[Show abstract][Hide abstract] ABSTRACT: 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 rather trivial: Spin glasses
on Chimera display a zero-temperature second-order transition. This means that
benchmarking classical and quantum optimization methods using spin glasses on
the Chimera graph represents a typically easy exercise in optimization. We
propose stronger benchmarks by embedding nontrivial 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.
[Show abstract][Hide abstract] ABSTRACT: Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system's state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment's effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system.
[Show abstract][Hide abstract] ABSTRACT: We present new MCMC algorithms for computing the posterior distributions and
expectations of the unknown variables in undirected graphical models with
regular structure. For demonstration purposes, we focus on Markov Random Fields
(MRFs). By partitioning the MRFs into non-overlapping trees, it is possible to
compute the posterior distribution of a particular tree exactly by conditioning
on the remaining tree. These exact solutions allow us to construct efficient
blocked and Rao-Blackwellised MCMC algorithms. We show empirically that tree
sampling is considerably more efficient than other partitioned sampling schemes
and the naive Gibbs sampler, even in cases where loopy belief propagation fails
to converge. We prove that tree sampling exhibits lower variance than the naive
Gibbs sampler and other naive partitioning schemes using the theoretical
measure of maximal correlation. We also construct new information theory tools
for comparing different MCMC schemes and show that, under these, tree sampling
is more efficient.
[Show abstract][Hide abstract] ABSTRACT: We propose a new Monte Carlo algorithm for complex discrete distributions.
The algorithm is motivated by the N-Fold Way, which is an ingenious
event-driven MCMC sampler that avoids rejection moves at any specific state.
The N-Fold Way can however get "trapped" in cycles. We surmount this problem by
modifying the sampling process. This correction does introduce bias, but the
bias is subsequently corrected with a carefully engineered importance sampler.
[Show abstract][Hide abstract] ABSTRACT: This paper addresses the problem of sampling from binary distributions with
constraints. In particular, it proposes an MCMC method to draw samples from a
distribution of the set of all states at a specified distance from some
reference state. For example, when the reference state is the vector of zeros,
the algorithm can draw samples from a binary distribution with a constraint on
the number of active variables, say the number of 1's. We motivate the need for
this algorithm with examples from statistical physics and probabilistic
inference. Unlike previous algorithms proposed to sample from binary
distributions with these constraints, the new algorithm allows for large moves
in state space and tends to propose them such that they are energetically
favourable. The algorithm is demonstrated on three Boltzmann machines of
varying difficulty: A ferromagnetic Ising model (with positive potentials), a
restricted Boltzmann machine with learned Gabor-like filters as potentials, and
a challenging three-dimensional spin-glass (with positive and negative
[Show abstract][Hide abstract] ABSTRACT: This paper introduces a new specialized algorithm for equilibrium Monte Carlo
sampling of binary-valued systems, which allows for large moves in the state
space. This is achieved by constructing self-avoiding walks (SAWs) in the state
space. As a consequence, many bits are flipped in a single MCMC step. We name
the algorithm SARDONICS, an acronym for Self-Avoiding Random Dynamics on
Integer Complex Systems. The algorithm has several free parameters, but we show
that Bayesian optimization can be used to automatically tune them. SARDONICS
performs remarkably well in a broad number of sampling tasks: toroidal
ferromagnetic and frustrated Ising models, 3D Ising models, restricted
Boltzmann machines and chimera graphs arising in the design of quantum
[Show abstract][Hide abstract] ABSTRACT: This paper proposes a new randomized strategy for adaptive MCMC using
Bayesian optimization. This approach applies to non-differentiable objective
functions and trades off exploration and exploitation to reduce the number of
potentially costly objective function evaluations. We demonstrate the strategy
in the complex setting of sampling from constrained, discrete and densely
connected probabilistic graphical models where, for each variation of the
problem, one needs to adjust the parameters of the proposal mechanism
automatically to ensure efficient mixing of the Markov chains.
[Show abstract][Hide abstract] ABSTRACT: Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins; however, finding such a ground state remains computationally difficult. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing. If it could be harnessed, quantum annealing might improve on known methods for solving certain types of problem. However, physical investigation of quantum annealing has been largely confined to microscopic spins in condensed-matter systems. Here we use quantum annealing to find the ground state of an artificial Ising spin system comprising an array of eight superconducting flux quantum bits with programmable spin-spin couplings. We observe a clear signature of quantum annealing, distinguishable from classical thermal annealing through the temperature dependence of the time at which the system dynamics freezes. Our implementation can be configured in situ to realize a wide variety of different spin networks, each of which can be monitored as it moves towards a low-energy configuration. This programmable artificial spin network bridges the gap between the theoretical study of ideal isolated spin networks and the experimental investigation of bulk magnetic samples. Moreover, with an increased number of spins, such a system may provide a practical physical means to implement a quantum algorithm, possibly allowing more-effective approaches to solving certain classes of hard combinatorial optimization problems.
