Related papers: Fast and Stable Pascal Matrix Algorithms
The stabiliser formalism plays a central role in quantum computing, error correction, and fault tolerance. Conversions between and verifications of different specifications of stabiliser states and Clifford gates are important components of…
We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…
We present an efficient and scalable algorithm for performing matrix-vector multiplications ("matvecs") for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving…
Inverse linear programming (LP) has received increasing attention due to its potential to generate efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and…
Non-stationary signals are ubiquitous in real life. Many techniques have been proposed in the last decades which allow decomposing multi-component signals into simple oscillatory mono-components, like the groundbreaking Empirical Mode…
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this…
We tackle the problems of computing the rightmost variant of the Lempel-Ziv factorizations in the online/sliding model. Previous best bounds for this problem are O(n log n) time with O(n) space, due to Amir et al. [IPL 2002] for the online…
We consider so-called $N$-fold integer programs (IPs) of the form $\max\{c^T x : Ax = b, \ell \leq x \leq u, x \in \mathbb Z^{nt}\}, where $A \in \mathbb Z^{(r+sn)\times nt} consists of $n$ arbitrary matrices $A^{(i)} \in \mathbb Z^{r\times…
We present a computationally efficient algorithm for stable numerical differentiation from noisy, uniformly-sampled data on a bounded interval. The method combines multi-interval Fourier extension approximations with an adaptive domain…
We present a fast algorithm for modular exponentiation when the factorization of the modulus is known. Let $a,n,m$ be positive integers and suppose $m$ factors canonically as $\prod_{i=1}^k p_i^{e_i}$. Choose integer parameters $t_i\in [1,…
Motivated by applications in single-cell biology and metagenomics, we investigate the problem of matrix reordering based on a noisy disordered monotone Toeplitz matrix model. We establish the fundamental statistical limit for this problem…
We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we…
Fourier series of smooth, non-periodic functions on $[-1,1]$ are known to exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of overcoming these problems is by using a Fourier series on a larger domain, say $[-T,T]$…
A recently discovered universal rank-based matrix method to extract trends from noisy time series is described in [1] but the formula for the output matrix elements, implemented there as an open-access supplement MATLAB computer code, is…
The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the…
Boolean matrix factorization is a natural and a popular technique for summarizing binary matrices. In this paper, we study a problem of Boolean matrix factorization where we additionally require that the factor matrices have consecutive…
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient…
Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…