Robust teleportation of a surface code and cascade of topological quantum phase transitions

TL;DR

This paper presents a protocol for teleporting a surface code state using Bell measurements, analyzing its robustness under errors, and revealing a cascade of topological phase transitions related to anyon condensation and statistical mechanics models.

Contribution

It introduces a practical teleportation protocol for surface codes with a detailed analysis of phase transitions and thresholds influenced by measurement angles and entanglement strength.

Findings

Threshold varies continuously with measurement angle.

Infinite threshold along the self-dual line for $X+Z$ angle.

Protocol feasible with Rydberg atom arrays.

Abstract

Teleportation is a facet where quantum measurements can act as a powerful resource in quantum physics, as local measurements allow to steer quantum information in a non-local way. While this has long been established for a single Bell pair, the teleportation of a many-qubit entangled state using non-maximally entangled resources presents a fundamentally different challenge. Here we investigate a tangible protocol for teleporting a long-range entangled surface code state using elementary Bell measurements and its stability in the presence of coherent errors that weaken the Bell entanglement. We relate the underlying threshold problem to the physics of anyon condensation under weak measurements and map it to a variant of the Ashkin-Teller model of statistical mechanics with Nishimori type disorder, which gives rise to a cascade of phase transitions. Tuning the angle of the local Bell measurements, we find a continuously varying threshold. Notably, the threshold moves to infinity for the $X+Z$ angle along the self-dual line -- indicating that infinitesimally weak entanglement is sufficient in teleporting a self-dual topological surface code. Our teleportation protocol, which can be readily implemented in dynamically configurable Rydberg atom arrays, thereby gives guidance for a practical demonstration of the power of quantum measurements.