Editing Quantum algorithm (section)

===Solving a linear system of equations===
{{main|Quantum algorithm for linear systems of equations}}

In 2009, [[Aram Harrow]], Avinatan Hassidim, and [[Seth Lloyd]], formulated a quantum algorithm for solving [[System of linear equations|linear systems]]. The [[Quantum algorithm for linear systems of equations|algorithm]] estimates the result of a scalar measurement on the solution vector to a given linear system of equations.<ref name="Quantum algorithm for solving linear systems of equations by Harrow et al.">{{Cite journal|arxiv = 0811.3171|last1 = Harrow|first1 = Aram W|title = Quantum algorithm for solving linear systems of equations|journal = Physical Review Letters|volume = 103|issue = 15|pages = 150502|last2 = Hassidim|first2 = Avinatan|last3 = Lloyd|first3 = Seth|year = 2008|doi = 10.1103/PhysRevLett.103.150502|pmid = 19905613|bibcode = 2009PhRvL.103o0502H|s2cid = 5187993}}</ref>

Provided that the linear system is [[sparse matrix|sparse]] and has a low [[condition number]] <math>\kappa</math>, and that the user is interested in the result of a scalar measurement on the solution vector (instead of the values of the solution vector itself), then the algorithm has a runtime of <math>O(\log(N)\kappa^2)</math>, where <math>N</math> is the number of variables in the linear system. This offers an exponential speedup over the fastest classical algorithm, which runs in <math>O(N\kappa)</math> (or <math>O(N\sqrt{\kappa})</math> for positive semidefinite matrices).