Quantum Linear System Algorithm for Dense Matrices

An approximate minimum degree algorithm for matrices with dense rows

We present a modified version of the approximate minimum degree algorithm for preordering a matrix with a symmetric sparsity pattern prior to the numerical factorization. The modification is designed to improve the efficiency of the algorithm when some of the rows and columns have significantly more entries than the average for the matrix. Numerical results are presented for problems arising fr...

Certified Dense Linear System Solving

Quantum algorithm for linear systems of equations.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g.,...

New Quantum Algorithm for Linear Regression

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 only yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new d...

Linear Time Algorithm for Quantum 2SAT

A well-known result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Πi j on a system of n qubits, and the task is to decide whether the Hamiltonian H = ∑Πi j has a 0-eigenvalue, or all eigenvalues are greater than 1/nα for some α = O(1). The...