A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits -the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test. arXiv:1608.00263v3 [quant-ph]

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.Comment: 13 pages, 7 figures. This revision is to correct an error in the coefficients of identity in Table

We present a new dataset of image caption annotations, Conceptual Captions, which contains an order of magnitude more images than the MS-COCO dataset (Lin et al., 2014) and represents a wider variety of both images and image caption styles. We achieve this by extracting and filtering image caption annotations from billions of webpages. We also present quantitative evaluations of a number of image captioning models and show that a model architecture based on Inception-ResNet-v2 (Szegedy et al., 2016) for image-feature extraction and Transformer (Vaswani et al., 2017) for sequence modeling achieves the best performance when trained on the Conceptual Captions dataset.

Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to Simulated Annealing (SA). For instances with 945 variables, this results in a time-to-99%success-probability that is ∼ 10 8 times faster than SA running on a single processor core. We also compared physical QA with Quantum Monte Carlo (QMC), an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to ∼ 10 8 times faster than an optimized implementation of QMC on a single core. We note that there exist heuristic classical algorithms that can solve most instances of Chimera structured problems in a timescale comparable to the D-Wave 2X. However, we believe that such solvers will become ineffective for the next generation of annealers currently being designed. To investigate whether finite range tunneling will also confer an advantage for problems of practical interest, we conduct numerical studies on binary optimization problems that cannot yet be represented on quantum hardware. For random instances of the number partitioning problem, we find numerically that QMC, as well as other algorithms designed to simulate QA, scale better than SA. We discuss the implications of these findings for the design of next generation quantum annealers.

Abstract. In this paper we study how to perform object classification in a principled way that exploits the rich structure of real world labels. We develop a new model that allows encoding of flexible relations between labels. We introduce Hierarchy and Exclusion (HEX) graphs, a new formalism that captures semantic relations between any two labels applied to the same object: mutual exclusion, overlap and subsumption. We then provide rigorous theoretical analysis that illustrates properties of HEX graphs such as consistency, equivalence, and computational implications of the graph structure. Next, we propose a probabilistic classification model based on HEX graphs and show that it enjoys a number of desirable properties. Finally, we evaluate our method using a large-scale benchmark. Empirical results demonstrate that our model can significantly improve object classification by exploiting the label relations.