Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines

Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a decohering thermal bath. Therefore, we study the finite-temperature critical behavior of the standard benchmark problem used to assess the computational capabilities of these complex machines. We simulate both random-bond Ising models and spin glasses with bimodal and Gaussian disorder on the D-Wave Chimera topology. Our results show that while the worst-case complexity of finding a ground state of an Ising spin glass on the Chimera graph is not polynomial, the finite-temperature phase space is likely rather simple: Spin glasses on Chimera have only a zero-temperature transition. This means that benchmarking optimization methods using spin glasses on the Chimera graph might not be the best benchmark problems to test quantum speedup. We propose alternative benchmarks by embedding potentially harder problems on the Chimera topology. Finally, we also study the (reentrant) disorder-temperature phase diagram of the random-bond Ising model on the Chimera graph and show that a finite-temperature ferromagnetic phase is stable up to 19.85(15)% antiferromagnetic bonds. Beyond this threshold the system only displays a zero-temperature spin-glass phase. Our results therefore show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealing machines.Comment: 8 pages, 5 figures, 1 tabl

The Random-bond Potts model in the large-q limit

We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random Potts model. The optimization problem of finding the dominant graph, is studied on the square lattice by simulated annealing and by a combinatorial algorithm. Critical exponents of the magnetization and the correlation length are estimated and conformal predictions are compared with numerical results.Comment: 7 pages, 6 figure

Optimization in random field Ising models by quantum annealing

We investigate the properties of quantum annealing applied to the random field Ising model in one, two and three dimensions. The decay rate of the residual energy, defined as the energy excess from the ground state, is find to be $e_{res}\sim \log(N_{MC})^{-\zeta}$ with $\zeta$ in the range $2...6$, depending on the strength of the random field. Systems with ``large clusters'' are harder to optimize as measured by $\zeta$. Our numerical results suggest that in the ordered phase $\zeta=2$ whereas in the paramagnetic phase the annealing procedure can be tuned so that $\zeta\to6$.Comment: 7 pages (2 columns), 9 figures, published with minor changes, one reference updated after the publicatio

A Method to Change Phase Transition Nature -- Toward Annealing Method --

In this paper, we review a way to change nature of phase transition with annealing methods in mind. Annealing methods are regarded as a general technique to solve optimization problems efficiently. In annealing methods, we introduce a controllable parameter which represents a kind of fluctuation and decrease the parameter gradually. Annealing methods face with a difficulty when a phase transition point exists during the protocol. Then, it is important to develop a method to avoid the phase transition by introducing a new type of fluctuation. By taking the Potts model for instance, we review a way to change the phase transition nature. Although the method described in this paper does not succeed to avoid the phase transition, we believe that the concept of the method will be useful for optimization problems.Comment: 27 pages, 3 figures, revised version will appear in proceedings of Kinki University Quantum Computing Series Vo.

Topological Defects, Orientational Order, and Depinning of the Electron Solid in a Random Potential

We report on the results of molecular dynamics simulation (MD) studies of the classical two-dimensional electron crystal in the presence disorder. Our study is motivated by recent experiments on this system in modulation doped semiconductor systems in very strong magnetic fields, where the magnetic length is much smaller than the average interelectron spacing $a_0$, as well as by recent studies of electrons on the surface of helium. We investigate the low temperature state of this system using a simulated annealing method. We find that the low temperature state of the system always has isolated dislocations, even at the weakest disorder levels investigated. We also find evidence for a transition from a hexatic glass to an isotropic glass as the disorder is increased. The former is characterized by quasi-long range orientational order, and the absence of disclination defects in the low temperature state, and the latter by short range orientational order and the presence of these defects. The threshold electric field is also studied as a function of the disorder strength, and is shown to have a characteristic signature of the transition. Finally, the qualitative behavior of the electron flow in the depinned state is shown to change continuously from an elastic flow to a channel-like, plastic flow as the disorder strength is increased.Comment: 31 pages, RevTex 3.0, 15 figures upon request, accepted for publication in Phys. Rev. B., HAF94MD

Cut Size Statistics of Graph Bisection Heuristics

We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve approximately the graph bisection problem. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by ``local'' algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends towards a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure which takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics.Comment: 17 pages, 5 figures, submitted to SIAM Journal on Optimization also available at http://ipnweb.in2p3.fr/~martin