Fault-tolerant photon-number selective phase gate in circuit quantum electrodynamics

In circuit quantum electrodynamics (QED), it has been demonstrated that universal control of the cavity states can be realized by quantum control of a transmon coupled to the cavity in the strongly dispersive regime. An important class of quantum gates are the selective number-dependent arbitrary phase (SNAP) gates, which impart arbitrary phases to the different Fock states of the cavity by cyclically driving the transmon. However, the SNAP gate fidelity is limited by the transmon relaxation and dephasing. Here we show that by using a multi-level transmon and conditional evolution on the transmon state after the gate, the SNAP gate can be made fault-tolerant to the dominant transmon relaxation and dephasing errors. The simulations show that the SNAP gate infidelity can be reduced compared to that of non-fault-tolerant SNAP gate. The fault-tolerant SNAP gates combined with the displacement operations on the cavity can realize fault-tolerant quantum computation in circuit QED.