# 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:

- 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):
- LA-UR-16-21218

Journal ID: ISSN 1533-7146

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: 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.

```
Chowdhury, Anirban Narayan, & Somma, Rolando D.
```*Quantum algorithms for Gibbs sampling and hitting-time estimation*. United States. doi:10.26421/QIC17.1-2.

```
Chowdhury, Anirban Narayan, and Somma, Rolando D. Wed .
"Quantum algorithms for Gibbs sampling and hitting-time estimation". United States.
doi: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 = {Wed Feb 01 00:00:00 EST 2017},

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

}