Logical Gates on Grid Codes with Gaussian Operations, Part 1: Analytical Approach for the Phase Gate

Grid codes are a promising approach to quantum error correction, offering intrinsic protection against certain types of noise. Realizing practical quantum computation also requires the implementation of a universal set of high-fidelity gates. In physical implementations of grid codes, finite-energy effects limit the fidelity of both single- and two-qubit gates. Recently, some strategies have been proposed to mitigate these effects, such as applying finite-energy stabilization cycles following the gate to recover the initial state [1].



In this work, we analytically investigate how the small-Big-small (sBs) stabilization protocol [2] can increase the state fidelity following the Clifford phase gate by modifying the small conditional displacements of each stabilization round. Our method finds an order of magnitude improvement over the standard stabilization protocol, with the use of only standard control pulses and without increasing the total gate time. This work paves the way for the implementation of high-fidelity universal gates on logical grid code qubits, while providing more insight into the inner workings of the sBs stabilization protocol.

[1] Rojkov et al., Phys. Rev. Lett. 133, 100601 (2024)

[2] Royer et al., Phys. Rev. Lett. 125, 260509 (2020)