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Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N. The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02 N) >> 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity., Comment: 8 pages, 5 figures

We simulate the quantum adiabatic algorithm (QAA) for the exact cover problem for sizes up to N=256 using quantum Monte Carlo simulations incorporating parallel tempering. At large N we find that some instances have a discontinuous (first order) quantum phase transition during the evolution of the QAA. This fraction increases with increasing N and may tend to 1 for N -> infinity., Comment: 5 pages, 3 figures. Replaced with published version; two figures slightly changed and some small changes to the text

We study the typical (median) value of the minimum gap in the quantum version of the Exact Cover problem using Quantum Monte Carlo simulations, in order to understand the complexity of the quantum adiabatic algorithm (QAA) for much larger sizes than before. For a range of sizes, N <= 128, where the classical Davis-Putnam algorithm shows exponential median complexity, the QAA shows polynomial median complexity. The bottleneck of the algorithm is an isolated avoided crossing point of a Landau-Zener type (collision between the two lowest energy levels only)., Comment: 4 pages, 5 figures

We study the quantum version of the random $K$-Satisfiability problem in the presence of the external magnetic field $\Gamma$ applied in the transverse direction. We derive the replica-symmetric free energy functional within static approximation and the saddle-point equation for the order parameter: the distribution $P[h(m)]$ of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in $\Gamma$ magnetizations $m \approx 0$ become relevant in addition to purely classical values of $m \approx \pm 1$. Self-consistency equations for the order parameter are solved numerically using Quasi Monte Carlo method for K=3. It is shown that for an arbitrarily small $\Gamma$ quantum fluctuations destroy the phase transition present in the classical limit $\Gamma=0$, replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random $K$-Satisfiability problem is briefly revisited. It is shown that the phase transition at T=0 predicted by the replica-symmetric theory is of continuous type with atypical critical exponents., Comment: 35 pages, 23 figures; changed abstract, improved discussion in the introduction, added references, corrected typos

In this paper we show that the performance of the quantum adiabatic algorithm is determined by phase transitions in underlying problem in the presence of transverse magnetic field $\Gamma$. We show that the quantum version of random Satisfiability problem with 3 bits in a clause (3-SAT) has a first-order quantum phase transition. We analyze the phase diagram $\gamma=\gamma(\Gamma)$ where $\gamma$ is an average number of clauses per binary variable in 3-SAT. The results are obtained in a closed form assuming replica symmetry and neglecting time correlations at small values of the transverse field $\Gamma$. In the limit of $\Gamma=0$ the value of $\gamma(0)\approx$ 5.18 corresponds to that given by the replica symmetric treatment of a classical random 3-SAT problem. We demonstrate the qualitative similarity between classical and quantum versions of this problem., Comment: 30 pages, 7 figures

We propose an approach suitable for solving NP-complete problems via adiabatic quantum computation with an architecture based on a lattice of interacting spins (qubits) driven by locally adjustable effective magnetic fields. Interactions between qubits are assumed constant and instance-independent, programming is done only by changing local magnetic fields. Implementations using qubits coupled by magnetic-, electric-dipole and exchange interactions are discussed., Comment: 10 pages, 10 figures, reference added

Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per clause. Determining the location of satisfiability threshold $\alpha_c=M/N$ for a number of difficult combinatorial problems is a major open problem in the theory of random graphs. The method is based on identification of the core -- a subexpression (subgraph) that has the same satisfiability properties as the original expression. We formulate self-consistency equations that determine macroscopic parameters of the core and compute an improved annealing bound. We illustrate the method for three sample problems: K-XOR-SAT, K-SAT and positive 1-in-K-SAT., Comment: 31 pages, 6 figures

In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, $\gamma=M/N$. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (versus only energy) is used, and are able to show the existence of a dynamic threshold $\gamma=\gamma_d$ starting with some value of K -- the number of variables in each clause. Beyond dynamic threshold, the algorithm should take exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz. We have been able to map the ensemble of random graphs onto another ensemble with fluctuations significantly reduced. This enabled us to obtain tight upper bounds on satisfiability transition and to recompute the dynamical transition using the extended set of landscapes., Comment: 41 pages, 10 figures; added a paragraph on paper's organization to the introduction, fixed references

In this paper we analyze the performance of the Quantum Adiabatic Evolution (QAE) algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, gamma = M / N. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (verses only energy) is used, and are able to show the existence of a dynamic threshold gamma = gammad, beyond which QAE should take an exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz.