The effects of alloy disorder on strongly-driven flopping mode qubits in Si/SiGe

TL;DR

This work analyzes how alloy disorder and charge noise affect strongly driven flopping-mode qubits in Si/SiGe, combining an 8-level Hamiltonian with a reduced two-level spin model to capture valley-induced leakage. A three-stage pulse-optimization framework is developed to maximize gate fidelity across varying valley configurations, including optimistic and pessimistic detuning-noise regimes. The results show that high-fidelity single-qubit gates are achievable across a wide range of valley parameters when valley splittings are large and valley-phase differences are small, while valley fluctuations can dominate infidelity in unfavorable configurations, especially at higher charge noise. To scale these devices, the authors propose strategies such as lateral double-dot displacement and sparse quantum-dot grids, which significantly raise the probability of finding high-fidelity qubits, particularly in Ge-modified wells that boost average valley splittings. Overall, the work underscores the importance of engineering large valley splittings and suppressing charge noise, and demonstrates practical avenues to mitigate valley-induced infidelity for scalable Si/SiGe spin-qubit architectures.

Abstract

In Si quantum dot systems, large magnetic field gradients are needed to implement spin rotations via electric dipole spin resonance (EDSR). By increasing the effective electron dipole, flopping mode qubits can provide faster gates with smaller field gradients. Moreover, operating in the strong-driving limit can reduce their sensitivity to charge noise. However, alloy disorder in Si/SiGe heterostructures randomizes the valley energy splitting and the valley phase difference between dots, enhancing the probably of valley excitations while tunneling between the dots, and opening a leakage channel. In this work, we analyze the performance of flopping mode spin qubits in the presence of charge noise and alloy disorder, and we optimize these qubits for a variety of valley configurations, in both weak and strong charge-noise regimes. When the charge noise is weak, high fidelity qubits can be implemented across a wide range of valley parameters, provided the electronic pulse is fine-tuned for a given valley configuration. When the charge noise is strong, high-fidelity pulses can still be engineered, provided the valley splittings in each dot are relatively large and the valley phase difference is relatively small. We analyze how charge noise-induced fluctuations of the inter-dot detuning, as well as small shifts in other qubit parameters, impact qubit fidelities. We find that strongly driven pulses are less sensitive to detuning fluctuations but more sensitive to small shifts in the valley parameters, which can actually dominate the qubit infidelities in some regimes. Finally, we discuss schemes to tune devices away from poor-performing configurations, enhancing the scalability of flopping-mode-based qubit architectures.

Figures (18)

