  Cluster states are useful in many quantum information processing
applications. In particular, universal measurement-based quantum computation
(MBQC) utilizes 2D cluster states, and topologically fault-tolerant MBQC
requires cluster states with three or higher dimensions. In this work, we
propose a protocol to deterministically generate multi-dimensional photonic
cluster states using a single atom-cavity system and time-delay feedback. The
dimensionality of the cluster state increases linearly with the number of
time-delay feedback. We analyze the protocol's performance for a 3D cluster
state under realistic experimental conditions using a tensor network formalism
that addresses both one-qubit and two-qubit errors. Through numerical
simulation, we demonstrate that our approach can generate small to medium-sized
cluster states with nearly unity fidelity. This work shows that a single
atom-cavity system can efficiently generate multi-dimensional cluster states
with only a small number of time-delay feedback.
