Guaranteed mapping from Simulated Bifurcation steady states to Ising ground states

Determine whether there exists a function f that, for any Ising instance H(s)=∑_{⟨i,j⟩} J_{ij} s_i s_j + ∑_i h_i s_i, maps the steady-state positions q_i^s produced by the modified Simulated Bifurcation Machine (with perfectly inelastic walls at |q_i|=1 and discretized interaction term ∑_j J_{ij} sign(q_j)) to a binary spin configuration s_i ∈ {−1,+1} that is guaranteed to be the ground state of H(s). If such a function exists, construct f and specify the conditions under which this guarantee holds.

Background

The Simulated Bifurcation Machine (SBM) evolves a classical nonlinear dynamical system derived from a Hamiltonian intended to approximate the Ising energy landscape via bifurcation. A modified SBM variant replaces the quartic term with perfectly inelastic walls at |q_i|=1 and discretizes the Ising coupling contribution using sign(q_j), enabling halting near a(t)=a_0 with the system in a local minimum.

In practice, solutions are extracted by discretizing continuous steady-state variables q_i^s to spins s_i using a mapping f, commonly f(q_i^s)=sign(q_i^s). However, there is no general guarantee that this mapping returns the ground state of the original Ising Hamiltonian for arbitrary instances, motivating the explicit open question of whether a guaranteed mapping exists and, if so, how to construct it.