Quantum-control approach to realizing a Toffoli gate in circuit QED

We study the realization of a Toffoli gate with superconducting qubits in a circuit-quantum-electrodynamics setup using quantum-control methods. Starting with optimized piecewise-constant control fields acting on all qubits and typical strengths of $XY$-type coupling between the qubits, we demonstrate that the optimal gate fidelities are affected only slightly by a “low-pass” filtering of these fields with the typical cutoff frequencies of microwave driving. Restricting ourselves to the range of control-field amplitudes for which the leakage to the noncomputational states of a physical qubit is heavily suppressed, we theoretically predict that in the absence of decoherence and leakage, within 75 ns a Toffoli gate can be realized with intrinsic fidelities higher than $90\%$ while fidelities above $99\%$ can be reached in about 140 ns.

Figure 1

Lumped-element circuit diagram of three transmon qubits coupled to a superconducting transmission-line resonator. The resonator serves as a coupling bus for the qubits and is also used for the readout of their states (red). Local flux lines (blue) allow for individual control of the qubit frequencies on the nanosecond time scale. Microwave lines (green) are used to create control fields, each acting on its corresponding qubit.

Figure 2

Piecewise-constant and filtered (${\omega }_0=500$ MHz) control fields acting on the three qubits for $J=30$ MHz, i.e., a gate time $t_g$ of $4.18J^{-1}=139$ ns.