Superconducting Grid-Bus Surface Code Architecture for Hole-Spin Qubits

We present a scalable hybrid architecture for the 2D surface code combining superconducting resonators and hole-spin qubits in nanowires with tunable direct Rashba spin-orbit coupling. The backbone of this architecture is a square lattice of capacitively coupled coplanar waveguide resonators each of which hosts a nanowire hole-spin qubit. Both the frequency of the qubits and their coupling to the microwave field are tunable by a static electric field applied via the resonator center pin. In the dispersive regime, an entangling two-qubit gate can be realized via a third order process, whereby a virtual photon in one resonator is created by a first qubit, coherently transferred to a neighboring resonator, and absorbed by a second qubit in that resonator. Numerical simulations with state-of-the-art coherence times yield gate fidelities approaching the 99% fault tolerance threshold.

Figure 1

Grid-bus surface code architecture. (a) and (c) Four-way capacitor design minimizing undesired cross-couplings. (b) and (e) A nanowire hole-spin qubit inside a capacitor in the trench of the resonator. The electric field perpendicular to the wire $E_z$ is controlled by voltage biasing the center conductor via the bias-tee shown in (f). (d) Resonator drive port placed at a node of the ac field $E_x$. (f) Resonator grid layout. The light gray areas represent the superconductor thin film on top of the dielectric substrate (dark blue). The red and blue dots at the center of each resonator indicate the positions of the hole-spin qubits. (g) Each resonator couples to four neighboring resonators. (h) Resonator (black lines) and qubits (red and blue bars) arranged in a square lattice. The red bars denote code qubits while the blue bars denote ancilla qubits. The colored rectangles represent the two types of plaquettes of the surface code (e.g., $XXXX$ or $ZZZZ$). The basis vectors on the lattice are indicated by dark green arrows labeled $n$ and $m$.

Figure 2

Frequency layout for parallelization of syndrome mapping in four steps. Red (blue) bars denote qubits at frequency ${\omega }_Z^{(\mathrm{r})}$ (${\omega }_Z^{(\mathrm{b})}$). The dashed black arrows indicate which couplings are resonant, i.e., active in a given configuration. The mapping of a $ZZZZ$ stabilizer is highlighted as an example (magenta arrows).

Figure 3

Gate fidelity averaged over initial states on the Bloch sphere: ${\mathcal{F}}_{\mathrm{av}}=(1/4\pi ){\int }_0^{2\pi }d\phi {\int }_0^{\pi }d\theta \sin (\theta )\mathcal{F}(\theta ,\phi )$, with $\mathcal{F}(\theta ,\phi )=⟨\theta ,\phi |\mathit{\rho }(T)|\theta ,\phi ⟩$ and $|\theta ,\phi ⟩=\cos (\theta /2)|0⟩+e^{i\phi }\sin (\theta /2)|1⟩$. (a) and (c) Single-qubit rotation around the $x$ axis by angle $\pi$. Here $|0⟩=|g{⟩}_{00}$, $|1⟩=|e{⟩}_{00}$, the remaining qubits are initialized in their ground state. Only the qubit at (0,0) is coupled to its resonator with $E_z=0.8\mathrm{V}/\mu \mathrm{m}$ and the drive strength is varied. The ideal gate unitary is $R_x(\pi )=e^{-i(\pi /2){\mathit{\sigma }}_{00}^x}$. (b) and (d) Two-qubit $\sqrt{i\mathrm{SWAP}}$ gate. Here $|0⟩=|g{⟩}_{00}|e{⟩}_{01}$, $|1⟩=|e{⟩}_{00}|g{⟩}_{01}$, the other qubits are initialized in their ground state. The qubits at (0,0) and (0,1) are coupled to their resonators with varying but equal field strength $E_z$. The ideal gate unitary is $\sqrt{i\mathrm{SWAP}}=e^{-i(\pi /4)({\mathit{\sigma }}_{00}^{+}{\mathit{\sigma }}_{01}^{-}+{\mathit{\sigma }}_{00}^{-}{\mathit{\sigma }}_{01}^{+})}$. Full curves are numerical results obtained by solving the master equation (ME) (9), and the dashed curves show analytic upper bounds for ideal gates [36].