CZ Gate on Tunable Transmons
The controlled-phase gate uses an avoided crossing with a non-computational level to imprint a state-dependent phase of pi.
The Mechanism
Consider two coupled transmons with states labeled by their excitation numbers. The state where both qubits hold an excitation, written as the eleven state, lies close in energy to states involving a transmon's second excited level, such as the state with two excitations in one transmon. Because the transmon is anharmonic, these levels sit at slightly different energies, and the coupling opens an avoided crossing between them.
By tuning the flux so the eleven state approaches this avoided crossing and then returns, the eleven state acquires an extra phase that the other computational states do not. When that extra phase equals pi, the operation is a controlled-Z, which flips the sign of only the eleven amplitude. Combined with single-qubit gates this is locally equivalent to a controlled-NOT.
Adiabatic Versus Fast Pulses
- Slow, adiabatic pulses keep population in the eleven state and accumulate phase smoothly, but leave more time for dephasing.
- Fast, diabatic pulses use a controlled excursion through the crossing; timing is tight but the qubit spends less time off its sweet spot.
- Net-zero pulse shapes send equal positive and negative flux so slow line distortions cancel over the gate.
Controlling Leakage
The whole scheme relies on the non-computational level, which means population can be lost there if the pulse is imperfect. Leakage into and out of that level is the dominant error channel for the CZ gate. Pulse shaping, careful choice of the approach speed, and calibration of the return trajectory all aim to bring every bit of amplitude back to the computational subspace at the end of the gate.
With modern shaping and calibration, tunable-transmon CZ gates reach some of the highest two-qubit fidelities demonstrated in superconducting hardware, which is why the gate anchors many surface-code experiments.