OAM-Induced Lattice Rotation Reveals a Fractional Optimum in Fault-Tolerant GKP Quantum Sensing

Abstract

Photon loss and dephasing rapidly degrade the sensitivity of quantum sensors, yet systematic methods for designing error-correcting codes whose geometry is simultaneously adapted to the sensing task and the noise channel do not exist. Here we establish that orbital-angular-momentum (OAM) encoding and Gottesman-Kitaev-Preskill (GKP) lattice geometry are structurally coupled: an OAM mode of topological charge $\ell$ induces a phase-space rotation ${\theta }_{\ell }=\ell \pi /{\ell }_{\max }$, corresponding to a family of twisted GKP stabilizer lattices. Using an end-to-end differentiable Strawberry Fields--TensorFlow circuit, we jointly optimise $\ell$, the lattice aspect ratio $r$, and the finite-energy envelope $\epsilon$ to maximise quantum Fisher information subject to $P_{\mathrm{err}}\le 10^{-3}$. The optimum occurs at the fractional charge $\ell =1.5$ ($\theta =67.5^{\circ }$), implementable with a half-integer spiral phase plate, which reduces $P_{\mathrm{err}}$ by $23.9\times$ relative to the square-lattice baseline while leaving ${\mathcal{F}}_Q$ unchanged to within $0.2\%$. This surpasses the best integer value ($\ell =2$, $15.7\times$) and arises from an exact $180^{\circ }$ periodicity of the $P_{\mathrm{err}}(\theta )$ landscape, confirmed analytically and numerically. We derive a transcendental balance equation for the optimal angle ${\theta }^{\ast }(\eta ,\gamma ,r)$ and prove that it decreases with both $\gamma$ and $\eta$. A Shannon-inspired metrological capacity $\mathcal{C}={\mathcal{F}}_Q\cdot (-\ln P_{\mathrm{err}})$, maximised at $\ell =1.5$ with a $41\%$ gain over the square lattice, quantifies the joint sensitivity--fault-tolerance resource. These results establish a geometric design principle for noise-adaptive quantum sensors and a fully open-source differentiable template extensible to other bosonic code families.

21 pages (18 main + 3 appendix), 16 figures, 14 tables, and 42 references. Code available at https://github.com/simanshukumar369/oam-gkp-quantum-metrology (Zenodo: doi:10.5281/zenodo.20099263)