Superconducting Qubit Based on Twisted Cuprate Van der Waals Heterostructures

Van-der-Waals assembly enables the fabrication of novel Josephson junctions featuring an atomically sharp interface between two exfoliated and relatively twisted ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\mathrm{x}}$ (Bi2212) flakes. In a range of twist angles around 45°, the junction provides a regime where the interlayer two-Cooper pair tunneling dominates the current-phase relation. Here we propose employing this novel junction to realize a capacitively shunted qubit that we call flowermon. The $d$-wave nature of the order parameter endows the flowermon with inherent protection against charge-noise-induced relaxation and quasiparticle-induced dissipation. This inherently protected qubit paves the way to a new class of high-coherence hybrid superconducting quantum devices based on unconventional superconductors.

Figure 1

Potential design for the flowermon qubit. (a) A relative twist of two $d$-wave flakes placed together to form a Josephson junction can suppress Cooper pair tunneling due to momentum mismatch. At 45° the mismatch completely suppresses single Cooper pair tunneling, and two-pair tunneling dominates the junction. (b) The design of the flowermon with a single $d$-wave junction shunted by a large capacitor similar to the transmon qubit. The 3D design shows a possible physical implementation with the capacitor pads of a conventional superconductor coupled to the junction. (c) Josephson’s potential for different values of the twist angle.

Figure 2

Flowermon low-energy spectrum. (a) Energy levels of the flowermon qubit as a function of the angle $\theta$. Even and odd levels are shown in red and blue lines respectively; as $\theta$ approaches $\pi /4$, the levels combine into quasidegenerate doublets. Here we assume $E_{\kappa }/E_J=0.1$ and $E_J/E_C=2000$. (b),(c) Level structure and potential energy for $\theta =30°$ and $\theta =43°$. (d),(e) The wave functions for the ground and first excited state $|{\mathrm{\Psi }}_0⟩$ and $|{\mathrm{\Psi }}_1⟩$ in the phase basis for $\theta =30°$ and $\theta =43°$. (f),(g) Evolution of the structure of $|{\mathrm{\Psi }}_0⟩$ and $|{\mathrm{\Psi }}_1⟩$ in the charge basis as a function of the twisting angle $\theta$.

Figure 3

Protection from decoherence. (a) The charge matrix elements $n_{01}^2$ and $|n_{11}-n_{00}{|}^2$, related to capacitive relaxation and dephasing rates respectively, plotted vs $E_J/E_C$ and the twisting angle $\theta$. Different regimes are shown, with the flowermon exhibiting protection from both relaxation and dephasing processes. (b) Charge matrix elements $n_{01}$, $n_{03}$, $n_{23}$, and $n_{12}$ vs the twisting angle $\theta$ and $E_J/E_C=2000$. Both $n_{01}$ and $n_{23}$ decay exponentially as $\theta \to 45°$ but with a significant offset. (c) Dependence of the quasiparticle relaxation rate on the ratio $\mathrm{\Delta }/k_{B}T$ for different values of the twisting angles ranging between $\theta =0°$ and $\theta =40°$.

Figure 4

Manipulation of the flowermon logical states. (a) Logical state manipulation can be performed via the 2nd and 3rd excited states in a regime where the 0-1 matrix element and frequency are near 0, but the amplitude of the $2\leftrightarrow 3$ transition is still finite. The control is performed by standard microwave $\pi$ pulses between the 1-2 and 0-3 states, the desired control gate $U$ pulse between the 2-3 states, and again $\pi$ pulses between the 1-2 and 0-3 states. Alternatively, the 0-1 transition can be driven indirectly by a Raman process via simultaneous off-resonant driving of the 0-3, 2-3, and 2-1 transitions (orange arrows). Note that this protocol works even when ${\omega }_{01}\to 0$ because the selection rule forbids 0-2 and 1-3 transitions. (b) A sketch of a possible implementation of the device with control and readout lines.