Circuit QED with hole-spin qubits in Ge/Si nanowire quantum dots

We propose a setup for universal and electrically controlled quantum information processing with hole spins in Ge/Si core/shell nanowire quantum dots (NW QDs). Single-qubit gates can be driven through electric-dipole-induced spin resonance, with spin-flip times shorter than 100 ps. Long-distance qubit-qubit coupling can be mediated by the cavity electric field of a superconducting transmission line resonator, where we show that operation times below 20 ns seem feasible for the entangling $\sqrt{i\text{SWAP}}$ gate. The absence of Dresselhaus spin-orbit interaction (SOI) and the presence of an unusually strong Rashba-type SOI enable precise control over the transverse qubit coupling via an externally applied, perpendicular electric field. The latter serves as an on-off switch for quantum gates and also provides control over the $g$ factor, so single- and two-qubit gates can be operated independently. Remarkably, we find that idle qubits are insensitive to charge noise and phonons, and we discuss strategies for enhancing noise-limited gate fidelities.

Figure 1

Proposed setup. (a) An electric (magnetic) field $E_y$ ($B_y$) is applied perpendicularly to the Ge/Si NW QD. Ac fields $E_z\left(t\right)$ shift the confining potential along the NW axis $z$. (b) When placed between the center conductor (C) and the ground plane (G) of a transmission line resonator, the hole-spin qubits (red arrows) can interact via the cavity field $E_c$, with the interaction strengths controlled by local electric fields $E_y^{\left(i\right)}$. In the sketch, a two-qubit gate is performed between qubits 2 and 4. The setup does not require equidistant QDs and is robust against misalignment.

Figure 2

Electrical tunability of $\overline{\nu }$ for the QD in the text. Solid blue (dotted red) curves result from the numerical calculation (effective model);[57] dashed black lines correspond to Eq. (6). The thin blue lines (RSOI only, $H_{\mathrm{DR}}=0$) illustrate that much stronger $E_y$ would be required for realizing a given $\overline{\nu }$ with the conventional RSOI. Top: Result for $|B_y|\lesssim \text{0.5}\text{T}$, where a linear expansion in $B_y$ applies. Bottom: $B_y=\text{1.5}\text{T}$, beyond the linear regime. Inset: Height and position of the peak as a function of $B_y$. For $E_y=0$, a level crossing in the numerics occurs at $B_y\simeq 1.8\text{T}$ (vertical dashed line). The achievable operation times scale with $1/\overline{\nu }$.

Figure 3

The $g$ factor [$|B_y|\lesssim \text{0.5}\text{T}$ (top)] and Zeeman splitting [$B_y=\text{1.5}\text{T}$ (bottom)] as a function of $E_y$ for the parameters in the text. Solid blue (dotted red) lines are calculated numerically (perturbatively);[57] dashed black lines result from Eq. (7). The thin blue lines (RSOI only, $H_{\mathrm{DR}}=0$) confirm that the strong electrical tunability results from the DRSOI, Eq. (4). Inset: $E_Z$ at the $E_y$ for which $\overline{\nu }$ is maximal (see inset of Fig. 2).

Figure 4

Basic operation scheme with the numerical results from Figs. 2 and 3. When $E_y=0$ (idle), the qubit features long lifetimes. Two-qubit operations are envisaged at $E_y$ with maximal $\overline{\nu }$ (cavity). Single-qubit gates can be performed independently by applying a different $E_y$ (EDSR) for which all cavity modes are far off-resonant. The associated change in the $g$ factor ($E_Z$) is indicated by $\Delta g$ ($\Delta E_Z$). For $|B_y|\lesssim \text{0.5}\text{T}$ (left), $\overline{\nu }$ is maximal at $E_y\simeq 1.8\text{V}/\mu \text{m}$, where $g\simeq 3.4$ and ${\partial }_{E_y}g\simeq -1.6\mu \mathrm{m}/\mathrm{V}$. For $B_y=\text{1.5}\text{T}$ (right), $\overline{\nu }$ peaks at $E_y\simeq 1.1\text{V}/\mu \text{m}$, where $E_Z\simeq 0.35\text{meV}$ and ${\partial }_{E_y}{E}_Z\simeq -0.13\text{nm}e$.