Quantum Algorithm for Linear Systems of Equations
Name
Harrow-2009-Quantum Algorithm fo.pdf
Size
96.98 KB
Format
Adobe PDF
Checksum (MD5)
8130f72c48159e35e64a4ad209ee3dc6
Author(s)
Lloyd, Seth
Hassidim, Avinatan
Date Issued
October 2009
Journal
Physical Review Letters
Publisher
American Physical Society
Citation
Harrow, Aram W., Avinatan Hassidim, and Seth Lloyd. “Quantum Algorithm for Linear Systems of Equations.” Physical Review Letters 103.15 (2009): 150502. © 2009 The American Physical Society.
Version
Final published version
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 b⃗, find a vector x⃗ such that Ax⃗=b⃗. We consider the case where one does not need to know the solution x⃗ itself, but rather an approximation of the expectation value of some operator associated with x⃗, e.g., x⃗†Mx⃗ 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 x⃗ and estimate x⃗†Mx⃗ in time scaling roughly as N√κ. Here, we exhibit a quantum algorithm for estimating x⃗†Mx⃗ 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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Massachusetts Institute of Technology. Research Laboratory of Electronics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
http://dx.doi.org/10.1103/PhysRevLett.103.150502
