Realistic GKP stabilizer states enable universal quantum computation

Abstract

Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.

Figures (2)

• Figure 1: Quantum circuit for gate teleportation
• Figure 2: Probability distribution functions for the output mode over angles $\theta$ and $\phi$ on the Bloch sphere. Parameters for the four panels are (a) $\beta=0.04$ and $\theta_r=\pi/4$, and yield highly localized Pauli-Y eigenstates; (b) $\beta=0.01$ and $\theta_r=0.0681\pi$; (c) $\beta=0.001$ and $\theta_r=0.38467\pi$ yielding non-Pauli eigenstates with high probability; and (d) $\beta=0.001$ and applied rotation angles in the range $\theta_r\in[0.38012\pi, 0.38248\pi]$ so that the four symmetry-connected points on the Bloch sphere form trajectories.