Propagation of Quantum Information in Tree Networks: Noise Thresholds for Infinite Propagation

We study quantum tree networks which propagate information from a root to leaves. At each node, the received qubit unitarily interacts with fresh ancilla qubits, after which each qubit is sent through a noisy channel to a different node in the next level. Therefore, as the tree depth grows, there is a competition between the effect of noise and the protection against such noise achieved by delocalization of information. In the classical setting, where each node copies the input bit into multiple output bits, this model has been studied as the tree broadcasting or reconstruction problem. In this work, we study its quantum version: consider a Clifford encoder at each node that encodes the input qubit in a stabilizer code. Such noisy quantum trees, for instance, provide a useful model for understanding the effect of noise within encoders of concatenated codes. We prove that above certain noise thresholds, which depend on properties of the code such as its distance, as well as properties of the encoder, information decays exponentially with tree depth. On the other hand, by studying certain efficient decoders, we prove that for codes with distance d>=2 and for sufficiently small (but non-zero) noise, classical information and entanglement propagate over noisy quantum trees with infinite depth. Indeed, we find that this remains true even for binary trees with certain 2-qubit encoders at each node, which encode the received qubit in the binary repetition code with distance d=1.