Quantum Algorithm for Linear Systems of Equations
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 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., x→^{†}Mx→ for some matrix M. In this case, when A is sparse, N×N and has condition number κ, the fastest known classical algorithms can find x→ and estimate x→^{†}Mx→ in time scaling roughly as Nκ. Here, we exhibit a quantum algorithm for estimating x→^{†}Mx→ whose runtime is a polynomial of log(N) and κ. Indeed, for small values of κ [i.e., polylog(N)], we prove (using some common complexitytheoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.
 Publication:

Physical Review Letters
 Pub Date:
 October 2009
 DOI:
 10.1103/PhysRevLett.103.150502
 arXiv:
 arXiv:0811.3171
 Bibcode:
 2009PhRvL.103o0502H
 Keywords:

 03.67.Ac;
 02.10.Ud;
 89.70.Eg;
 Quantum algorithms protocols and simulations;
 Linear algebra;
 Computational complexity;
 Quantum Physics
 EPrint:
 15 pages. v2 is much longer, with errors fixed, runtime improved and a new BQPcompleteness result added. v3 is the final published version and mostly adds clarifications and corrections to v2