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Title: Quantum algorithms for Gibbs sampling and hitting-time 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/(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 hitting-time estimation. Finally, both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.

Authors:
 [1]; ORCiD logo [2]
  1. Univ. of New Mexico, Albuquerque, NM (United States). Center for Quantum Information and Control; New Mexico Consortium, Los Alamos, NM (United States)
  2. 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):
LA-UR-16-21218
Journal ID: ISSN 1533-7146
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Quantum Information & Computation
Additional Journal Information:
Journal Volume: 17; Journal Issue: 1-2; Journal ID: ISSN 1533-7146
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 hitting-time estimation. United States: N. p., 2017. Web. https://doi.org/10.26421/QIC17.1-2.
Chowdhury, Anirban Narayan, & Somma, Rolando D. Quantum algorithms for Gibbs sampling and hitting-time estimation. United States. https://doi.org/10.26421/QIC17.1-2
Chowdhury, Anirban Narayan, and Somma, Rolando D. Wed . "Quantum algorithms for Gibbs sampling and hitting-time estimation". United States. https://doi.org/10.26421/QIC17.1-2. https://www.osti.gov/servlets/purl/1360697.
@article{osti_1360697,
title = {Quantum algorithms for Gibbs sampling and hitting-time 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 hitting-time 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/QIC17.1-2},
journal = {Quantum Information & Computation},
number = 1-2,
volume = 17,
place = {United States},
year = {2017},
month = {2}
}

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