Symmetrically Threaded Superconducting Quantum Interference Devices As Next Generation Kerr-cat Qubits

Abstract

We theoretically explore an alternative circuit for Kerr-cat qubits based on symmetrically threaded Superconducting Quantum Interference Devices (SQUID). The Symmetrically Threaded SQUIDs (STS) architecture employs a simplified flux-pumped design that suppresses two-photon dissipation, a dominant loss mechanism in high-Kerr regimes, by engineering the drive Hamiltonian's flux operator to generate only even-order harmonics. By fulfilling two critical criteria for practical Kerr-cat qubit operation, the STS emerges as an ideal platform: (1) a static Hamiltonian with diluted Kerr nonlinearity (achieved via the STS's middle branch) and (2) a drive Hamiltonian restricted to even harmonics, which ensures robust two-photon driving with reduced dissipation. For weak Kerr nonlinearity, we find that the coherent state lifetime ($T_α$) is similar between STS and SNAIL circuits. However, STS Kerr-cat qubits exhibit enhanced resistance to higher-order photon dissipation, enabling significantly extended $T_α$ even with stronger Kerr nonlinearities ($\sim$10 MHz). In contrast to SNAIL, STS Kerr-cat qubits display a $T_α$ dip under weak two-photon driving for high Kerr coefficient. We demonstrate that this dip can be suppressed by applying drive-dependent detuning, enabling Kerr-cat qubit operation with only eight Josephson junctions (of energies 80 GHz); fewer junctions suffice for higher junction energies. We further validate the robustness of the STS design by studying the impact of strong flux driving and asymmetric Josephson junctions on $T_α$. With the proposed design and considering a cat size of 10 photons, we predict $T_α$ of the order of tens of milliseconds, even in the presence of multi-photon heating and dephasing effects.

Figures (14)

• Figure 1: (a) STS design for Kerr-cat qubit. The junctions ${\rm J}_{1}$ and ${\rm J}_{3}$ compose the "SQUID branch" whereas the junction ${\rm J}_{2}$ gives the "transmon branch". The loop between ${\rm J}_{1}$ and ${\rm J}_{2}$ is threaded by an external flux $\phi_e$ whereas the loop between ${\rm J}_{2}$ and ${\rm J}_{3}$ is threaded by an external flux $\phi_e^\prime$. We consider $\phi_e^\prime = \phi_e = -\pi/2 + \delta\phi \cos(\omega_{\rm d}t)$. $C_{\rm T}$ is the shunt capacitor associated with the transmon branch. (b) Bloch sphere representation of the Kerr-cat qubit states.
• Figure 2: Classical phase space surface of the total effective energy of the Kerr-cat Hamiltonian in Eq. (\ref{['eq:ham_main']}) for $\Delta/K = 4$, (a) $\epsilon_2/K=\Lambda/K=0$, and (b) $\epsilon_2 /K= 0.3$, $\Lambda = 0$ (yellow plot) and $\Lambda/K = 0.2$ (grey plot). Energy spectrum of Eq. (\ref{['eq:ham_main']}) as a function of $\Delta$ for $\epsilon_2/K=2$ and (c) $\Lambda/K = 0$ and (d) $\Lambda/K = 0.12$. Red stars denote the first degeneracies of the first three excited state pairs with opposite parity.
• Figure 3: (a) The Floquet quasi-energy level diagram of the STS lab Hamiltonian of Eq. (\ref{['eq:ham_unexp']}) (solid black curves) compared to the eigenenergy level diagrams of the STS effective Hamiltonian of Eq. (\ref{['eq:ham_main']}) (dashed red curves) plotted as a function of the two-photon drive strength for $K/h=7.81$ MHz. (b) The overlap between the eigenstates of the STS lab Hamiltonian and the cat states (eigenstates of the static effective Hamiltonian) plotted as a function of time when the two-photon drive is adiabatically switched on from $\epsilon_2/K = 0$ to $\epsilon_2/K = 4$ (solid red curve). The solid orange curve gives the photon population in the STS at each time $t$.
• Figure 4: $T_\alpha$ of the Kerr-cat qubit as a function of the two-photon drive strength $\epsilon_2/K$ for 10 STS connected in series with a single junction in each branch setting the Kerr nonlinearity to $K/h=1.25\textrm{~MHz}$. The RWA calculation, which takes the system environment coupling up to order $\varphi_{\rm zps}^{0}$ and keeps only single-photon processes in the master equation, is given by the solid blue line. The dashed orange curve gives the leading order beyond RWA ${\cal O}(\varphi_{\rm zps}^{2})$ calculation and keeps multiphoton processes in the master equation. In the inset, we plot the energy spectra of the Kerr-cat qubit as a function of the two-photon drive strength $\epsilon_2/K$.
• Figure 5: $T_\alpha$ of the Kerr-cat qubit as a function of the two-photon drive strength $\epsilon_2/K$ for different values of Kerr nonlinearity. In order to obtain the mentioned Kerr coefficient, we took $M=2,N=2$ (solid blue curve), $M=2,N=4$ (dashed orange curve) and $M=3,N=6$ (dotted yellow curve) and $M=10,N=10$ (dot-dashed purple curve). In the inset, we show how the initial dip in the lifetime observed for $K/h = 31.3 {\rm MHz}$ can be actively canceled by introducing a drive-dependent detuning.