TY - JOUR

T1 - Quantum algorithm for linear systems of equations

AU - Harrow, Aram W.

AU - Hassidim, Avinatan

AU - Lloyd, Seth

PY - 2009/10/7

Y1 - 2009/10/7

N2 - 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 complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.

AB - 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 complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.

UR - http://www.scopus.com/inward/record.url?scp=70349862042&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.103.150502

DO - 10.1103/PhysRevLett.103.150502

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C2 - 19905613

AN - SCOPUS:70349862042

SN - 0031-9007

VL - 103

JO - Physical Review Letters

JF - Physical Review Letters

IS - 15

M1 - 150502

ER -