Coupling superconducting flux qubits at optimal point via dynamic decoupling with the quantum bus

We propose a scheme with dc control of finite bandwidth to implement a two-qubit gate for superconducting flux qubits at the optimal point. We provide a detailed nonperturbative analysis on the dynamic evolution of the qubits interacting with a common quantum bus. An effective qubit-qubit coupling is induced while decoupling the quantum bus with proposed pulse sequences. The two-qubit gate is insensitive to the initial state of the quantum bus and applicable to nonperturbative coupling regime which enables rapid two-qubit operation. This scheme can be scaled up to multiqubit coupling.

Figure 1

(Color online) The schematic of a single qubit with four Josephson junctions (denoted by crosses) connected in two superconducting loops. The upper loop forms a dc SQUID with two identical junctions, while the lower loop encloses three junctions similar to the conventional 3-JJ flux qubit. Each loop can be controlled separately by external magnetic fluxes ${\Phi }_d^{\left(i\right)}$ and ${\Phi }_m^{\left(i\right)}$, respectively.

Figure 2

(Color online) The energy spectrum of the lowest six energy levels of the superconducting loop with respect to the total magnetic flux ${\Phi }_t^{\left(i\right)}$. The energy is in the unit of $E_J$, while the magnetic flux is in the unit of ${\Phi }_0$. We take $E_J^{\left(i\right)}/E_C^{\left(i\right)}=35$ and ${\alpha }^{\left(i\right)}=0.8$.

Figure 3

(Color online) The energy gap of single qubit ${\Delta }^{\left(i\right)}$ and its derivative $d{\Delta }^{\left(i\right)}/d{\alpha }^{\left(i\right)}$ as a function of ${\alpha }^{\left(i\right)}$ for (a) $E_J^{\left(i\right)}/E_C^{\left(i\right)}=35$ and (b) $E_J^{\left(i\right)}/E_C^{\left(i\right)}=50$ [for simplicity, the superscript $\left(i\right)$ is omitted in the figure]. The solid line (red) is obtained from the exact diagonalization of the original 4-JJ qubit Hamiltonian, while the dashed line (black) is obtained from the analytical solution of the tight-binding model with the WKB approximation which breaks down at the low barrier regime. The energy is in the unit of $E_J$.

Figure 4

(Color online) The circuit design examples to implement the required coupling. Two 4JJ-2L flux qubits are coupled with each other through the inductive coupling with a resonator as the data bus: (a) a twisted $LC$ resonator and (b) a 1D superconducting transmission line resonator in a separate layer. The current of the data bus induces magnetic fluxes both in the upper loop and in the lower loop of each qubit. The directions of two magnetic fluxes are opposite.

Figure 5

Schematics of the pulse sequence to realize two-qubit gate operation.

Figure 6

(Color online) Schematics to scale up the coupling system. (a) Each qubit is coupled with the nearest neighbors by twisted $LC$ resonators. (b) All qubits are interacting with a common TLR resonator on top of the qubit array.