  We study quantum networks with tree structures, where information propagates from a root to leaves: at each node in the network, the received qubit unitarily interacts with fresh ancilla qubits, and then each qubit is sent through a noisy channel to a different node in the next level. As the tree's depth grows, there is a competition between the decay of quantum information due to the noisy channels and the additional protection against noise that is achieved by further delocalizing information. In the classical setting, where each node just copies the input bit into multiple output bits, this model has been studied as the broadcasting or reconstruction problem on trees, which has broad applications. In this work, we study the quantum version of this problem, where the encoder at each node is a Clifford unitary 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 the encoders of concatenated codes. We prove that above certain noise thresholds, which depend on the properties of the code such as its distance, as well as the properties of the encoder, information decays exponentially with the depth of the tree. 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 a noisy tree 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. 
Propagation of Quantum Information in Tree Networks: Noise Thresholds   for Infinite Propagation