Stellar Representation of Gottesman Kitaev Preskill States on the Double Discrete Cylinder

Abstract

Covariant integral quantization is developed on the double discrete cylinder phase space, with the torus as configuration space, and yields the Wigner distribution and the stellar representation. This phase space is particularly well suited to translationally invariant states, such as the Gottesman Kitaev Preskill (GKP) states. These states are central to fault-tolerant quantum computing architectures, yet their quantum resources are still under investigation. The stellar representations of ideal GKP logical states, which behave as phase eigenstates of the double discrete cylinder, have no zeros on this phase space, whereas those of their superpositions have infinitely many. The vacuum, although Gaussian, has zeros on this phase space, and the zeros of finitely squeezed GKP states move with the widths of the peaks and of the envelope, and reach this phase space for fine-tuned widths. These features echo the role of the vacuum as the resource that makes GKP-based architectures hard to simulate.

The paper asks how phase-space geometry adapted to GKP translation symmetry changes the stellar representation, particularly whether Husimi zeros distinguish ideal GKP states, finite-squeezing corrections, and the oscillator vacuum. It develops covariant integral quantization on the double discrete cylinder $\mathbb Z^2\times\mathbb T^2$, using modular position and momentum and the Zak transform. Particular quantization weights yield Wigner and Husimi distributions; wrapped Gaussian coherent states define an entire stellar function whose zeros at integer cylinder heights define a proposed stellar rank. Theta-function identities locate the zeros analytically for position eigenstates and ideal GKP states. Position eigenstates and ideal computational-basis GKP states have no zeros at integer heights, whereas the equal plus/minus logical superpositions have infinitely many. Exact integer-basis coefficients and numerical phase-winding searches show that the vacuum also has zeros on this phase space. For finitely squeezed GKP states, a large-comb approximation factorizes the modular wavefunction and gives closed-form stellar functions. Their zeros reach integer heights only at fine-tuned peak and envelope widths within this approximation. Finite squeezing generally moves the ideal superposition zeros off the height lattice, but four zeros of the minus superposition remain protected by parity without approximation. The construction depends on the modular-position unit-cell choice. Most finite-squeezing results neglect envelope-induced coupling between modular variables; exponentially small boundary corrections can also dominate at sufficiently large heights. Unit-cell-independent constructions and quantitative relations between this stellar rank, Wigner negativity, and classical simulation cost remain open.