• Figure 1: Schematic illustration of the flopping mode qubit. (a) Structure of the double quantum dot system, where the left and right dots are labeled L and R. Here, the detuning $\varepsilon$ controls the relative energy levels of the two dots. The valley splittings of the left and right dots are defined as $E_{vL}$ and $E_{vR}$. If the valley phase difference between the two dots satisfies $\delta \phi \neq 0$, then tunneling between the dots does not preserve the valley index, and both valley-preserving and valley-flipping tunneling processes are allowed. Two such processes are indicated in red, where $t_{--}$ and $t_{-+}$ are defined in Eq. (\ref{['eq:tunneling_matrix_elements']}). Differences in the local magnetic fields of the two dots, labeled $B_L$ and $B_R$, enable spin rotations via EDSR as the electron is driven back and forth between the dots. (b) Example energy spectra of the double dot system as a function of detuning $\varepsilon$, with representative valley splittings given by $E_{vL} = E_{vR} = 100$ µeV. The top panel shows the spectrum when the difference in the valley phase is given by $\delta \phi = 0$, while for the bottom panel, $\delta \phi = 0.9 \pi$. Here, we use solid blue and dashed red to indicate the different spins states, which appear degeneratre for the parameters considered here. All other parameters used in these calculates are specified in Sec. \ref{['sec:flopping_mode_summary']}.
• Figure 2: Performance comparison of the three pulse families described in Sec. \ref{['sec:gate_op']}, for the valley splitting values $E_{vL} = E_{vR} = 100$ µeV, as indicated in the inset of (a). (a) Infidelities of the optimized pulses as a function of the valley phase difference $\delta \phi$, including the effects of charge noise, for the three pulse families: rectangular (blue), cosine (green), and charge-cosine (orange). We consider two charge noise regimes: an optimistic regime given by $\sigma_{\varepsilon} = 1$ µeV (solid lines, circle markers), and a pessimistic regime given by $\sigma_\varepsilon = 15$ µeV (dashed lines, square markers). (b) One period of the optimized pulse shapes, for the three pulse families indicated in the insets. Results here are shown for the case $\sigma_\varepsilon = 1$ µeV and for the $\delta \phi$ values indicated by the grayscale. (c) The total pulse time, in units of $T_\text{res}$, for the same optimized pulse results shown in (a), using the same color-coding scheme. The rectangular pulse is found to yield better results for most $\delta \phi$ values, especially for large noise amplitudes, as consistent with Ref. [Teske:2023p035302].
• Figure 3: Evaluation of the cosine pulse family for a range of valley configurations. (a) The average infidelity $\mathcal{I}_\text{avg}$, and (b) the corresponding pulse lengths, for the cases $\sigma_\varepsilon = 1$ µeV (circles and solid lines) and $\sigma_\varepsilon = 15$ µeV (squares and dashed lines). Here, we take $E_{vL} = 100$ and $E_{vR} = 20$ µeV (pink), $E_{vL} = 20$ and $E_{vR} = 100$ µeV (blue), and $E_{vL} = E_{vR} = 20$ µeV (black). We also show the results for $E_{vL} = E_{vR} = 100$ µeV (green), previously reported in Fig. \ref{['fig:phase_dependence']}(a). Note that the insets in (b) indicate the valley splitting configurations. Although the case of high valley splittings (green data) generally gives better results, the other cases also achieve good fidelities, except when valley phase differences and charge noise fluctuations are large.
• Figure 4: Optimized wavefunction solutions, including the excited states, for each of the three pulse families (indicated at the top), assuming the parameters $E_{vL} = E_{vR} = 20$ µeV and $\delta \phi = 0.5\pi$. Here, we plot the occupations (i.e., the weights, $W$, defined in the main text) of the instantaneous eigenstates, including the ground (dark blue) and first excited (light blue) spin states, which constitute the logical states, and the third (dark red) and fourth (light red) states, which are excited valley states and constitute leakage. Results are shown for the noise levels (a) $\sigma_\varepsilon = 1$ µeV and (b) $\sigma_\varepsilon = 15$ µeV. The lower panels in (a) and (b) show zoomed-in views of the excited valley states. To highlight the relative length of the various pulses, we plot them all on the same $x$-axis scale. For the parameters considered here, the charge-cosine pulse shape consistently minimizes excited valley leakage, since it drives more slowly through the valley anticrossing.
• Figure 5: Locally varying valley splittings and valley phases require individualized pulse optimizations. (a) Energy dispersion of a double dot as a function of the detuning $\varepsilon$, for valley splittings $E_{vL} = 20$ and $E_{vR} = 200$ µeV (light gray), or $E_{vL} = E_{vR} = 100$ µeV (black), where the valley phase difference is $\delta \phi = 0$ in both cases. (Note that the spin splitting is too small to be resolved here.) (b) The relative dot occupation, $\langle \tau_z \rangle$, defined in Eq. (\ref{['eq:tau_z_ep']}), for the same two cases and color codings shown in (a). (Note that the gray and black lines lie beneath the orange dotted lines.) Here, the orange dotted lines represent the theoretical results obtained in Eq. (\ref{['eq:tau_z_ep_theory_supp']}). (c), (d) The same quantities as (a) and (b), where we now set $\delta \phi = 0.9\pi$. The differences in the dot occupations in (b) and (d) lead to different spin evolutions via Eq. (\ref{['eq:ham_2level']}).