Loss-aware optimization of realistic GKP gid states

Abstract

Realistic preparation of Gottesman-Kitaev-Preskill (GKP) states, the prime candidates for quantum computation with bosonic systems, is inevitably burdened by photon loss. We show how taking this loss into account can lead to more robust quantum states, thereby diminishing its negative influence. Using GKP squeezing as an operational figure of merit, we formulate a loss-aware optimization procedure based on the ground state representation of GKP states. Numerical simulations for square, hexagonal, and asymmetric GKP states show that the optimal preparation generally requires a different initial grid spacing than that of the desired ideal code state. For symmetric and hexagonal lattices, the optimal parameters asymptotically follow the simple scaling relation. The optimized states exhibit better GKP squeezing and greater resilience to losses. These results provide a simple and experimentally relevant strategy for improving realistic photonic GKP state preparation in the presence of known photon loss.

The paper asks how finite-energy GKP states should be chosen when a known photon-loss channel acts after preparation. It optimizes the output’s “GKP squeezing,” an expectation value of a positive operator built from two displacement stabilizers, rather than logical error probability or fidelity. The method transfers loss to the squeezing operator using the channel’s adjoint, then finds its ground state after projection onto a finite Fock space. Numerical calculations cover square, hexagonal, and rectangular grids, with representative cutoffs $N=20,40,80,150$. For square and hexagonal cases, equal attenuation of the two displacement terms gives an exact prescription for the input operator’s grid coefficients, $\mathbf{k}=\eta\mathbf{g}$, where $\eta^2$ is intensity transmission. This prescription is approximately optimal for the rectangular case at weak loss; the approximate and fully optimized curves nearly coincide for $\eta^2>0.9$. At $N=80$ and $\eta^2=0.9$, optimization reduces squeezing from $0.61$ to $0.54$ for square grids, $0.67$ to $0.57$ for hexagonal grids, and $0.67$ to $0.59$ for rectangular grids; smaller values are better. At higher loss, nearly degenerate low-energy eigenstates develop different phase-space structures. The paper uses a gap threshold $\Delta<10^{-3}$ to mark this regime. Benefits increase with Fock cutoff and are stated to be small for near-term traveling-light preparations. The analysis assumes known loss, excludes dephasing and thermal noise, and does not demonstrate preparation circuits or improvements in logical error-correction performance.