Three level control for fault tolerant echoed conditional displacement

3D superconducting cavities have been shown to be good memories for quantum states. For example, a Gottesman-Kitaev-Preskill (GKP) code encoded in a cavity has achieved lifetime beyond break even [1]. This system uses a transmon for quantum control of the cavity but errors in the transmon during operations can alter the information stored in the cavity. In principle, the use of the first three levels of the transmon as control allows the e state of the transmon to act as a buffer state. Three level control, combined with other techniques, has been used for fault tolerant parity measurements [2] and SNAP gates [3].

Here, we study a scheme that uses a dissipative element, such as the readout resonator, to correct errors in a three level qubit used for quantum control of a cavity. We test the scheme on the echoed conditional displacement (ECD) operation used for GKP stabilisation. Using drives on the transmon and the dissipative element, we engineer a dissipator that acts as a pump from the buffer state e to the state f. We demonstrate the properties necessary for the scheme to correct errors during an ECD and propose an ancilla circuit design that can reduce errors during GKP stabilisation and augment GKP lifetime inside the cavity.

[1] Sivak, V.V. et al. Real-time quantum error correction beyond break-even. Nature 616 (2023).

[2] Rosenblum, S. et al. Fault-tolerant detection of a quantum error, Science 361 (2018).

[3] Reinhold, P., Rosenblum, S., Ma, WL. et al. Error-corrected gates on an encoded qubit. Nat. Phys. 16 (2020).