Fig. 8: Time to solution scaling for 3-SAT under averaged and heralded quantum algorithms.
From: Solving k–SAT problems with generalized quantum measurement
The 3-SAT time to solution TTS99% defined by Eq. (42) is plotted as a function of the number of qubits, for different values of the total dragging time Tf and for different time duration Δt of a single clause measurement under both Algorithm 1 with average dynamics (upper (a), (b), (c)) and Algorithm 3 with heralded dynamics (lower (d), (e), (f)). The 3-SAT instances are randomly generated at α ≈ αc with a unique solution. We have set τ = 1 and used a linear schedule θ(t) = πt/2Tf for all simulations. We use Δt = 0.01 (a, d), Δt = 0.1 (b, e), and Δt = 1 (c, f), all in units of τ. For the heralded algorithm, we have additionally set \({T}_{be}=\max \{2\tau ,0.1{T}_{f}\}\), Tmin = 5τ, and \({r}_{th}=-2.5/\sqrt{{T}_{be}}\). The data for the heralded algorithm are averaged over more than 10,000 trajectories while the data for averaged algorithm are averaged over more than 150 trajectories. We see that TTS99% generally increases exponentially as a function of the number of qubits. Notice the appearance of line-crossings, which is a signature of non-monotonic dependence of TTS99% on Tf that indicates the possible existence of optimal values of TTS99%. This figure continues with larger values of Δt in Fig. 9.