[Show abstract][Hide abstract] ABSTRACT: Adiabatic quantum optimization offers a new method for solving hard
optimization problems. In this paper we calculate median adiabatic times (in
seconds) determined by the minimum gap during the adiabatic quantum
optimization for an NP-hard Ising spin glass instance class with up to 128
binary variables. Using parameters obtained from a realistic superconducting
adiabatic quantum processor, we extract the minimum gap and matrix elements
using high performance Quantum Monte Carlo simulations on a large-scale
Internet-based computing platform. We compare the median adiabatic times with
the median running times of two classical solvers and find that, for the
considered problem sizes, the adiabatic times for the simulated processor
architecture are about 4 and 6 orders of magnitude shorter than the two
classical solvers' times. This shows that if the adiabatic time scale were to
determine the computation time, adiabatic quantum optimization would be
significantly superior to those classical solvers for median spin glass
problems of at least up to 128 qubits. We also discuss important additional
constraints that affect the performance of a realistic system.
Quantum Information Processing 06/2010; · 1.75 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: CUDA and OpenCL are two different frameworks for GPU programming. OpenCL is
an open standard that can be used to program CPUs, GPUs, and other devices from
different vendors, while CUDA is specific to NVIDIA GPUs. Although OpenCL
promises a portable language for GPU programming, its generality may entail a
performance penalty. In this paper, we use complex, near-identical kernels from
a Quantum Monte Carlo application to compare the performance of CUDA and
OpenCL. We show that when using NVIDIA compiler tools, converting a CUDA kernel
to an OpenCL kernel involves minimal modifications. Making such a kernel
compile with ATI's build tools involves more modifications. Our performance
tests measure and compare data transfer times to and from the GPU, kernel
execution times, and end-to-end application execution times for both CUDA and
[Show abstract][Hide abstract] ABSTRACT: This paper describes an algorithm for selecting parameter values (e.g. temperature values) at which to measure equilibrium properties with Parallel Tempering Monte Carlo simulation. Simple approaches to choosing parameter values can lead to poor equilibration of the simulation, especially for Ising spin systems that undergo $1^st$-order phase transitions. However, starting from an initial set of parameter values, the careful, iterative respacing of these values based on results with the previous set of values greatly improves equilibration. Example spin systems presented here appear in the context of Quantum Monte Carlo. Comment: Accepted in International Journal of Modern Physics C 2010, http://www.worldscinet.com/ijmpc
International Journal of Modern Physics C 04/2010; · 0.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper presents two conceptually simple methods for parallelizing a
Parallel Tempering Monte Carlo simulation in a distributed volunteer computing
context, where computers belonging to the general public are used. The first
method uses conventional multi-threading. The second method uses CUDA, a
graphics card computing system. Parallel Tempering is described, and challenges
such as parallel random number generation and mapping of Monte Carlo chains to
different threads are explained. While conventional multi-threading on CPUs is
well-established, GPGPU programming techniques and technologies are still
developing and present several challenges, such as the effective use of a
relatively large number of threads. Having multiple chains in Parallel
Tempering allows parallelization in a manner that is similar to the serial
algorithm. Volunteer computing introduces important constraints to high
performance computing, and we show that both versions of the application are
able to adapt themselves to the varying and unpredictable computing resources
of volunteers' computers, while leaving the machines responsive enough to use.
We present experiments to show the scalable performance of these two
approaches, and indicate that the efficiency of the methods increases with
bigger problem sizes.
International Journal of High Performance Computing Applications 03/2010; 24(3). · 1.30 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Much of the current focus in high-performance computing is on multi-threading, multi-computing, and graphics processing unit (GPU) computing. However, vectorization and non-parallel optimization techniques, which can often be employed additionally, are less frequently discussed. In this paper, we present an analysis of several optimizations done on both central processing unit (CPU) and GPU implementations of a particular computationally intensive Metropolis Monte Carlo algorithm. Explicit vectorization on the CPU and the equivalent, explicit memory coalescing, on the GPU are found to be critical to achieving good performance of this algorithm in both environments. The fully-optimized CPU version achieves a 9× to 12× speedup over the original CPU version, in addition to speedup from multi-threading. This is 2× faster than the fully-optimized GPU version, indicating the importance of optimizing CPU implementations.
Journal of Computational Physics 01/2010; · 2.14 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation (LDA). Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a "population-based" sequential Monte Carlo (SMC) method, but unlike ex- isting SMC methods there is no need to design the artificial sequence of dis- tributions. Significantly, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. We conduct experiments on a difficult inference prob- lem in population genetics, a problem that is related to inference for LDA. The results of these experiments suggest that our method can offer improvements in stability and accuracy over existing methods, and at a comparable cost.
Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009. Proceedings of a meeting held 7-10 December 2009, Vancouver, British Columbia, Canada.; 01/2009
[Show abstract][Hide abstract] ABSTRACT: This paper presents a new sampling algorithm for approximating func- tions of variables representable as undirected graphical models of arbi- trary connectivity with pairwise potentials, as well as for estimating the notoriously difcult partition function of the graph. The algorithm ts into the framework of sequential Monte Carlo methods rather than the more widely used MCMC, and relies on constructing a sequence of in- termediate distributions which get closer to the desired one. While the idea of using ìtemperedî proposals is known, we construct a novel se- quence of target distributions where, rather than dropping a global tem- perature parameter, we sequentially couple individual pairs of variables that are, initially, sampled exactly from a spanning tree of the variables. We present experimental results on inference and estimation of the parti- tion function for sparse and densely-connected graphs.
[Show abstract][Hide abstract] ABSTRACT: This paper proposes numerical algorithms for reducing the computational cost of semi-supervised and active learning procedures for visually guided mobile robots from O(M<sup>3</sup>to O(M), while reducing the storage requirements from M<sup>2</sup>to M . This reduction in cost is essential for real-time interaction with mobile robots. The considerable speed ups are achieved using Krylov subspace methods and the fast Gauss transform. Although these state-of-the-art numerical algorithms are known, their application to semi-supervised learning, active learning and mobile robotics is new and should be of interest and great value to the robotics community. We apply our fast algorithms to interactive object recognition on Sony’s ERS-7 Aibo. We provide comparisons that clearly demonstrate remarkable improvements in computational speed.
Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on; 05/2005
[Show abstract][Hide abstract] ABSTRACT: This thesis contains the author’s work in and contributions to the field of Monte Carlo sampling for undirected graphical models, a class of statistical model commonly used in machine learning, computer vision, and spatial statistics; the aim is to be able to use the methodology and resultant samples to estimate integrals of functions of the variables in the model. Over the course of the study, three different but related methods were proposed and have appeared as research papers. The thesis consists of an introductory chapter discussing the models considered, the problems involved, and a general outline of Monte Carlo methods. The three subsequent chapters contain versions of the published work. The second chapter, which has appeared in (Hamze and de Freitas 2004), is a presentation of new MCMC algorithms for computing the posterior distributions and expectations of the unknown variables in undirected graphical models with regular structure. For demonstration purposes, we focus on Markov Random Fields (MRFs). By partitioning the MRFs into non-overlapping trees, it is possible to compute the posterior distribution of a particular tree exactly by conditioning on the remaining tree. These exact solutions allow us to construct efficient blocked and Rao-Blackwellised MCMC algorithms. We show empirically that tree sampling is considerably more efficient than other partitioned sampling schemes and the naive Gibbs sampler, even in cases where loopy belief propagation fails to converge. We prove that tree sampling exhibits lower variance than the naive Gibbs sampler and other naive partitioning schemes using the theoretical measure of maximal correlation. We also construct new information theory tools for comparing different MCMC schemes and show that, under these, tree sampling is more efficient. Although the work discussed in Chapter 2 exhibited promise on the class of graphs to which it was suited, there are many cases where limiting the topology is quite a handicap. The work in Chapter 3 was an exploration in an alternative methodology for approximating functions of variables representable as undirected graphical models of arbitrary connectivity with pairwise potentials, as well as for estimating the notoriously difficult partition function of the graph. The algorithm, published in (Hamze and de Freitas 2005), fits into the framework of sequential Monte Carlo methods rather than the more widely used MCMC, and relies on constructing a sequence of intermediate distributions which get closer to the desired one. While the idea of using “tempered” proposals is known, we construct a novel sequence of target distributions where, rather than dropping a global temperature parameter, we sequentially couple individual pairs of variables that are, initially, sampled exactly from a spanning treeof the variables. We present experimental results on inference and estimation of the partition function for sparse and densely-connected graphs. The final contribution of this thesis, presented in Chapter 4 and also in (Hamze and de Freitas 2007), emerged from some empirical observations that were made while trying to optimize the sequence of edges to add to a graph so as to guide the population of samples to the high-probability regions of the model. Most important among these observations was that while several heuristic approaches, discussed in Chapter 1, certainly yielded improvements over edge sequences consisting of random choices, strategies based on forcing the particles to take large, biased random walks in the state-space resulted in a more efficient exploration, particularly at low temperatures. This motivated a new Monte Carlo approach to treating complex discrete distributions. The algorithm is motivated by the N-Fold Way, which is an ingenious event-driven MCMC sampler that avoids rejection moves at any specific state. The N-Fold Way can however get “trapped” in cycles. We surmount this problem by modifying the sampling process to result in biased state-space paths of randomly chosen length. This alteration does introduce bias, but the bias is subsequently corrected with a carefully engineered importance sampler.
[Show abstract][Hide abstract] ABSTRACT: We propose efficient MCMC tree samplers for random fields and factor graphs. Our tree sampling approach combines elements of Monte Carlo simulation as well as exact belief propagation. It requires that the graph be partitioned into trees first. The partition can be generated by hand or automatically using a greedy graph algorithm. The tree partitions allow us to perform exact inference on each tree. This enables us to implement efficient Rao-Blackwellised blocked Gibbs samplers, where each tree is sampled by conditioning on the other trees. We use information theory tools to rank MCMC algorithms corresponding to different partitioning schemes.