Enlarging the GKP stabilizer group for enhanced noise protection

Encoding a qubit in a larger Hilbert space of an oscillator is an efficient way to protect its quantum information against decoherence [1]. Promising examples of such bosonic encodings are the Gottesman-Kitaev-Preskill (GKP) codes [2–4], where the usage of quantum error correction has been shown to enhance the lifetime of the qubit beyond break-even [5]. In this work, we investigate how redefining the stabilizer group to include any operation for which the action commutes on the codespace can help find an optimal implementation of a logical circuit, when it is affected by noise. We show that the gaussian stabilizer group allows one to choose between different physical implementations of a Clifford operation. As a result, we propose an algorithm that finds the optimal implementation of a given logical Clifford circuit on GKP qubits, such that the state is less affected by loss errors during the computation. Finally, we evaluate the performance of the algorithm using a method inspired on logical randomized benchmarking [6].

[1] A. Joshi et al., Quantum Science and Technology 6, 033001 (2021)

[2] D. Gottesman et al., Physical Review A 64 (2001)

[3] B. Royer et al., PRX Quantum 3, 10.1103-010335 (2022)

[4] J. Conrad et al., Quantum 6, 648 (2022)

[5] V. V. Sivak et al., Nature 616, 50–55 (2023)

[6] J. Combes et al., arXiv:1702.03688 (2017)