相关论文: Pauli Decomposition via the Fast Walsh-Hadamard Tr…

We present a practical algorithm to decode erasures of Reed-Solomon codes over the q elements binary field in O(q \log_2^2 q) time where the constant implied by the O-notation is very small. Asymptotically fast algorithms based on fast…

This paper provides an accurate method to obtain the bidiagonal factorization of many generalized Pascal matrices, which in turn can be used to compute with high relative accuracy the eigenvalues, singular values and inverses of these…

A quantum computer has a clear advantage over a classical computer for exhaustive search. The quantum mechanical algorithm for exhaustive search was originally derived by using subtle properties of a particular quantum mechanical operation…

For every fixed constant $\alpha > 0$, we design an algorithm for computing the $k$-sparse Walsh-Hadamard transform of an $N$-dimensional vector $x \in \mathbb{R}^N$ in time $k^{1+\alpha} (\log N)^{O(1)}$. Specifically, the algorithm is…

We consider an n-qubit quantum system with a Hamiltonian, defined by an expansion in the Pauli basis, and propose a new algorithm for classical computing the exponential of the Hamiltonian. The algorithm is based on the representation of…

The Clifford+$T$ quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group $\langle H, T\rangle$. The matrix $T$ can be considered the fourth root of Pauli $Z$, since $T^4 = Z$ or…

The quantum Fourier transform (QFT) is a powerful tool in quantum computing. The main ingredients of QFT are formed by the Walsh-Hadamard transform H and phase shifts P(.), both of which are 2x2 unitary matrices as operators on the…

We propose a novel hybrid classical-quantum approach for image processing based on polar Walsh basis functions. Using this approach, we present an algorithm for the removal of the circular banding noise (including Airy pattern noise) and…

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…

Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum…

We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to…

In this work, we study the Hamiltonian Decoded Quantum Interferometry (HDQI) for the general Hamiltonians $H=\sum_ic_iP_i$ on an $n$-qubit system, where the coefficients $c_i\in \mathbb{R}$ and $P_i$ are Pauli operators. We show that, given…

We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…

Any square matrix can be transformed into a doubly stochastic matrix via Sinkhorn scaling with diagonal matrices or completing to a larger dimensional matrix. Standard Birkhoff-von Neumann and Pauli decompositions represent such matrices as…

In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a…

The discretized Poisson equation matrix (DPEM) in 1D has been shown to require an exponentially large number of terms when decomposed in the Pauli basis when solving numerical linear algebra problems on a quantum computer. Additionally,…

We investigate the dividing line between classical and quantum computational power in estimating properties of matrix functions. More precisely, we study the computational complexity of two primitive problems: given a function $f$ and a…

A quantum computer directly manipulates information stored in the state of quantum mechanical systems. The available operations have many attractive features but also underly severe restrictions, which complicate the design of quantum…

Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately,…

Cross-platform verification is the task of comparing the output states produced by different physical platforms using solely local quantum operations and classical communication. While protocols have previously been suggested for this task,…