits. (a, b) Map of high (green) and low (blue) fidelity regions against rescaled resonator frequency and drive amplitudes. Dashed red lines are linecuts taken and plotted in panels (c) and (d). (c, d) Fidelity linecuts at $\tilde{\omega}_r = 531.5$ and $292.3$, against rescaled drive amplitude, with (blue) and without (red) decoherence, for rates $\tilde{\kappa}_s = \tilde{\kappa}_c = \tilde{\gamma}_s = \tilde{\gamma}_c = 0.008$. Highest fidelities for spin and charge qubits are $99.20\%$ and $99.95\%$ (red circles in panels c, a), at optimal drive amplitudes $2\tilde{\Omega} = 38.5$ and $106.75$ respectively, indicated by the central dashed vertical line. With decoherence, optimal fidelities drop to $76\%$ and $83\%$ (blue circles in panels d, b) for the spin and charge qubits. The boundaries of high-fidelity regions identified in panels (a) and (b) are shown as vertical dashed lines at the two sides computed from the infidelity approximation of Eq. \ref{['eq:infidelity formula']}. Since these boundaries intersect with the 98% fidelity line, the analytical approximation shows excellent agreement with the numerical results. Arrows denoting $n=1 \dots 4$ indicate locations of spurious resonances from the multiple frequency components of the charge qubit detuning drive, as identified in Eq. \ref{['eq:charge_qubit_RWA']}. The effect on fidelity is negligible beyond $n=4$. (e, f) Fidelity against rescaled photon loss and dephasing rates at optimal drive amplitudes and resonator frequencies used in panels (a--d), showing that the charge qubit is more resilient against noise than the spin qubit.
• Figure 3: (a) Spin qubit fidelity for $\tilde{\omega}_r = 158$ which is smaller than the linecut in Fig. \ref{['fig:F2']}(c). Red (blue) curves represent fidelity with (without) gate time correction Appendix \ref{['supp:gate time correction']}. The curve is smoothed but the best fidelity (circles) is not improved. (b) Exact $\epsilon(t)$ waveform (blue) compared against realistic waveform (orange) composed with the first 8 Fourier components. The latter produces a good approximation of constant qubit frequency $\omega_c(t)$ (green).
• Figure 4: (a) Plot of final photon population of each Fock state $\ket{n}_r$ for various initial states. Only for an initial resonator vacuum state $\ket{0}_r$ does the gate sequence achieve a CZ gate and return to the vacuum subspace (see Appendix \ref{['supp:gate evolution']}). (b) Plot of the purity of the final two-qubit state against initial resonator state. The two-qubit state is pure only with an initial vacuum state $\ket{0}_r$, and becomes increasingly mixed as the initial resonator state contains a greater number of photons.
• Figure 5: Upper bound $N(r, 2m)$ of the $2m$-th term in the infidelity expansion against the order $m$ of the terms, for various values of $r$. Convergence requires monotonically decreasing values as the order $m$ increases. Here, with the exception of $r=7/12$ (pink line), all other values ensure convergence, enabling the set of $r$ values in Eq. \ref{['eq:r_values']} to be obtained.