Explicit Wigner Function of Prime-Comb GKP States and Rigidity Analysis of Phase-Space Grids

The Gottesman–Kitaev–Preskill (GKP) code encodes discrete quan- tum information on a periodic lattice in the phase space of a continuous- variable harmonic oscillator. In this paper, we propose a prime-comb GKP state, which replaces the uniform integer sequence with a se- quence of prime numbers to construct a non-uniform phase-space grid. We present an explicit expression for the Wigner function of this state and analyze its grid structure in phase-space projection. By compar- ing it with the standard square GKP grid, we demonstrate that the prime-comb grid exhibits enhanced rigidity against resonant displace- ment noise, rooted in the irrationality of prime gaps and the stability mechanism of KAM theory. This construction provides a number- theoretic blueprint for the physical realization of both quantum error correction and post-quantum cryptography.