Fawzi, Hamza ;
Fawzi, Omar ;
Scalet, Samuel O.
A SubpolynomialTime Algorithm for the Free Energy of OneDimensional Quantum Systems in the Thermodynamic Limit
Abstract
We introduce a classical algorithm to approximate the free energy of local, translationinvariant, onedimensional quantum systems in the thermodynamic limit of infinite chain size. While the ground state problem (i.e., the free energy at temperature T = 0) for these systems is expected to be computationally hard even for quantum computers, our algorithm runs for any fixed temperature T > 0 in subpolynomial time, i.e., in time O((1/ε)^c) for any constant c > 0 where ε is the additive approximation error. Previously, the best known algorithm had a runtime that is polynomial in 1/ε where the degree of the polynomial is exponential in the inverse temperature 1/T. Our algorithm is also particularly simple as it reduces to the computation of the spectral radius of a linear map. This linear map has an interpretation as a noncommutative transfer matrix and has been studied previously to prove results on the analyticity of the free energy and the decay of correlations. We also show that the corresponding eigenvector of this map gives an approximation of the marginal of the Gibbs state and thereby allows for the computation of various thermodynamic properties of the quantum system.
BibTeX  Entry
@InProceedings{fawzi_et_al:LIPIcs.ITCS.2023.49,
author = {Fawzi, Hamza and Fawzi, Omar and Scalet, Samuel O.},
title = {{A SubpolynomialTime Algorithm for the Free Energy of OneDimensional Quantum Systems in the Thermodynamic Limit}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {49:149:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17552},
URN = {urn:nbn:de:0030drops175520},
doi = {10.4230/LIPIcs.ITCS.2023.49},
annote = {Keywords: Onedimensional quantum systems, Free energy}
}
01.02.2023
Keywords: 

Onedimensional quantum systems, Free energy 
Seminar: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Issue date: 

2023 
Date of publication: 

01.02.2023 