Abstract
Tensor network methods are routinely used in approximating various equilibrium and non-equilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature. These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size. To address this problem, we construct PEPOs that approximate, for all local observables, $i)$ their thermal expectation values and $ii)$ their Heisenberg time evolution. The bond dimension required does not depend on system size, but only on the temperature or time. We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.
Highlights
The classical simulation of quantum many-body systems is an important challenge for many different fields, including condensed-matter physics, quantum chemistry, quantum information, and high-energy physics
Tensor-network methods based on the densitymatrix renormalization group (DMRG) algorithm [1] are routinely used for the simulation of many important physical situations
We introduce a tensor-network construction of a linear map that outputs different projected entangled-pair operator (PEPO)
Summary
The classical simulation of quantum many-body systems is an important challenge for many different fields, including condensed-matter physics, quantum chemistry, quantum information, and high-energy physics. For thermal states and time evolution, previous results indicate that it is possible to simulate specific local properties in an efficient way [37,40] It was previously not known whether there exist particular tensor networks that approximate all the local properties of a system with a bond dimension independent of system size. This is because said guarantees show that the methods target quantities that can, in principle, be computed efficiently We prove these guarantees with the aid of previous results on global approximations [35–39], combined with ideas that allow us to exploit the locality of the problem. For thermal states, this is the principle usually known as the local indistinguishability [37,47–49], which relies on the clustering of correlations [49,50]. The technical proofs and further background are placed in the Appendices
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