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arXiv · 0811.3171

Quantum algorithm for solving linear systems of equations

Aram W. Harrow·Avinatan Hassidim·Seth Lloyd
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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.

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Aram W. Harrow, Avinatan Hassidim, Seth Lloyd. 2009-09-30. Quantum algorithm for solving linear systems of equations. https://doi.org/10.1103/physrevlett.103.150502

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