Linear time algorithm for quantum 2SAT

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors $\Pi_{ij}$ on a system of $n$ qubits, and the task is to decide whether the Hamiltonian $H=\sum \Pi_{ij}$ has a 0-eigenvalue, or it is larger than $1/n^\alpha$ for some $\alpha=O(1)$. The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin $\frac{1}{2}$, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is $O(n^4)$. In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.Comment: 20 page

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

Renyi entropies as a measure of the complexity of counting problems

Counting problems such as determining how many bit strings satisfy a given Boolean logic formula are notoriously hard. In many cases, even getting an approximate count is difficult. Here we propose that entanglement, a common concept in quantum information theory, may serve as a telltale of the difficulty of counting exactly or approximately. We quantify entanglement by using Renyi entropies S(q), which we define by bipartitioning the logic variables of a generic satisfiability problem. We conjecture that S(q\rightarrow 0) provides information about the difficulty of counting solutions exactly, while S(q>0) indicates the possibility of doing an efficient approximate counting. We test this conjecture by employing a matrix computing scheme to numerically solve #2SAT problems for a large number of uniformly distributed instances. We find that all Renyi entropies scale linearly with the number of variables in the case of the #2SAT problem; this is consistent with the fact that neither exact nor approximate efficient algorithms are known for this problem. However, for the negated (disjunctive) form of the problem, S(q\rightarrow 0) scales linearly while S(q>0) tends to zero when the number of variables is large. These results are consistent with the existence of fully polynomial-time randomized approximate algorithms for counting solutions of disjunctive normal forms and suggests that efficient algorithms for the conjunctive normal form may not exist.Comment: 13 pages, 4 figure

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

High Fidelity Adiabatic Quantum Computation via Dynamical Decoupling

We introduce high-order dynamical decoupling strategies for open system adiabatic quantum computation. Our numerical results demonstrate that a judicious choice of high-order dynamical decoupling method, in conjunction with an encoding which allows computation to proceed alongside decoupling, can dramatically enhance the fidelity of adiabatic quantum computation in spite of decoherence.Comment: 5 pages, 4 figure

Picturing counting reductions with the ZH-calculus

Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting problems in terms of diagrams, to give graphical reductions from #SAT to several related counting problems. Some of these graphical reductions, like to #2SAT, are substantially simpler than known reductions via the matrix permanent. Additionally, our approach allows us to consider the case of counting solutions modulo an integer on equal footing. Finally, since the ZH-calculus was originally introduced to reason about quantum computing, we show that the problem of evaluating ZH-diagrams in the fragment corresponding to the Clifford+T gateset, is in $FP^{\#P}$. Our results show that graphical calculi represent an intuitive and useful framework for reasoning about counting problems

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

On the complexity of trial and error for constraint satisfaction problems

In 2013 Bei, Chen and Zhang introduced a trial and error model of computing, and applied to some constraint satisfaction problems. In this model the input is hidden by an oracle which, for a candidate assignment, reveals some information about a violated constraint if the assignment is not satisfying. In this paper we initiate a systematic study of constraint satisfaction problems in the trial and error model, by adopting a formal framework for CSPs, and defining several types of revealing oracles. Our main contribution is to develop a transfer theorem for each type of the revealing oracle. To any hidden CSP with a specific type of revealing Oracle, the transfer theorem associates another CSP in the normal setting, such that their complexities are polynomial-time equivalent. This in principle transfers the study of a large class of hidden CSPs to the study of normal CSPs. We apply the transfer theorems to get polynomial-time algorithms or hardness results for several families of concrete problems. (C) 2017 Elsevier Inc. All rights reserved