Related papers: A quantum algorithm for linear autonomous differen…
We introduce a quantum approximate optimization algorithm (QAOA) for continuous optimization. The algorithm is based on the dynamics of a quantum system moving in an energy potential which encodes the objective function. By approximating…
Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four…
Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that…
Linear regression is a basic and widely-used methodology in data analysis. It is known that some quantum algorithms efficiently perform least squares linear regression of an exponentially large data set. However, if we obtain values of the…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the…
Generally, discretization of partial differential equations (PDEs) creates a sequence of linear systems $A_k x_k = b_k, k = 0, 1, 2, ..., N$ with well-known and structured sparsity patterns. Preconditioners are often necessary to achieve…
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A…
Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$ matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of…
The QLP decomposition is one of the effective algorithms to approximate singular value decomposition (SVD) in numerical linear algebra. In this paper, we propose some single-pass randomized QLP decomposition algorithms for computing the…
This paper focuses on numerical approximation for fractional powers of elliptic operators on $2$-d manifolds. Firstly, parametric finite element method is employed to discretize the original problem. We then approximate fractional powers of…
Quantum amplitude estimation is one of the core subroutines in quantum algorithms. This paper gives a parallelized amplitude estimation (PAE) algorithm that simultaneously achieves near-Heisenberg scaling in the total number of queries and…
In this paper, we introduce a quantum-enhanced algorithm for simulation-based optimization. Simulation-based optimization seeks to optimize an objective function that is computationally expensive to evaluate exactly, and thus, is…
Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…
Variational quantum algorithms constitute one of the most widespread methods for using current noisy quantum computers. However, it is unknown if these heuristic algorithms provide any quantum-computational speedup, although we cannot…
Consider the following optimization problem: Given $n \times n$ matrices $A$ and $\Lambda$, maximize $\langle A, U\Lambda U^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix…
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of…
We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics. We map a…
Quantum linear-system algorithms (QLSAs) have rigorous worst-case complexity guarantees, but their runtimes are often chosen from spectral information assumed in advance. What is largely lacking is an a posteriori progress flag: most QLSA…
We present a quantum algorithm for fitting a linear regression model to a given data set using the least squares approach. Different from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs…