Geometric quantum gates with superconducting qubits

We suggest a scheme to implement a universal set of non-Abelian geometric transformations for a single logical qubit composed of three superconducting qubits coupled to a single cavity. The scheme utilizes an adiabatic evolution in a rotating frame induced by the effective tripod Hamiltonian which is achieved by longitudinal driving of the qubits. The proposal is experimentally feasible with the current state of the art and could serve as a first proof of principle for geometric quantum computing.

Figure 1

The level structures of the lowest energy states of the Hamiltonian equation (2) are shown. The photon excitation number of the cavity is labeled by $|n\rangle$, while $|g\rangle$ and $|e\rangle$ are the ground and excited states of the three qubits, respectively. The couplings between the one-excitation levels $g_i$ are small compared to the detunings ${\Delta }_i$.

Figure 2

The parameters $\beta$ (black) and $\alpha$ [blue (gray in the print version)] as a function of time during the adiabatic control cycle.

Figure 3

The population of the states $|1\rangle$ (black) and $|2\rangle$ [blue (gray in the print version)], obtained from the numerical integration of the Schrödinger equation using the effective Hamiltonian in Eq. (1) with $\Omega /2\pi =10.5$ MHz. (a) The gate time $T=0.5$ $\mu$s shows good population transfer, while (b) for $T=0.3$ $\mu$s, the smaller fidelity indicates the invalidity of the adiabatic approximation. (c) The gate fidelity as a function of the gate time $T$.

Figure 4

(a and b) The population of the states $|1\rangle$ (black) and $|2\rangle$ [blue (gray in the print version)], obtained from the numerical integration of the Schrödinger equation using the exact Hamiltonian equation (3). The parameters are as follows: $\omega /2\pi =5$ GHz, $g_1^{(0)}/2\pi =60$ MHz, $g_2^{(0)}/2\pi =-80$ MHz, $g_3^{(0)}/2\pi =100$ MHz, ${\Delta }_1^{(0)}/2\pi =-300$ MHz, ${\Delta }_2^{(0)}/2\pi =-400$ MHz, and ${\Delta }_3^{(0)}/2\pi =-500$ MHz. The longitudinal driving has a restriction of $\max (L_i)=100$ MHz, resulting in an effective coupling $\Omega /2\pi =10.5$ MHz. The gate times are $T=0.5$ $\mu$s in panel (a) and $T=0.3$ $\mu$s in panel (b). (c) The fidelity as a function of the gate time $T$.

Figure 5

Gate fidelity as a function of gate time $T$. Parameters are chosen as in Fig. 4, but with stronger effective coupling achieved by using half the detuning ${\Delta }_i^{(0)}$ (a), double the qubit-cavity coupling $g_i^{(0)}$ (b), double the driving $L_i$ (c), and double of all of them (d). This results in the effective couplings $\Omega /2\pi =$ 20.5 MHz (a), 21 MHz (b), 21 MHz (c), and 22 MHz (d).