Parametric coupling for superconducting qubits

We propose a scheme to couple two superconducting charge or flux qubits biased at their symmetry points with unequal energy splittings. Modulating the coupling constant between two qubits at the sum or difference of their two frequencies allows to bring them into resonance in the rotating frame. Switching on and off the modulation amounts to switching on and off the coupling which can be realized at a nanosecond speed. We discuss various physical implementations of this idea, and find that our scheme can lead to rapid operation of a two-qubit gate.

Figure 1

(Color online) (a) Two flux qubits [shown coupled to their read-out SQUIDs and to their flux control line $C_i$ $(i=1,2)$] coupled by a fixed coupling constant $g_0$. (b) Parametric coupling scheme: the two flux qubits are now coupled through a circuit that allows to modulate the coupling constant $g$ through the control parameter $\lambda$.

Figure 2

(a) Circuit proposed in (Ref. 12) to implement a tunable coupling between two flux qubits. The two qubits are directly coupled by a mutual inductance $M_{qq}$, and also via the dynamical inductance of a dc-SQUID which depends on the bias current $I_b$ at fixed flux bias. The total coupling constant $g$ is shown in (b) for the same parameters as those considered in Ref. 12 as a function of $s=I_b∕2I_0$. The dashed line indicates $g=0$. Inset: $(dg∕ds)$ as a function of $s$.

Figure 3

Calculated evolution of the density matrix under the application of an entangling microwave pulse at the frequency $\mid {\Delta }_1-{\Delta }_2\mid$ in the SQUID bias current, with ${\Delta }_1=5\mathrm{GHz},{\Delta }_2=7\mathrm{GHz}$, $g\left(t\right)=\delta g\cos \left[2\pi ({\Delta }_2-{\Delta }_1)t\right]$ and $\delta g=100\mathrm{MHz}$. For the black curve, $g_0=0$; for the grey curve, $g_0=200\mathrm{MHz}$.

Figure 4

(Color online) Flux qubit parametrically coupled to an LC oscillator via a dc SQUID. (a) Electrical scheme: the qubit (blue loop) is inductively coupled to a dc SQUID shunted by a capacitor and thus forming a LC oscillator. (b) Dependence of the coupling constant $g_1$ as a function of the bias current $I_b$ (Ref. 20). At the current $I_b^{*}$ the coupling constant vanishes. (c) Derivative $d{g}_1∕d{I}_b$ as a function of $I_b$. It stays nearly at a constant value on the current range considered.