A note on analyzing the stability of oscillator Ising machines

The rich non-linear dynamics of the coupled oscillators (under second harmonic injection) can be leveraged to solve computationally hard problems in combinatorial optimization such as finding the ground state of the Ising Hamiltonian. While prior work on the stability of the so-called Oscillator Ising Machines (OIMs) has used the linearization method, in this letter, the authors present a complementary method to analyze stability using the second-order derivative test of the energy/cost function. The authors establish the equivalence between the two methods, thus augmenting the tool kit for the design and implementation of OIMs. While prior work has focused on the use of linearization methods to analyze stability of oscillator Ising machines (OIMs), here, the authors introduce an alternative approach to analyze the stability of the fixed points using the second-order derivative test of the energy/cost function. The authors' work is uniquely enabled by a novel theoretical relationship between the eigenvalues of the Jacobian matrix and the eigenvalues of the second-order Hessian in OIMs, elucidated in this work. Moreover, the authors' approach is applicable to a broader class of gradient descent systems.