Quantum algorithm for solving linear systems of equations
arXiv:0811.3171 · doi:10.1103/PhysRevLett.103.150502
Abstract
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 doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.
15 pages. v2 is much longer, with errors fixed, run-time improved and a new BQP-completeness result added. v3 is the final published version and mostly adds clarifications and corrections to v2