Optimal operation of hole spin qubits

TL;DR

This work tackles the challenge of balancing fast, all-electrical control of hole-spin qubits with resilience to charge noise in silicon MOS quantum dots by identifying a continuum of magnetic-field orientations, called sweetlines, where the longitudinal spin-electric susceptibility vanishes ($\beta_\parallel=0$). Using a g-matrix formalism to map $f_L$, $\beta_\parallel$, and $\beta_\perp$ as functions of field orientation and gate voltages, the authors demonstrate reciprocal sweetness: the transverse susceptibility $\beta_\perp$ peaks align with the sweetlines, enabling fast driving with high coherence. In experiments with two neighboring qubits, they align Q3 and Q4 to a common sweetspot, achieving $f_R$ up to $\sim$24 MHz, $T_2^R$ around $18$–$25 \mu$s, and single-qubit fidelities of $\sim$99.5–99.7%, corresponding to gate-fidelity benchmarks suitable for fault-tolerant operation. The results imply a scalable path toward multi-qubit hole-spin processors with electrostatic tunability to harmonize sweetspots across qubit arrays, with broad relevance to other spin-orbit qubits and materials such as Ge/SiGe.

Abstract

Hole spins in silicon or germanium quantum dots have emerged as a compelling solid-state platform for scalable quantum processors. Besides relying on well-established manufacturing technologies, hole-spin qubits feature fast, electric-field-mediated control stemming from their intrinsically large spin-orbit coupling [1, 2]. This key feature is accompanied by an undesirable susceptibility to charge noise, which usually limits qubit coherence. Here, by varying the magnetic-field orientation, we experimentally establish the existence of ``sweetlines'' in the polar-azimuthal manifold where the qubit is insensitive to charge noise. In agreement with recent predictions [3], we find that the observed sweetlines host the points of maximal driving efficiency, where we achieve fast Rabi oscillations with quality factors as high as 1200. Furthermore, we demonstrate that moderate adjustments in gate voltages can significantly shift the sweetlines. This tunability allows multiple qubits to be simultaneously made insensitive to electrical noise, paving the way for scalable qubit architectures that fully leverage all-electrical spin control. The conclusions of this experimental study, performed on a silicon metal-oxide-semiconductor device, are expected to apply to other implementations of hole spin qubits.

Figures (6)

• Figure 1: Device and measurement of the longitudinal spin electric susceptibility (LSES)a False-color scanning electron micrograph of the silicon metal–oxide–semiconductor device used in the experiment. It consists of a silicon nanowire channel (yellow) with p-doped source (S) and drain (D) contacts, and six gates on each side of the nanowire (T1,..,T6 and B1,..,B6). The blue gates are negatively biased to accumulate holes, while the green gates are used to tune tunnel rates and confinement potentials. The investigated single-hole qubits, $Q_3$ and $Q_4$, are located next to the gates T3 and T4, respectively. $(\vec{n}, \vec{o}, \vec{p})$ defines the coordinate system. b LSES of $Q_3$ to gate T3 denoted $\beta_\parallel($T$3)\,$, as a function of magnetic-field orientation in the three device symmetry planes. The red arrows mark the sweetspot orientations where $\beta_\parallel($T$3)\,$$= 0$. The solid line is a fit based on the g-matrix formalism (see Supplementary Information). c Full angular dependence of $\beta_\parallel($T$3)\,$ as derived from the fits in b. The LSES is represented in color scale on the sphere defined by the magnetic-field unit vector, $(b_n, b_o, b_p)=\vec{B}/|\bm{B}|$. The LSES vanishes all along the white lines circling around the poles (sweetlines).
• Figure 2: Reciprocal sweetness. Experimentally reconstructed relevant quantity for qubit $Q_3$, as function of magnetic field angle. a Color plot of the transverse spin electric susceptibility (TSES) of qubit $Q_3$ to gate T3, $\beta_\perp(\text{T}3)$, as a function of the magnetic-field angles $\theta$ and $\psi$, and for a constant qubit frequency $f_L = 18G Hz$. The $\beta_\perp(\text{T}3)$ maxima (dark blue), which correspond to the highest Rabi frequencies, align with the sweetlines where $\beta_\parallel($T$3)\,$$=0$ (dashed lines). Conversely, the $\beta_\perp(\text{T}3)$ minima (light blue) coincide with the $\beta_\parallel($T$3)\,$ maxima. b The angular dependence of $\beta_\parallel($T$3)\,$ is reproduced here from Fig. 1 for direct comparison with $\beta_\perp(\text{T}3)$ in a. As in a, the sweetlines are indicated by dashed lines.
• Figure 3: Gate tunability of the sweetlines. Angular dependence of $\beta_\parallel($T$3)\,$ for three different gate-voltage settings at $f_L = 17.99G Hz$). Increasing the voltage on gate B$3$ pushes the hole wave function against the left-side facet of the silicon nanowire as schematically represented in the top insets. The deformation of the wave function results in a shift of the sweetlines (blue lines).
• Figure 4: Simultaneous tuning of two qubits to noise-resilient and fast operation points.a LSES of qubit $Q_4$ to gate T4 as a function of magnetic-field orientation, with sweetlines indicated by dashed red lines. The previously measured sweetlines of qubit $Q_3$ (shown in Fig. 3) are reproduced here as dashed blue lines, with the shaded blue region in between highlighting the range over which qubit $Q_3$ can be tuned to a noise-resilient regime. b-c Rabi oscillations measured at the common fast-operation sweetspot marked by the red cross in panel a. The different panels correspond to widely spaced time intervals. Solid lines are fits to a sinusoidal function. The decay of the oscillation amplitudes are shown in the rightmost panels. Fitting to a generalized Gaussian decay function of the type $e^{-(t/T_R)^\gamma}$ (solid line) yields the Rabi decay time $T_R$ and hence the quality factor, $Q$, for single-qubit operation. d Randomized benchmarking measurements for the two qubits, $Q_3$ (red) and $Q_4$ (blue), averaged over 40 repetitions shown as light red (blue) at the common sweetspot. $P_{Qi}$ is the recovery contrast of qubit $i$ measured after $N$ random Clifford gates (See Methods). $F_c$ is the fitted Clifford fidelity, and $F_1$ is the single qubit gate fidelity.
• Figure 5: Hole effective g-factor comparison between qubits located below gates T3 and T4: Blue (red) points are g-factor values evaluated by EDSR relating to the qubit located underneath gate T3 (T4). Solid lines are the fitted $g$-factor using the $g$-matrix formalism as presented in Methods. The $g$-factor configuration presented for the qubit $Q_3$ correspond to the dotted line configuration of Figure 1.