Exploring connections between continuous and discrete variable theorems via the Gottesman-Kitaev-Preskill code

Bosonic codes protect quantum information by embedding discrete-variable (DV) logical Hilbert subspaces into the larger continuous-variable (CV) physical Hilbert space of bosonic modes. Besides the practical use of these codes in the lab, the mathematical connection between DV and CV systems allows applying theorems from one area to the other, leading us to two main results. First, we discriminate circuits—defined with input, operations, and measurements—that cannot provide a quantum computational advantage in CV [1]. With rotation-symmetric bosonic (RSB) codes and the Gottesman-Kitaev-Preskill (GKP) code, families of CV circuits with large Wigner negativity—a necessary resource for computational advantage—correspond to Clifford circuits in DV. Thus, by applying known DV results, we conclude that we can efficiently simulate them with classical computers. Second, we define a measure for magic resource in multiqubit pure states [2], a desired property in fault-tolerant quantum computation. With the GKP code and the resource theory of Wigner logarithmic negativity in CV, we construct the measure and prove its properties for the DV system. The analytical expression of the magic measure allows easy computations with large systems and connects to the st-norm, a previously regarded one-way magic witness.

[1] L. García-Álvarez, C. Calcluth, A. Ferraro, G. Ferrini, Efficient simulatability of continuous-variable circuits with large Wigner negativity, Phys. Rev. Research 2, 043322 (2020).

[2] O. Hahn, A. Ferraro, L. Hultquist, G. Ferrini, and L. García-Álvarez, Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding, Phys. Rev. Lett. 128, 210502 (2022).