Description
We show that when one half of a logical Bell pair is transported through a gravitational field, the differential proper time accumulated between two error-corrected memories manifests as a relative unitary $U_{\rm rel} = U_A^\dagger U_B$ in the reference frame of the stationary memory. This evolution is indistinguishable from free precession within each memory's local inertial frame, and is therefore invisible to local QEC throughout the journey: each syndrome record sees only stochastic noise, and no correction is applied to the gravitational phase. The accumulated relative unitary becomes observable only when the two memories are reunited and their local quantum reference frames are compared via a joint logical Bell measurement. The effect is code-dependent: for the bit-flip repetition code, $R_z(\delta\theta)^{\otimes n}$ is a logical operator and the gravitational phase accumulates as a logical phase amplified linearly by the code distance $n$; for the $[[3,1,3]]$ phase-flip code the response is nonlinear, with the logical phase $\Phi_L = \arctan(\tan^3(\delta\theta/2))$ giving quartic suppression of the logical error rate at small $\delta\theta$ but recovering a clean Bell signal near $\delta\theta = \pi/2$. We verify these regimes analytically and via Tsim simulation. For trapped-ion hyperfine memories, where $T_1 \gg T_2^*$ by many orders of magnitude, the phase-flip code's $T_1$-limited logical lifetime is far longer, allowing the gravitational phase to accumulate over minutes to days at terrestrial height differences. We further propose a gravitational interferometry protocol based on encoding-basis switching, converting the gravitational signal into a logically amplified $\bar{R}_z(n\,\delta\theta)$ rotation with $1/(n\sqrt{M})$ phase sensitivity over $M$ Bell-pair shots. We characterise noise floors from local-oscillator phase noise, Pauli-frame bookkeeping requirements, and trapped-ion feasibility, connecting the two protocols' experimental requirements to existing and near-term ion-trap QEC demonstrations.
| I am the presenting author | Yes |
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