Performance of quantum annealing for 2-SAT problems with multiple satisfying assignments

Abstract

Using a specially constructed set of hard 2-SAT problems with four satisfying assignments, we study the scaling and sampling performance of numerical simulation of quantum annealing as well as that of the physical quantum annealers offered by D-Wave. To this end, we use both the standard quantum annealing and reverse annealing protocols in both our simulations and on the D-Wave quantum annealer. In the case of ideal quantum annealing the sampling behavior can be explained by perturbation theory and the scaling behavior of the time to solution depends on the scaling behavior of the minimum energy gap between the ground state and the first excited state of the annealing Hamiltonian. The corresponding results from the D-Wave quantum annealers do not fit to this ideal picture, but suggest that the scaling of the time to solution from the quantum annealers matches those calculated from the equilibrium probability distribution.

Figures (13)

• Figure 1: (Color online) Average number of satisfying assignments $\mu$ for the 2-SAT problems as a function of the problem size where $6 \leq N \leq 20$ and $M = N+1$.
• Figure 2: (Color online) Average number of satisfying assignments $\mu$ for the 2-SAT problems as a function of the number of clauses of the problem size where $M=N+1$ for problems with 1, 2, and 4 satisfying assignments.
• Figure 3: (Color online) Average degeneracy of the first excited states (FES) of the problem Hamiltonians corresponding to the 2-SAT problems as a function of the problem size where $6 \leq N \leq 20$ and $M = N+1$.
• Figure 4: (Color online) Energy spectrum of a 14-variable 2-SAT problem Hamiltonian labeled as the problem "230" with four degenerate ground states.
• Figure 5: (Color online) Numerically obtained scaling of median TTS99 as a function of the number of variables for $T_A=10$ (square), $T_A=100$ (circle), and $T_A=1000$ (triangle).