Related papers: Computing eigenvalues of diagonalizable matrices i…
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These…
Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation"…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
Non-Hermitian operators naturally arise in the description of open quantum systems, which exhibit features such as resonances and decay processes, where the associated eigenvalues are complex. Standard quantum algorithms, including the…
Laplacian eigenmap algorithm is a typical nonlinear model for dimensionality reduction in classical machine learning. We propose an efficient quantum Laplacian eigenmap algorithm to exponentially speed up the original counterparts. In our…
Nonlinear equations are challenging to solve due to their inherently nonlinear nature. As analytical solutions typically do not exist, numerical methods have been developed to tackle their solutions. In this article, we give a quantum…
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their…
The vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using…
We report an attempt to calculate energy eigenvalues of large quantum systems by the diagonalization of an effectively truncated Hamiltonian matrix. For this purpose we employ a specific way to systematically make a set of orthogonal states…
We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive…
Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time…
The paper describes several efficient parallel implementations of the one-sided hyperbolic Jacobi-type algorithm for computing eigenvalues and eigenvectors of Hermitian matrices. By appropriate blocking of the algorithms an almost ideal…
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…
In contexts where relevant problems can easily attain configuration spaces of enormous sizes, solving Linear Differential Equations (LDEs) can become a hard achievement for classical computers; on the other hand, the rise of quantum…
Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The Variational Quantum Eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we…
The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from…
Quantum computing enables the efficient resolution of complex problems, often outperforming classical methods across various applications. In 2009, Harrow, Hassidim and Lloyd proposed an algorithm for solving linear systems of equations,…
In the era of quantum computing, the emergence of quantum computers and subsequent advancements have led to the development of various quantum algorithms capable of solving linear equations and eigenvalues, surpassing the pace of classical…
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,…
State-of-the-art noisy intermediate-scale quantum devices (NISQ), although imperfect, enable computational tasks that are manifestly beyond the capabilities of modern classical supercomputers. However, present quantum computations are…