Fast two-bit operations in inductively coupled flux qubits

A central problem for implementing efficient quantum computing is how to realize fast operations (both one- and two-bit ones). However, this is difficult to achieve for a collection of qubits, especially for those separated far away, because the interbit coupling is usually much weaker than the intrabit coupling. Here we present an experimentally feasible method to effectively couple two flux qubits via a common inductance and treat both single and coupled flux qubits with more realistic models, which include the loop inductance. The main advantage of our proposal is that a strong interbit coupling can be achieved using a small inductance, so that two-bit as fast as one-bit operations can be easily realized. We also show the flux dependence of the transitions between states for the coupled flux qubits.

Figure 1

(Color online) (a) A flux qubit, where an external magnetic flux ${\Phi }_e$ pierces the superconducting loop that contains three Josephson junctions and an inductance $L$. The Josephson energies and capacitances of the junctions are $E_{J1}=E_{J2}=E_J$, $C_1=C_2=C$, $E_{J3}=\alpha E_J$, and $C_3=\alpha C$. Here we choose $\alpha =0.8$ and $E_J=35E_c$, where $E_c=e^2∕2C$. (b) Two flux qubits coupled by a common inductance $L_c$, where the external flux ${\Phi }_e$ is applied within the left loop $A_1{L}_c{B}_1{A}_1$. The parameters of each flux qubit are $E_{J1}^{\left(i\right)}=E_{J2}^{\left(i\right)}=E_J^{\left(i\right)}$, $C_1^{\left(i\right)}=C_2^{\left(i\right)}=C^{\left(i\right)}$, $E_{J3}^{\left(i\right)}={\alpha }_i{E}_J^{\left(i\right)}$, and $C_3^{\left(i\right)}={\alpha }_i{C}^{\left(i\right)}$, with $i=1,2$. Here we choose ${\alpha }_i=0.8$ and $E_J^{\left(i\right)}=35E_c^{\left(i\right)}$, where $E_c^{\left(i\right)}=e^2∕2C^{\left(i\right)}$. To implement a readout of the flux-qubit states, a switchable superconducting flux transformer is employed to couple the dc-SQUID magnetometer with the inductance $L$ in (a) or $L_c$ in (b) during the quantum measurement. However, this coupling is switched off in the absence of a readout.

Figure 2

(Color online) Contour plots of the potential energy $U({\varphi }_p,{\varphi }_m)$, in units of $E_J$, for $\alpha =0.8$ and $f=0.5$. Here ${\beta }_L\equiv 2\pi I_{0}L∕{\Phi }_0=$ (a) 0, (b) 1, (c) 4, and (d) 10. Notice that the well-defined double-well potential structure vanishes in (d), and thus the flux qubit breaks down.

Figure 3

(Color online) Energy levels of a single flux qubit vs reduced flux $f$ for different values of ${\beta }_L$, where only the levels of the states $\mid i⟩$, $i=0$ to 5, are shown. Here the energy $E$ is in units of $E_J$. Notice the robustness of the two lowest levels for wide changes in the loop inductance $L$.

Figure 4

Energy difference $\Delta$ between qubit states ∣1⟩ and ∣0⟩ as a function of ${\beta }_L$ for $f=0.5$. Here $\Delta$ is in units of $E_J$. Notice that the energy difference varies $\sim 0.003E_J$ when varying the loop inductance $L$.

Figure 5

(Color online) Energy levels of two coupled flux qubits vs reduced flux $f$ for $L_{b1}∕L_c=0.1$ and $(L_{12}+L_{b2})∕L_c=0.2$. The parameters ${\beta }_{Li}\equiv 2\pi I_{0i}{L}_c∕{\Phi }_0$ are (a) ${\beta }_{L1}={\beta }_{L2}=0.03$; (b) ${\beta }_{L1}=0.03$, ${\beta }_{L2}=0.04$; (c) ${\beta }_{L1}={\beta }_{L2}=0.07$; and (d) ${\beta }_{L1}=0.07$, ${\beta }_{L2}=0.1$. Here the energy $E$ is in units of $E_J^{\left(1\right)}$. Near $f=0.5$, the energy levels are robust with respect to the asymmetry between ${\beta }_{L1}$ and ${\beta }_{L2}$.

Figure 6

(Color online) (a) Moduli of the transition matrix elements $t_{ij}$ between single-qubit states $\mid i⟩$ and $\mid j⟩$ vs reduced flux $f$. (b) and (c) Moduli of the transition matrix elements between coupled-qubit states $\mid {ϵ}_i⟩$ and $\mid {ϵ}_j⟩$ vs $f$ for $L_{b1}∕L_c=0.1$ and $(L_{12}+L_{b2})∕L_c=0.2$. Here $\mid t_{ij}\mid$ is in units of $I_0{\Phi }_X$ in (a) and $I_{01}{\Phi }_X$ in (b) and (c). Note that, in (b) and (c), some transitions for the coupled two qubits are sensitive with respect to the asymmetry of ${\beta }_{L1}$ and ${\beta }_{L2}$.

Figure 7

(Color online) (a) Supercurrents $I$ vs reduced flux $f$ at eigenstates ∣0⟩ and ∣1⟩ of a single flux qubit for ${\beta }_L=0.03$. (b) Supercurrents $I_1$ vs $f$ at eigenstates $\mid {ϵ}_k⟩$, $k=1\text{to}4$, of two coupled flux qubits for the symmetric circuit with ${\beta }_{L1}={\beta }_{L2}=0.03$. (c) Supercurrents $I_1$, $I_2$, and $I$ vs $f$ at eigenstates $\mid {ϵ}_k⟩$, $k=1\text{to}4$, of two coupled flux qubits for an asymmetric circuit with ${\beta }_{L1}=0.03$ and ${\beta }_{L2}=0.04$. Here we choose $L_{b1}∕L_c=0.1$ and $(L_{12}+L_{b2})∕L_c=0.2$ for the coupled flux qubits. The supercurrents are in units of $I_0$ in (a) and $I_{01}$ in (b) and (c). For the coupled two qubits, the total current $I=I_1+I_2$ is robust with respect to the asymmetry of ${\beta }_{L1}$ and ${\beta }_{L2}$.