Hybrid quantum-classical algorithms provide ways to use noisy intermediate-scale quantum computers for practical applications. Expanding the portfolio of such techniques, we propose a quantum circuit learning algorithm that can be used to assist the characterization of quantum devices and to train shallow circuits for generative tasks. The procedure leverages quantum hardware capabilities to its fullest extent by using native gates and their qubit connectivity. We demonstrate that our approach can learn an optimal preparation of the Greenberger-Horne-Zeilinger states, also known as "cat states". We further demonstrate that our approach can efficiently prepare approximate representations of coherent thermal states, wave functions that encode Boltzmann probabilities in their amplitudes. Finally, complementing proposals to characterize the power or usefulness of near-term quantum devices, such as IBM's quantum volume, we provide a new hardware-independent metric called the qBAS score. It is based on the performance yield in a specific sampling task on one of the canonical machine learning data sets known as Bars and Stripes. We show how entanglement is a key ingredient in encoding the patterns of this data set; an ideal benchmark for testing hardware starting at four qubits and up. We provide experimental results and evaluation of this metric to probe the trade off between several architectural circuit designs and circuit depths on an ion-trap quantum computer.
Generative modeling is a flavor of machine learning with applications ranging from computer vision to chemical design. It is expected to be one of the techniques most suited to take advantage of the additional resources provided by near-term quantum computers. Here we implement a data-driven quantum circuit training algorithm on the canonical Bars-and-Stripes data set using a quantum-classical hybrid machine. The training proceeds by running parameterized circuits on a trapped ion quantum computer, and feeding the results 1 arXiv:1812.08862v2 [quant-ph] 31 Oct 2019 to a classical optimizer. We apply two separate strategies, Particle Swarm and Bayesian optimization to this task. We show that the convergence of the quantum circuit to the target distribution depends critically on both the quantum hardware and classical optimization strategy. Our study represents the first successful training of a high-dimensional universal quantum circuit, and highlights the promise and challenges associated with hybrid learning schemes. One Sentence SummaryWe train generative modeling circuits on a quantum-classical hybrid computer showing optimization strategy and resource trade-off.
Delaunay showed in 1841 that any surface of revolution of constant mean curvature in ޒ 3 has as its profile curve a roulette -specifically, the curve described by the focus of a quadric rolling on a line. Here we introduce a notion similar to the roulette that we call the treadmill sled, and we use it to provide a dynamical interpretation for the profile curves of twizzlershelicoidal surfaces of nonzero constant mean curvature.The treadmill sled is connected with a change of variables that allows us to solve the ordinary differential equation that produces twizzlers in a fairly easy way. This allows us to prove that all twizzlers are isometric to Delaunay surfaces; this is similar to work done by do Carmo and Dajczer.We also provide a moduli space for twizzlers and Delaunay surfaces that shows the connection of each surface with its dynamical interpretation, and we explicitly show the foliation of our moduli space by curves of locally isometric CMC "associated surfaces" analogous to the well-known helicoidto-catenoid deformation. Our dynamical interpretation for twizzlers also allows us to naturally define the notion of a fundamental piece of the profile curve of a twizzler, which yields the fact that, whenever a twizzler is not properly immersed, it is dense in the region bounded by two concentric cylinders if the twizzler does not contain the axis of symmetry, or dense in the region bounded by a cylinder otherwise.Using the change of coordinates induced by the notion of the treadmill sled, we also provide a dynamical interpretation for helicoidal surfaces with constant Gauss curvature, and we find an easy way to describe Delaunay surfaces by a relatively simple first integral.
Let m ≥ 2 and n ≥ 2 be any pair of integers. In this paper we prove that if H lies between cot( π m ) and bm,n = (m 2 −2), there exists a non isoparametric, compact embedded hypersurface in S n+1 with constant mean curvature H that admits O(n) × Zm in its group of isometries. These hypersurfaces therefore have exactly 2 principal curvatures. When m = 2 and H is close to the boundary value 0 = cot( π 2 ), such a hypersurface looks like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m > 2 and H is close to cot( π m ), it looks like a necklace made out of m spheres with m + 1 catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to bm,n the hypersurface is close to an isoparametric hypersurface with the same mean curvature. For hyperbolic spaces we prove that every H ≥ 0 can be realized as the mean curvature of an embedded CMC hypersurface in H n+1 . Moreover we prove that when H > 1 this hypersurface admits O(n)× Z in its group of isometries. As a corollary of the properties we prove for these hypersurfaces, we construct, for any n ≥ 6, non-isoparametric compact minimal hypersurfaces in S n+1 whose cones in R n+2 are stable. Also, we prove that the stability index of every non-isoparametric minimal hypersurface with two principal curvatures in S n+1 exceeds n + 3.
Diabetic macular edema (DME) is one of the most common eye complication caused by diabetes mellitus, resulting in partial or total loss of vision. Optical Coherence Tomography (OCT) volumes have been widely used to diagnose different eye diseases, thanks to their sensitivity to represent small amounts of fluid, thickness between layers and swelling. However, the lack of tools for automatic image analysis for supporting disease diagnosis is still a problem. Convolutional neural networks (CNNs) have shown outstanding performance when applied to several medical images analysis tasks. This paper presents a model, OCT-NET, based on a CNN for the automatic classification of OCT volumes. The model was evaluated on a dataset of OCT volumes for DME diagnosis using a leave-one-out cross-validation strategy obtaining an accuracy, sensitivity, and specificity of 93.75%.
Abstract. Let M be a compact oriented non-equatorial minimal hypersurface of the unit ndimensional sphere. Suppose that for any non-zero vector in w 6 R n+1 there exists an orthogonal matrix B such that B(M) = M and B{w) ^ w. Since all known examples of minimal hypersurfaces have antipodal symmetry, they satisfy this condition.We prove that: i) the stability index of M is greater than or equal to n-f 2 with strict inequality, unless M is a Clifford hypersurface; ii) the difference between the first two eigenvalues of the Jacobi operator is less than or equal to n -1 with strict inequality, unless the norm of the second fundamental form is constant; and iii) if M hats antipodal symmetry and is not a Clifford hypersurface then the index is greater than n + 3. Moreover if the unit normal vector is even, the index is greater than 2n + 2.
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