Graphs have provided an expressive mathematical tool to model quantum-mechanical devices and systems. In particular, it has been recently discovered that graph theory can be used to describe and design quantum components, devices, setups and systems, based on the two-dimensional lattice of parametric nonlinear optical crystals and linear optical circuits, different to the standard quantum photonic framework. Realizing such graph-theoretical quantum photonic hardware, however, remains extremely challenging experimentally using conventional technologies. Here we demonstrate a graph-theoretical programmable quantum photonic device in very-large-scale integrated nanophotonic circuits. The device monolithically integrates about 2,500 components, constructing a synthetic lattice of nonlinear photon-pair waveguide sources and linear optical waveguide circuits, and it is fabricated on an eight-inch silicon-on-insulator wafer by complementary metal-oxide-semiconductor processes. We reconfigure the quantum device to realize and process complex-weighted graphs with different topologies and to implement different tasks associated with the perfect matching property of graphs. As two non-trivial examples, we show the generation of genuine multipartite multidimensional quantum entanglement with different entanglement structures, and the measurement of probability distributions proportional to the modulus-squared hafnian (permanent) of the graph's adjacency matrices. This work realizes a prototype of graph-theoretical quantum photonic devices manufactured by very-large-scale integration technologies, featuring arbitrary programmability, high architectural modularity and massive manufacturing scalability.

A graph-theoretical programmable quantum photonic device composed of about 2,500 components is fabricated on a silicon substrate within a 12 mm x 15 mm footprint. It shows the generation, manipulation and certification of genuine multiphoton multidimensional entanglement, as well as the implementations of scattershot and Gaussian boson sampling.