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Finite correlation length implies efficient preparation of quantum thermal states

Brandão, Fernando G. S. L. and Kastoryano, Michael J. (2019) Finite correlation length implies efficient preparation of quantum thermal states. Communications in Mathematical Physics, 365 (1). pp. 1-16. ISSN 0010-3616. http://resolver.caltech.edu/CaltechAUTHORS:20170726-092817023

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Abstract

Preparing quantum thermal states on a quantum computer is in general a difficult task. We provide a procedure to prepare a thermal state on a quantum computer with a logarithmic depth circuit of local quantum channels assuming that the thermal state correlations satisfy the following two properties: (i) the correlations between two regions are exponentially decaying in the distance between the regions, and (ii) the thermal state is an approximate Markov state for shielded regions. We require both properties to hold for the thermal state of the Hamiltonian on any induced subgraph of the original lattice. Assumption (ii) is satisfied for all commuting Gibbs states, while assumption (i) is satisfied for every model above a critical temperature. Both assumptions are satisfied in one spatial dimension. Moreover, both assumptions are expected to hold above the thermal phase transition for models without any topological order at finite temperature. As a building block, we show that exponential decay of correlation (for thermal states of Hamiltonians on all induced subgraphs) is sufficient to efficiently estimate the expectation value of a local observable. Our proof uses quantum belief propagation, a recent strengthening of strong sub-additivity, and naturally breaks down for states with topological order.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00220-018-3150-8DOIArticle
https://rdcu.be/OC24PublisherFree ReadCube access
https://arxiv.org/abs/1609.07877arXivDiscussion Paper
ORCID:
AuthorORCID
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© Springer-Verlag GmbH Germany, part of Springer Nature 2018. Received: 5 December 2017; Accepted: 28 February 2018; First Online: 15 May 2018. We thank Angelo Lucia and David Perez-Garcia for helpful discussions.We thank Isaac Kim for pointing out the unpublished result of Kitaev to us. MJK was supported by the Carlsberg fund and the Villum fund.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Carlsberg FundUNSPECIFIED
Villum FundUNSPECIFIED
Record Number:CaltechAUTHORS:20170726-092817023
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170726-092817023
Official Citation:Brandão, F.G.S.L. & Kastoryano, M.J. Commun. Math. Phys. (2019) 365: 1. https://doi.org/10.1007/s00220-018-3150-8
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79402
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Jul 2017 16:32
Last Modified:18 Jan 2019 17:25

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