  The need for solving optimization problems is prevalent in a wide range of
physical applications, including neuroscience, network design, biological
systems, socio-economics, and chemical reactions. Many of these are classified
as non-deterministic polynomial-time (NP) hard and thus become intractable to
solve as the system scales to a large number of elements. Recent research
advances in photonics have sparked interest in using a network of coupled
degenerate optical parametric oscillators (DOPO's) to effectively find the
ground state of the Ising Hamiltonian, which can be used to solve other
combinatorial optimization problems through polynomial-time mapping. Here,
using the nanophotonic silicon-nitride platform, we propose a network of
on-chip spatial-multiplexed DOPO's for the realization of a photonic coherent
Ising machine. We demonstrate the generation and coupling of two
microresonator-based DOPO's on a single chip. Through a reconfigurable phase
link, we achieve both in-phase and out-of-phase operation, which can be
deterministically achieved at a fast regeneration speed of 400 kHz with a large
phase tolerance. Our work provides the critical building blocks towards the
realization of a chip-scale photonic Ising machine.
