Quantum error correction, quantum advantage and magic measures with the Gottesman-Kitaev-Preskill code

Since when it was introduced in 2001, the Gottesman-Kitaev-Preskill (GKP) code [1] has attracted considerable attention, due to its capability of making bosonic quantum computation fault-tolerant. In this talk, I will review several results around GKP codes. First, I will argue that the performance of GKP codes for quantum error correction is hindered when dephasing noise, or finite detection efficiencies in the state recovery procedure, are present, while other bosonic codes such as binomial codes are less affected by these effects [2]. Then, using the stabiliser formalism I will show that the ingredients composing GKP quantum error correction circuits (excluding the data qubit), i.e. stabiliser GKP states together with Gaussian operations and quadrature measurements, are classically efficiently simulatable [3]. Combined with the result that GKP error-correction of the vacuum yields magic states [4] this allows us to conclude that vacuum provides quantum advantage in restricted quantum computing architectures made of stabiliser GKP states together with Gaussian operations and quadrature measurements. Finally, I will show that using the GKP encoding allows for defining a magic measure for qubits, easier to compute than other measures.

[1] D. Gottesman et al, Phys.Rev.A 64, 01231 (2001)

[2] T. Hillmann et al, PRX Quantum 3, 020334 (2022)

[3] C. Calcluth et al, arXiv:2205.09781 (2022)

[4] B. Baragiola et al, Phys. Rev. Lett. 123, 200502 (2019)

[5] O. Hahn et al, Phys. Rev. Lett. 128, 210502 (2022)