Quantum algorithms for Gibbs sampling and hittingtime estimation
Abstract
In this paper, we present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in √Nβ/Ζ and polynomial in log(1/ϵ), where N is the Hilbert space dimension, β is the inverse temperature, Ζ is the partition function, and ϵ is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on 1/ϵ and quadratically improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix Ρ, it runs in time almost linear in 1/(ϵΔ ^{3/2}), where ϵ is the absolute precision in the estimation and Δ is a parameter determined by Ρ, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on 1/ϵ and 1/Δ of the analog classical algorithm for hittingtime estimation. Finally, both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.
 Authors:
 Univ. of New Mexico, Albuquerque, NM (United States). Center for Quantum Information and Control; New Mexico Consortium, Los Alamos, NM (United States)
 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
 Contributing Org.:
 New Mexico Consortium, Los Alamos, NM (United States)
 OSTI Identifier:
 1360697
 Report Number(s):
 LAUR1621218
Journal ID: ISSN 15337146
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Quantum Information & Computation
 Additional Journal Information:
 Journal Volume: 17; Journal Issue: 12; Journal ID: ISSN 15337146
 Publisher:
 Rinton Press
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Computer Science; Information Science; Quantum algorithms
Citation Formats
Chowdhury, Anirban Narayan, and Somma, Rolando D. Quantum algorithms for Gibbs sampling and hittingtime estimation. United States: N. p., 2017.
Web. doi:10.26421/QIC17.12.
Chowdhury, Anirban Narayan, & Somma, Rolando D. Quantum algorithms for Gibbs sampling and hittingtime estimation. United States. doi:10.26421/QIC17.12.
Chowdhury, Anirban Narayan, and Somma, Rolando D. Wed .
"Quantum algorithms for Gibbs sampling and hittingtime estimation". United States.
doi:10.26421/QIC17.12. https://www.osti.gov/servlets/purl/1360697.
@article{osti_1360697,
title = {Quantum algorithms for Gibbs sampling and hittingtime estimation},
author = {Chowdhury, Anirban Narayan and Somma, Rolando D.},
abstractNote = {In this paper, we present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in √Nβ/Ζ and polynomial in log(1/ϵ), where N is the Hilbert space dimension, β is the inverse temperature, Ζ is the partition function, and ϵ is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on 1/ϵ and quadratically improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix Ρ, it runs in time almost linear in 1/(ϵΔ3/2), where ϵ is the absolute precision in the estimation and Δ is a parameter determined by Ρ, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on 1/ϵ and 1/Δ of the analog classical algorithm for hittingtime estimation. Finally, both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.},
doi = {10.26421/QIC17.12},
journal = {Quantum Information & Computation},
number = 12,
volume = 17,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}
}

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