  Measurement-based quantum computing is a promising paradigm of quantum
computation, where universal computing is achieved through a sequence of local
measurements. The backbone of this approach is the preparation of multipartite
entanglement, known as cluster states. While a cluster state with
two-dimensional (2D) connectivity is required for universality, a
three-dimensional (3D) cluster state is necessary for additionally achieving
fault tolerance. However, the challenge of making 3D connectivity has limited
cluster state generation up to 2D. Here we demonstrate deterministic generation
of a 3D cluster state based on the photonic continuous-variable platform. To
realize 3D connectivity, we harness a crucial advantage of time-frequency modes
of ultrafast quantum light: an arbitrary complex mode basis can be accessed
directly, enabling connectivity as desired. We demonstrate the versatility of
our method by generating cluster states with 1D, 2D, and 3D connectivities. For
their complete characterization, we develop a quantum state tomography method
for multimode Gaussian states. Moreover, we verify the cluster state generation
by nullifier measurements as well as full inseparability tests. Our work paves
the way toward fault-tolerant and universal measurement-based quantum
computing.
