Quantum algorithms for Gibbs sampling and hitting-time estimation

Chowdhury, Anirban Narayan; Somma, Rolando D.

doi:10.26421/QIC17.1-2

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/(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 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:

Chowdhury, Anirban Narayan ^[1];

^[2]

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:: Wed Feb 01 00:00:00 EST 2017

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.  doi:10.26421/QIC17.1-2.

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                    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

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                    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.

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@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         = {Wed Feb 01 00:00:00 EST 2017},

  month        = {Wed Feb 01 00:00:00 EST 2017}

}

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Works referencing / citing this record:

Optimising Matrix Product State Simulations of Shor's Algorithm
journal, January 2019

Dang, Aidan; Hill, Charles D.; Hollenberg, Lloyd C. L.
Quantum, Vol. 3
DOI: 10.22331/q-2019-01-25-116

Optimising Matrix Product State Simulations of Shor's Algorithm
text, January 2017

Dang, Aidan; Hill, Charles D.; Hollenberg, Lloyd C. L.
arXiv
DOI: 10.48550/arxiv.1712.07311

Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing
text, January 2018

Subasi, Yigit; Somma, Rolando D.; Orsucci, Davide
arXiv
DOI: 10.48550/arxiv.1805.10549

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Title: Quantum algorithms for Gibbs sampling and hitting-time estimation

Abstract

Citation Formats

Optimising Matrix Product State Simulations of Shor's Algorithm journal, January 2019

Optimising Matrix Product State Simulations of Shor's Algorithm text, January 2017

Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing text, January 2018

Optimising Matrix Product State Simulations of Shor's Algorithm
journal, January 2019

Optimising Matrix Product State Simulations of Shor's Algorithm
text, January 2017

Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing
text, January 2018