Fig. 4: Properties of the Lindblad dynamics for the 2–SAT problem of Eq. (33b) illustrated in Fig. 3, on scanning through a range of values of measurement strength Γ = 1/(4τ).
From: Solving k–SAT problems with generalized quantum measurement
The same measurement strengths and linear schedule θ = π t/2 Tf are used through all four panels. The time axis is displayed here in units of the total evolution time Tf. a Illustrates the reduced density matrix evolution for qubit 1 (teal) and qubit 2 (purple), shown for both in the xz Bloch plane; results are shown for a variety of measurement strengths spanning from Γ Tf = 1 (pale) to Γ Tf = 1 × 104 (dark). b Shows the corresponding time traces of the local z coordinates for the two qubits. c shows the behavior of the separability (\(1-{\mathcal{C}}\), where \({\mathcal{C}}\) is the concurrence127) throughout the Lindblad evolution. d Plots the purity of the two-qubit state as it undergoes Lindblad evolution. These results show that in the adiabatic regime Γ Tf ≫ 1, we converge towards a pure-state and completely separable evolution (see Eq. (35)) that deterministically generates the solution of this very simple 2-SAT problem, i.e., z1 = 1 and z2 = −1, corresponding to (b1, b2) = (0, 1) as expected from Eq. (33b). We showed in section “Convergence in the Zeno limit” that this is in fact a general feature of k-SAT problems with a unique solution in this measurement--driven algorithm. Even far from the adiabatic regime (Γ T ~ 1), we still see that some information about this solution manifests itself on average, since the reduced density matrix traces (local z coordinates in (b)) still drift discernibly apart and move towards their respective solution states.
