Just as the brain must infer 3D structure from 2D retinal images, radiologists are tasked with inferring 3D densities from 2D X-rays. Computer simulations suggest that V1 simple cells use lateral inhibition to generate sparse representations that are selective for 3D depth when presented with 2D stereo images and video. Analogously, we cast radiographic inference as a sparse coding problem employing lateral inhibition between binary neurons, resulting in a quadratic unconstrained binary optimization (QUBO)problem suitable for implementation on a quantum annealing D-Wave 2X (1152-qubit)computer. We generated synthetic radiographs by performing discrete Abel transforms on mathematically-defined objects possessing axial (cylindrical)symmetry and whose radially density profile was given by the sum of a randomly-chosen, sparse set of (nearly binary)Fourier components. We used embedding tools to map the above QUBO problem, which involved dense connections between up to 47 Fourier coefficients, onto the very sparsely connected D-Wave chimera. Using quantum inference, we were able to reconstruct reasonably accurate radial density profiles even after adding sufficiently noise to our synthetic radiographs to make inverse Abel transforms untenable. Compared to state-of-the-art classical QUBO solvers, GUROBI and the Hamze-Freitas-Selby algorithm, the quantum D-Wave 2X was orders of magnitude faster for the same final accuracy. Our results indicate a potential strategy for integrating neuromorphic and quantum computing techniques.
Just as the brain must infer 3D structure from 2D retinal images, radiologists are tasked with inferring 3D densities from 2D X-rays. Computer simulations suggest that V1 simple cells use lateral inhibition to generate sparse representations that are selective for 3D depth when presented with 2D stereo images and video. Analogously, we cast radiographic inference as a sparse coding problem employing lateral inhibition between binary neurons, resulting in a quadratic unconstrained binary optimization (QUBO)problem suitable for implementation on a quantum annealing D-Wave 2X (1152-qubit)computer. We generated synthetic radiographs by performing discrete Abel transforms on mathematically-defined objects possessing axial (cylindrical)symmetry and whose radially density profile was given by the sum of a randomly-chosen, sparse set of (nearly binary)Fourier components. We used embedding tools to map the above QUBO problem, which involved dense connections between up to 47 Fourier coefficients, onto the very sparsely connected D-Wave chimera. Using quantum inference, we were able to reconstruct reasonably accurate radial density profiles even after adding sufficiently noise to our synthetic radiographs to make inverse Abel transforms untenable. Compared to state-of-the-art classical QUBO solvers, GUROBI and the Hamze-Freitas-Selby algorithm, the quantum D-Wave 2X was orders of magnitude faster for the same final accuracy. Our results indicate a potential strategy for integrating neuromorphic and quantum computing techniques.
We use a quantum annealing D-Wave 2X (1,152-qubit) computer to generate sparse representations of Canny-filtered, center-cropped 30×30 CIFAR-10 images. Each binary neuron (qubit) represents a feature kernel obtained initially by imprinting on a randomly chosen 5×5 image patch and then adapted via an off-line Hebbian learning protocol using the sparse solutions generated by the D-Wave. When using binary neurons, the energy function is non-convex (multiple local-minima) and finding a global minimum is NP-hard. Quantum annealing provides a strategy for finding sparse representations that correspond to good local minima of a non-convex cost function. To overcome the severe coupling restrictions between physical qubits on the D-Wave Chimera graph, we use embedding tools to achieve approximately all-to-all connectivity across a reduced number of logical qubits. We assess the sparse representations generated by the D-Wave using both the total energy as well as classification accuracy on a subset of the CIFAR-10 database. The D-Wave 2X outperforms two classical state-of-the-art binary solvers, GUROBI and Chimera-inspired algorithm Hamze-Freitas-Selby (HFS). Specifically, the D-Wave 2X yields lower energy sparse solutions within seconds while the largest problems take over 10 hours for both GUROBI and HFS. We obtained cross-validation classification of 31.02% for the first 4K images using 47 features on the quantum D- Wave 2X.
