Fig. 1: Four methods of incorporating external fields applied to an exemplary COP.

a Example of a 3-spin COP with ground state configuration (–1,1,–1). The spins are indicated as σ₀, σ₁, and σ₂. Spin-spin couplings J_ij are shown as wavy lines, and external fields h_i as red arrows. The ground state is indicated with blue arrows. (b)-(e) Spin evolution under the methods described in Eqs. (3) and (6) to (8), respectively. In b, the initial direction of the spins (dotted lines) is given by h_i/(1 − α), independently of the spin-spin couplings J_ij. Methods in c–e are designed to prevent the external fields from dominating the spin-spin couplings. The inset in e shows a zoom-in near β = 0.