  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. This work proposes a
protocol to deterministically generate multidimensional 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 firstly give a diagrammatic derivation of the tensor network states, which
is valuable in simulating matrix product states and projected entangled pair
states generated from sequential photons. Our method also provides a simple way
to bridge and analyze the experimental imperfections and the logical errors of
the generated states. In this method, we analyze the generated cluster states
under realistic experimental conditions and address both one-qubit and
two-qubit errors. Through numerical simulation, we observe an optimal
atom-cavity cooperativity for the fidelity of the generated states, which is
surprising given the prevailing assumption that higher cooperativity systems
are inherently better for photonic applications.
