Quantum eigenvalue estimation via time series analysis
Abstract
We present an efficient method for estimating the eigenvalues of a Hamiltonian H from the expectation values of the evolution operator for various times. For a given quantum state ρ, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of ρ in those eigenstates of H associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter ε, which is the gap between eigenvalue estimates. Unlike the wellknown quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for nearterm applications. The output of our method can be used to estimate spectral properties of H and other expectation values efficiently, within additive error proportional to ε.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1634958
 Report Number(s):
 LAUR1926913
Journal ID: ISSN 13672630; TRN: US2201320
 Grant/Contract Number:
 89233218CNA000001
 Resource Type:
 Accepted Manuscript
 Journal Name:
 New Journal of Physics
 Additional Journal Information:
 Journal Volume: 21; Journal Issue: 12; Journal ID: ISSN 13672630
 Publisher:
 IOP Publishing
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; computer science; information science; mathematics; quantum computing; quantum simulation; phase estimation
Citation Formats
Somma, Rolando Diego. Quantum eigenvalue estimation via time series analysis. United States: N. p., 2019.
Web. doi:10.1088/13672630/ab5c60.
Somma, Rolando Diego. Quantum eigenvalue estimation via time series analysis. United States. https://doi.org/10.1088/13672630/ab5c60
Somma, Rolando Diego. Mon .
"Quantum eigenvalue estimation via time series analysis". United States. https://doi.org/10.1088/13672630/ab5c60. https://www.osti.gov/servlets/purl/1634958.
@article{osti_1634958,
title = {Quantum eigenvalue estimation via time series analysis},
author = {Somma, Rolando Diego},
abstractNote = {We present an efficient method for estimating the eigenvalues of a Hamiltonian H from the expectation values of the evolution operator for various times. For a given quantum state ρ, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of ρ in those eigenstates of H associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter ε, which is the gap between eigenvalue estimates. Unlike the wellknown quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for nearterm applications. The output of our method can be used to estimate spectral properties of H and other expectation values efficiently, within additive error proportional to ε.},
doi = {10.1088/13672630/ab5c60},
journal = {New Journal of Physics},
number = 12,
volume = 21,
place = {United States},
year = {2019},
month = {12}
}
Figures / Tables:
Works referenced in this record:
Simulated Quantum Computation of Molecular Energies
journal, September 2005
 AspuruGuzik, A.
 Science, Vol. 309, Issue 5741
Optimal Hamiltonian Simulation by Quantum Signal Processing
journal, January 2017
 Low, Guang Hao; Chuang, Isaac L.
 Physical Review Letters, Vol. 118, Issue 1
Quantum Algorithm for Linear Systems of Equations
journal, October 2009
 Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth
 Physical Review Letters, Vol. 103, Issue 15
Optimal quantum measurements of expectation values of observables
journal, January 2007
 Knill, Emanuel; Ortiz, Gerardo; Somma, Rolando D.
 Physical Review A, Vol. 75, Issue 1
Hamiltonian Simulation by Qubitization
journal, July 2019
 Low, Guang Hao; Chuang, Isaac L.
 Quantum, Vol. 3
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
journal, December 2006
 Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
 Communications in Mathematical Physics, Vol. 270, Issue 2
A variational eigenvalue solver on a photonic quantum processor
journal, July 2014
 Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter
 Nature Communications, Vol. 5, Issue 1
Higher order decompositions of ordered operator exponentials
journal, January 2010
 Wiebe, Nathan; Berry, Dominic; Høyer, Peter
 Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 6
Fractal decomposition of exponential operators with applications to manybody theories and Monte Carlo simulations
journal, June 1990
 Suzuki, Masuo
 Physics Letters A, Vol. 146, Issue 6
Simulating Hamiltonian Dynamics with a Truncated Taylor Series
journal, March 2015
 Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
 Physical Review Letters, Vol. 114, Issue 9
Learning the quantum algorithm for state overlap
journal, November 2018
 Cincio, Lukasz; Subaşı, Yiğit; Sornborger, Andrew T.
 New Journal of Physics, Vol. 20, Issue 11
Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors
journal, December 1999
 Abrams, Daniel S.; Lloyd, Seth
 Physical Review Letters, Vol. 83, Issue 24
A TrotterSuzuki approximation for Lie groups with applications to Hamiltonian simulation
journal, June 2016
 Somma, Rolando D.
 Journal of Mathematical Physics, Vol. 57, Issue 6
An efficient quantum algorithm for spectral estimation
journal, March 2017
 Steffens, Adrian; Rebentrost, Patrick; Marvian, Iman
 New Journal of Physics, Vol. 19, Issue 3
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
journal, October 1997
 Shor, Peter W.
 SIAM Journal on Computing, Vol. 26, Issue 5
Using the matrix pencil method to estimate the parameters of a sum of complex exponentials
journal, February 1995
 Sarkar, T. K.; Pereira, O.
 IEEE Antennas and Propagation Magazine, Vol. 37, Issue 1
Quantum algorithms revisited
journal, January 1998
 Cleve, R.; Ekert, A.; Macchiavello, C.
 Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 454, Issue 1969
General theory of fractal path integrals with applications to many‐body theories and statistical physics
journal, February 1991
 Suzuki, Masuo
 Journal of Mathematical Physics, Vol. 32, Issue 2
Hardwareefficient variational quantum eigensolver for small molecules and quantum magnets
journal, September 2017
 Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan
 Nature, Vol. 549, Issue 7671
Semiclassical Fourier Transform for Quantum Computation
journal, April 1996
 Griffiths, Robert B.; Niu, ChiSheng
 Physical Review Letters, Vol. 76, Issue 17
Spectral quantum tomography
journal, September 2019
 Helsen, Jonas; Battistel, Francesco; Terhal, Barbara M.
 npj Quantum Information, Vol. 5, Issue 1
Simulating physical phenomena by quantum networks
journal, April 2002
 Somma, R.; Ortiz, G.; Gubernatis, J. E.
 Physical Review A, Vol. 65, Issue 4
Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision
journal, January 2017
 Childs, Andrew M.; Kothari, Robin; Somma, Rolando D.
 SIAM Journal on Computing, Vol. 46, Issue 6
Entanglementfree Heisenberglimited phase estimation
journal, November 2007
 Higgins, B. L.; Berry, D. W.; Bartlett, S. D.
 Nature, Vol. 450, Issue 7168
Error mitigation extends the computational reach of a noisy quantum processor
journal, March 2019
 Kandala, Abhinav; Temme, Kristan; Córcoles, Antonio D.
 Nature, Vol. 567, Issue 7749
On the Convergence Rate of Generalized Fourier Expansions
journal, January 1973
 Mead, K. O.; Delves, L. M.
 IMA Journal of Applied Mathematics, Vol. 12, Issue 3
Quantum phase estimation of multiple eigenvalues for smallscale (noisy) experiments
journal, February 2019
 O’Brien, Thomas E.; Tarasinski, Brian; Terhal, Barbara M.
 New Journal of Physics, Vol. 21, Issue 2
Figures / Tables found in this record: