  Quantum walks, in virtue of the coherent superposition and quantum
interference, possess exponential superiority over its classical counterpart in
applications of quantum searching and quantum simulation. The quantum enhanced
power is highly related to the state space of quantum walks, which can be
expanded by enlarging the photon number and/or the dimensions of the evolution
network, but the former is considerably challenging due to probabilistic
generation of single photons and multiplicative loss. Here we demonstrate a
two-dimensional continuous-time quantum walk by using the external geometry of
photonic waveguide arrays, rather than the inner degree of freedoms of photons.
Using femtosecond laser direct writing, we construct a large-scale
three-dimensional structure which forms a two-dimensional lattice with up to
49X49 nodes on a photonic chip. We demonstrate spatial two-dimensional quantum
walks using heralded single photons and single-photon-level imaging. We analyze
the quantum transport properties via observing the ballistic evolution pattern
and the variance profile, which agree well with simulation results. We further
reveal the transient nature that is the unique feature for quantum walks of
beyond one dimension. An architecture that allows a walk to freely evolve in
all directions and a large scale, combining with defect and disorder control,
may bring up powerful and versatile quantum walk machines for classically
intractable problems.
