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Title: Eigenvalue estimation of differential operators with a quantum algorithm

Abstract

We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions {psi}(x{sub 1},x{sub 2},...,x{sub D}) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy {theta}(1/N{sup 2}) is {theta}((2(S+1)(1+1/{nu})+D)ln N) qubits and O(N{sup 2(S+1)(1+1/{nu})}ln{sup c} N{sup D}) gate operations, where N is the number of points to which each argument is discretized, {nu} and c are implementation dependent constants of O(1). Optimal classical methods require {theta}(N{sup D}) bits and {omega}(N{sup D}) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D>2(S+1)(1+1/{nu}). In the case of Schroedinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.

Authors:
; ; ;  [1];  [2]
  1. Department of Electrical Engineering, University of California Los Angeles, Los Angeles, California 90095 (United States)
  2. (United States)
Publication Date:
OSTI Identifier:
20786281
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.72.062318; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ALGORITHMS; EIGENFUNCTIONS; EIGENVALUES; GATING CIRCUITS; GROUND STATES; INFORMATION THEORY; POLYNOMIALS; QUANTUM MECHANICS; QUANTUM OPERATORS; QUBITS; SCHROEDINGER EQUATION

Citation Formats

Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., and Luminescent Technologies, Inc., Mountain View, California 94041. Eigenvalue estimation of differential operators with a quantum algorithm. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., & Luminescent Technologies, Inc., Mountain View, California 94041. Eigenvalue estimation of differential operators with a quantum algorithm. United States. doi:10.1103/PHYSREVA.72.0.
Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., and Luminescent Technologies, Inc., Mountain View, California 94041. Thu . "Eigenvalue estimation of differential operators with a quantum algorithm". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786281,
title = {Eigenvalue estimation of differential operators with a quantum algorithm},
author = {Szkopek, Thomas and Roychowdhury, Vwani and Yablonovitch, Eli and Abrams, Daniel S. and Luminescent Technologies, Inc., Mountain View, California 94041},
abstractNote = {We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions {psi}(x{sub 1},x{sub 2},...,x{sub D}) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy {theta}(1/N{sup 2}) is {theta}((2(S+1)(1+1/{nu})+D)ln N) qubits and O(N{sup 2(S+1)(1+1/{nu})}ln{sup c} N{sup D}) gate operations, where N is the number of points to which each argument is discretized, {nu} and c are implementation dependent constants of O(1). Optimal classical methods require {theta}(N{sup D}) bits and {omega}(N{sup D}) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D>2(S+1)(1+1/{nu}). In the case of Schroedinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.},
doi = {10.1103/PHYSREVA.72.0},
journal = {Physical Review. A},
number = 6,
volume = 72,
place = {United States},
year = {Thu Dec 15 00:00:00 EST 2005},
month = {Thu Dec 15 00:00:00 EST 2005}
}