Related papers: Quantum Approximation II. Sobolev Embeddings
In this paper we combine two existing approaches for approximating attractors. One of them approximates the attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside…
We introduce a local concept of speed-up applicable to intermediate stages of a quantum algorithm. We use it to analyse the complementary roles played by quantum parallel computation and quantum measurement in yielding the speed-up. A…
We present several quantum algorithms for performing nearest-neighbor learning. At the core of our algorithms are fast and coherent quantum methods for computing distance metrics such as the inner product and Euclidean distance. We prove…
Encoding classical data into quantum states is considered a quantum feature map to map classical data into a quantum Hilbert space. This feature map provides opportunities to incorporate quantum advantages into machine learning algorithms…
A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…
The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it allows to derive the ultimate bounds of the achievable precision. We show a relation between the statistical distance…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
We prove a rate of convergence for the $N$-particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of mean-field control problem under common…
We compare quantum and classical machines designed for learning an N-bit Boolean function in order to address how a quantum system improves the machine learning behavior. The machines of the two types consist of the same number of…
The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…
We study the query complexity of computing a function f:{0,1}^n-->R_+ in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f(x), using as few queries to the input x as…
We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the…
We consider the problem of computing the q->p norm of a matrix A, which is defined for p,q \ge 1, as |A|_{q->p} = max_{x !=0 } |Ax|_p / |x|_q. This is in general a non-convex optimization problem, and is a natural generalization of the…
In order to qualify quantum algorithms for industrial NP-Hard problems, comparing them to available polynomial approximate classical algorithms and not only to exact ones -- exponential by nature -- , is necessary. This is a great challenge…
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of…
Parameterized complexity theory was developed in the 1990s to enrich the complexity-theoretic analysis of problems that depend on a range of parameters. In this paper we establish a quantum equivalent of classical parameterized complexity…
We prove $L^p$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-type theorem holds in this context: the $L^p$ norm of the…
This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,\gamma)$ with dominating mixed smoothness defined via kernel related to the fractional…
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…
Quantum annealing (QA) holds promise for optimization problems in quantum computing, especially for combinatorial optimization. This analog framework attracts attention for its potential to address complex problems. Its gate-based…