  Multidimensional photonic graph states, such as cluster states, have
prospective applications in quantum metrology, secure quantum communication,
and measurement-based quantum computation. However, to date, generation of
multidimensional cluster states of photonic qubits has relied on probabilistic
methods that limit the scalability of typical generation schemes in optical
systems. Here we present an experimental implementation in the microwave domain
of a resource-efficient scheme for the deterministic generation of 2D photonic
cluster states. By utilizing a coupled resonator array as a slow-light
waveguide, a single flux-tunable transmon qubit as a quantum emitter, and a
second auxiliary transmon as a switchable mirror, we achieve rapid, shaped
emission of entangled photon wavepackets, and selective time-delayed feedback
of photon wavepackets to the emitter qubit. We leverage these capabilities to
generate a 2D cluster state of four photons with 70\% fidelity, as verified by
tomographic reconstruction of the quantum state. We discuss how our scheme
could be straightforwardly extended to the generation of even larger cluster
states, of even higher dimension, thereby expanding the scope and practical
utility of such states for quantum information processing tasks.
