  With photonics, the quantum computational advantage has been demonstrated on
the task of boson sampling. Next, developing quantum-enhanced approaches for
practical problems becomes one of the top priorities for photonic systems.
Quantum walks are powerful kernels for developing new and useful quantum
algorithms. Here we realize large-scale quantum walks using a fully
programmable photonic quantum computing system. The system integrates a silicon
quantum photonic chip, enabling the simulation of quantum walk dynamics on
graphs with up to 400 vertices and possessing full programmability over quantum
walk parameters, including the particle property, initial state, graph
structure, and evolution time. In the 400-dimensional Hilbert space, the
average fidelity of random entangled quantum states after the whole on-chip
circuit evolution reaches as high as 94.29$\pm$1.28$\%$. With the system, we
demonstrated exponentially faster hitting and quadratically faster mixing
performance of quantum walks over classical random walks, achieving more than
two orders of magnitude of enhancement in the experimental hitting efficiency
and almost half of the reduction in the experimental evolution time for mixing.
We utilize the system to implement a series of quantum applications, including
measuring the centrality of scale-free networks, searching targets on
Erd\"{o}s-R\'{e}nyi networks, distinguishing non-isomorphic graph pairs, and
simulating the topological phase of higher-order topological insulators. Our
work shows one feasible path for quantum photonics to address applications of
practical interests in the near future.
