Related papers: Epsilon-nets, unitary designs and random quantum c…
Training large and highly accurate deep learning (DL) models is computationally costly. This cost is in great part due to the excessive number of trained parameters, which are well-known to be redundant and compressible for the execution…
We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically…
We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the "growth" of certain operator spaces: It implies asymptotically…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line…
Tensor networks establish an adaptable framework for the emulation of quantum circuits. By partitioning exponentially large registers and gates into smaller tensors, this unlocks fast transformations through tensor algebra, and grants fine…
The concept of spherical $t$-design, which is a finite subset of the unit sphere, was introduced by Delsarte-Goethals-Seidel (1977). The concept of Euclidean $t$-design, which is a two step generalization of spherical design in the sense…
Uniform random rotations are a useful primitive in applications such as fast Johnson-Lindenstrauss embeddings, kernel approximation, communication-efficient learning, and recent AI compression pipelines, but they are computationally…
Approximation of non-linear kernels using random feature maps has become a powerful technique for scaling kernel methods to large datasets. We propose $\textit{Tensor Sketch}$, an efficient random feature map for approximating polynomial…
We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
We consider continuous structures which are obtained from finite dimensional Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum automata and circuits are naturally interpretable in such structures. We consider…
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many…
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate…
In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and $\pi/2$-phase rotations,…
We describe a new method for approximating an arbitrary $n$ qubit unitary with precision $\varepsilon$ using a Clifford and T circuit with $O(4^{n}n(\log(1/\varepsilon)+n))$ gates. The method is based on rounding off a unitary to a unitary…
Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in…
Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely,…
In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the…
Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…
We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution…