Fig. 3: Constant gain, adiabatic gain, and chaotic annealing methods for solving an Ising problem instance.
From: Combinatorial optimization with photonics-inspired clock models
The optimization results for the Ising problem on the “g05 100.6” graph with 100 nodes for constant gain (a–c), adiabatic gain (d–f), and chaotic annealing (g–i). The first column depicts the time-evolution of the Hamiltonian \(H=\sum {\kappa }_{ij}\cos ({\phi }_{i}-{\phi }_{j})/2\) (where κij is the coupling strength of the ith and jth oscillators and ϕi is the ith oscillator’s steady-state phase) for 1000 random initial conditions. The second column shows the distribution of the Ising energy H = ∑κijδ(ϕi, ϕj)/2 associated with the steady-state solutions. The third column exemplifies the dynamics of the phases ϕ for one simulation instant. For better visibility, the phase of each oscillator is shown with a different color. In (a–c), the gain is set to g = 2.5 and the step size is set to Δ = 0.01. In (d–f), the gain is linearly increased from g = 0 to g = 2.5 and the step size is set to Δ = 0.01. In (g–i), the gain is set to g = 2.5 and the step size is linearly decreased from Δ = 10 × 0.01 to Δ = 0.01. In the first column, the transient energy is calculated based on the XY Hamiltonian which transforms to the Ising Hamiltonian as the phases become binary after several iterations. These figures show the the minimum-to-maximum range of the energy distribution for all random initializations (light shaded area), the first standard deviation ('1 std dev') of the associated distribution (dark shaded area), and the average energy Have (blue line) as a function of the iteration number.
