Related papers: Classical and Quantum Algorithms for Tensor Princi…
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum…
These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques which can produce speed up versus classical…
A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC'19], we introduce such a model, where we assume we…
Classical machine learning theory and theory of quantum computations are among of the most rapidly developing scientific areas in our days. In recent years, researchers investigated if quantum computing can help to improve classical machine…
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for…
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known…
Tensor network methods strike a middle ground between fully-fledged quantum computing and classical computing, as they take inspiration from quantum systems to significantly speed up certain classical operations. Their strength lies in…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
This paper presents a sublinear classical algorithm for principal component regression. The algorithm uses quantum-inspired linear algebra, an idea developed by Tang. Using this technique, her algorithm for recommendation systems achieved…
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical…
Quantum algorithms are built enabling to find Poincar\'e recurrence times and periodic orbits of classical dynamical systems. It is shown that exponential gain compared to classical algorithms can be reached for a restricted class of…
We apply numerical optimization and linear algebra algorithms for classical computers to the problem of automatically synthesizing algorithms for quantum computers. Using our framework, we apply several common techniques from these…
We consider whether trainable quantum unitaries can be used to discover quantum speed-ups for classical problems. Using methods recently developed for training quantum neural nets, we consider Simon's problem, for which there is a known…
Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyze and classify data in machine learning applications. We present an improved…
Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks.…
Over the past few years several quantum machine learning algorithms were proposed that promise quantum speed-ups over their classical counterparts. Most of these learning algorithms either assume quantum access to data -- making it unclear…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
Demonstrating quantum advantage has been a pressing challenge in the field. Most claimed quantum speedups rely on a subroutine in which classical information can be accessed in a coherent quantum manner, which imposes a crucial constraint…
Running quantum algorithms often involves implementing complex quantum circuits with such a large number of multi-qubit gates that the challenge of tackling practical applications appears daunting. To date, no experiments have successfully…