On the complexity of probabilistic trials for hidden satisfiability problems

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC '13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.Comment: 24 pages, 2 figures. To appear in the 41st International Symposium on Mathematical Foundations of Computer Scienc

Quantum Searches in a Hard 2SAT Ensemble

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.Comment: non

Monte Carlo Search for Very Hard KSAT Realizations for Use in Quantum Annealing

Using powerful Multicanonical Ensemble Monte Carlo methods from statistical physics we explore the realization space of random K satisfiability (KSAT) in search for computational hard problems, most likely the 'hardest problems'. We search for realizations with unique satisfying assignments (USA) at ratio of clause to spin number $\alpha=M/N$ that is minimal. USA realizations are found for $\alpha$-values that approach $\alpha=1$ from above with increasing number of spins $N$. We consider small spin numbers in $2 \le N \le 18$. The ensemble mean exhibits very special properties. We find that the density of states of the first excited state with energy one $\Omega_1=g(E=1)$ is consistent with an exponential divergence in $N$: $\Omega_1 \propto {\rm exp} [+rN]$. The rate constants for $K=2,3,4,5$ and $K=6$ of KSAT with USA realizations at $\alpha=1$ are determined numerically to be in the interval $r=0.348$ at $K=2$ and $r=0.680$ at $K=6$. These approach the unstructured search value ${\rm ln}2$ with increasing $K$. Our ensemble of hard problems is expected to provide a test bed for studies of quantum searches with Hamiltonians that have the form of general Ising models.Comment: non