Quantum Searches in a Hard 2SAT Ensemble

Abstract

Using a recently constructed ensemble of hard 2SAT realizations, that has a unique ground-state we calculate for the quantized theory the median gap correlation length values $\xi_{GAP}$ along the direction of the quantum adiabatic control parameter $\lambda$. We use quantum annealing (QA) with transverse field and a linear time schedule in the adiabatic control parameter $\lambda$. The gap correlation length diverges exponentially $\xi_{\rm GAP} \propto {\rm exp} [+r_{\rm GAP}N]$ in the median with a rate constant $r_{\rm GAP}=0.553(6)$, while the run time diverges exponentially $\tau_{\rm QA} \propto {\rm exp} [+r_{\rm QA}N]$ with $r_{\rm QA}=1.184(16)$. Simulated classical annealing (SA) exhibits a run time rate constant $r_{\rm SA}=0.340(5)$ that is small and thus finds ground-states exponentially faster than QA. There are no quantum speedups in ground state searches on constant energy surfaces that have exponentially large volume. We also determine gap correlation length distribution functions $P(\xi_{\rm GAP})d\xi_{\rm GAP} \approx W_k$ over the ensemble that at $N=18$ are close to Weibull functions $W_k$ with $k \approx 1.2$ i.e., the problems show thin catastrophic tails in $\xi_{\rm GAP}$. The inferred success probability distribution functions of the quantum annealer turn out to be bimodal.