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Deep neural networks have been very successful in applications ranging from computer vision and natural language processing to strategy optimization in games. Recently neural networks with sinusoidal activation functions (SinNN) were found to be ideally suited for representing complex natural signals and their fine spatial and temporal details, which makes them effective representations of images, sound, and video, and good solvers of differential equations. However, training SinNN via gradient descent often results in bad local minima, posing a significant challenge when optimizing their weights. Furthermore, when the weights are discretized for better memory and inference efficiency on small devices, we find that a vanishing gradient problem appears on the resulting discrete SinNN (DSinNN). Brute force search provides an alternative way to find the best weights for DSinNN but is intractable for a large number of parameters. We here provide a qualitatively different training method: an algorithm for quantum training of DSinNNs. The quantum training evolves an initially uniform superposition over weight values to one that is guaranteed to peak on the best weights. We demonstrate the algorithm on toy examples and show that it indeed outperforms gradient descent in optimizing the loss function and outperforms brute force search in the time required.

Remote detectability is often taken as a physical assumption in the study of topologically ordered systems, and it is a central axiom of mathematical frameworks of topological quantum field theories. We show under the entanglement bootstrap approach that remote detectability is a necessary property; that is, we derive it as a theorem. Starting from a single wave function on a topologically-trivial region satisfying the entanglement bootstrap axioms, we can construct states on closed manifolds. The crucial technique is to immerse the punctured manifold into the topologically trivial region and then heal the puncture. This is analogous to Kirby's torus trick. We then analyze a special class of such manifolds, which we call pairing manifolds. For each pairing manifold, which pairs two classes of excitations, we identify an analog of the topological $S$-matrix. This pairing matrix is unitary, which implies remote detectability between two classes of excitations. These matrices are in general not associated with the mapping class group of the manifold. As a by-product, we can count excitation types (e.g., graph excitations in 3+1d). The pairing phenomenon occurs in many physical contexts, including systems in different dimensions, with or without gapped boundaries. We provide a variety of examples to illustrate its scope., Comment: 110+20 pages. Many figures

Quantum neural networks have been widely studied in recent years, given their potential practical utility and recent results regarding their ability to efficiently express certain classical data. However, analytic results to date rely on assumptions and arguments from complexity theory. Due to this, there is little intuition as to the source of the expressive power of quantum neural networks or for which classes of classical data any advantage can be reasonably expected to hold. Here, we study the relative expressive power between a broad class of neural network sequence models and a class of recurrent models based on Gaussian operations with non-Gaussian measurements. We explicitly show that quantum contextuality is the source of an unconditional memory separation in the expressivity of the two model classes. Additionally, as we are able to pinpoint quantum contextuality as the source of this separation, we use this intuition to study the relative performance of our introduced model on a standard translation data set exhibiting linguistic contextuality. In doing so, we demonstrate that our introduced quantum models are able to outperform state of the art classical models even in practice., Comment: 30 pages, 9 figures

We extend the entanglement bootstrap approach to (3+1)-dimensions. We study knotted excitations of (3+1)-dimensional liquid topological orders and exotic fusion processes of loops. As in previous work in (2+1)-dimensions, we define a variety of superselection sectors and fusion spaces from two axioms on the ground state entanglement entropy. In particular, we identify fusion spaces associated with knots. We generalize the information convex set to a new class of regions called immersed regions, promoting various theorems to this new context. Examples from solvable models are provided; for instance, a concrete calculation of knot multiplicity shows that the knot complement of a trefoil knot can store quantum information. We define spiral maps that allow us to understand consistency relations for torus knots as well as spiral fusions of fluxes., Comment: 84+43 pages, 46 figures, published version

For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with another quantum system (called memory); the quantum correlation between the systems can reduce the measurement uncertainty. On the other hand, in a curved spacetime, the presence of the Hawking radiation can increase the uncertainty in quantum measurement. The interplay of quantum correlation in the curved spacetime has become an interesting arena for studying quantum uncertainty relations. Here we demonstrate that the bounds of the entropic uncertainty relations, in the presence of memory, can be formulated in terms of the Holevo quantity, which limits how much information can be encoded in a quantum system. Specifically, we considered two examples with Dirac fields, near the event horizon of a Schwarzschild black hole, the Holevo bound provides a better bound than the previous bound based on the mutual information. Furthermore, if the memory moves away from the black hole, the difference between the total uncertainty and the Holevo bound remains a constant, not depending on any property of the black hole., Comment: 6 pages, 4 figures