We use a quantum annealing D-Wave 2X computer to obtain solutions to NP-hard sparse coding problems. To reduce the dimensionality of the sparse coding problem to fit on the quantum D-Wave 2X hardware, we passed downsampled MNIST images through a bottleneck autoencoder. To establish a benchmark for classification performance on this reduced dimensional data set, we built two deep convolutional neural networks (DCNNs). The first DCNN used an AlexNet-like architecture and the second a state-of-the-art residual network (RESNET)model, both implemented in TensorFlow. The two DCNNs yielded classification scores of 94.54 ± 0.7% and 98.8 ± 0.1%, respectively. As a control, we showed that both DCNN architectures produced near-state-of-the-art classification performance ( ~ 99%) on the original MNIST images. To obtain a set of optimized features for inferring sparse representations of the reduced dimensional MNIST dataset, we imprinted on a random set of 47 image patches followed by an off-line unsupervised learning algorithm using stochastic gradient descent to optimize for sparse coding. Our single-layer of sparse coding matched the stride and patch size of the first convolutional layer of the AlexNet-like DCNN and contained 47 fully-connected features, 47 being the maximum number of dictionary elements that could be embedded onto the D-Wave 2X hardware. When the sparse representations inferred by the D-Wave 2X were passed to a linear support vector machine, we obtained a classification score of 95.68%. We found that the classification performance supported by quantum inference was maximal at an optimal level of sparsity corresponding to a critical value of the sparsity/reconstruction error trade-off parameter that previous work has associated with a second order phase transition, an observation supported by a free energy analysis of D-Wave energy states. We mimicked a transfer learning protocol by feeding the D-Wave representations into a multilayer perceptron (MLP), yielding 98.48% classification performance. The classification performance supported by a single-layer of quantum inference was superior to that supported by a classical matching pursuit algorithm set to the same level of sparsity. Whereas the classification performance of both DCNNs declined as the number of training examples was reduced, the classification performance supported by quantum inference was insensitive to the number of training examples. We thus conclude that quantum inference supports classification of reduced dimensional MNIST images exceeding that of a size-matched AlexNet-like DCNN and nearly equivalent to a state-of-the-art RESNET DCNN.
Quantum annealers such as the D-Wave 2X computer are designed to natively solve Quadratic Unconstrained Binary Optimization (QUBO) problems to optimality or near optimality. Most NP-hard problems, which are hard for classical computers, can be naturally described as quadratic binary problems that contain a quadratic binary objective function and one or more constraints. Since a QUBO cannot have constraints, each such constraint has to be added to the objective function as a penalty, in order to solve on D-Wave. For a minimization problem, for instance, such penalty can be a quadratic term that gets a value zero, if the constraint is satisfied, and a large value, if it is not. In many cases, however, the penalty can significantly increase the number of quadratic terms in the resulting QUBO and make it too large to embed into the D-Wave hardware. In this paper, we develop an alternative method for formulating and embedding constraints of the type Σi=1sxi=1, which is much more scalable than the existing ones, and analyze the properties of the resulting embeddings.
The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger-sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved using a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per sub-image. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-Wave Systems's quantum annealer, the D-Wave 2000Q. The software-based solvers obtain high-quality solutions comparable to state-of-the-art phase-unwrapping solvers. We are currently working on optimally mapping the problem onto the restricted topology of the quantum annealer to improve the quality of the solution.
Abstract We consider the problem of computing a sparse binary representation of an image. Given an image and an overcomplete, non-orthonormal basis, we aim to find a sparse binary vector indicating the minimal set of basis vectors that when added together best reconstruct the given input. We formulate this problem with an L 2 loss on the reconstruction error, and an L 0 loss on the binary vector enforcing sparsity. First, we solve the sparse representation QUBOs by solving them both on a D-Wave quantum annealer with Pegasus chip connectivity, as well as on the Intel Loihi 2 spiking neuromorphic processor using a stochastic Non-equilibrium Boltzmann Machine (NEBM). Second, using Quantum Evolution Monte Carlo with Reverse Annealing and iterated warm starting on Loihi 2 to evolve the solution quality from the respective machines. We demonstrate that both quantum annealing and neuromorphic computing are suitable for solving binary sparse coding QUBOs.
Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.