Electronic inductorless parametric oscillators for compact large-scale Ising machines

Parametric Oscillator (PO)-based Ising machines (IMs) are promising candidates for electronic solvers. They are able to solve nondeterministic polynomial-time hard (NP-hard) combinatorial optimization problems that current von Neumann architectures struggle to address. PO-based IMs relying on coupled networks of parametric frequency divider (PFD) circuits were recently demonstrated, showing the capability to achieve power levels per spin in the range of hundreds of nanowatts. This has created an unprecedented opportunity to realize electronic Ising machines capable of solving NP-hard problems with a million variables while requiring <1 W. However, a major limitation still remains: due to its nonlinear dynamics, a typical PFD circuit requires multiple inductors to enable low-power activation of the period-doubling mechanism that is the basis of its PO-like operation, making it difficult to achieve wafer-scale Ising machines suited to solve problems with millions of variables. To overcome this limitation and enable large-scale PFD-based IMs without passive inductors, we propose a novel inductorless PFD architecture. This architecture leverages gyrator-based circuits to achieve inductive elements out of active networks of resistors and capacitors, allowing to achieve a low-power threshold for the activation of the period-doubling mechanism without requiring inductors. The design approach creates a path to significantly reduce circuit area and improve integration capability. While being demonstrated using lumped electronic components on a printed circuit board, this architecture can be replicated using complementary metal–oxide–semiconductor electronics in the future, providing a promising path toward large-scale, fully integrated PFD-based Ising machines.