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 AbramsLloyd 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 AbramsLloyd 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:
 Department of Electrical Engineering, University of California Los Angeles, Los Angeles, California 90095 (United States)
 (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 AbramsLloyd 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 AbramsLloyd 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}
}

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