English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We give a new proof for the area law for general 1D gapped systems, whichexponentially improves Hastings' famous result \cite{ref:Has07}. Specifically,we show that for a chain of d-dimensional spins, governed by a 1D localHamiltonian with a spectral gap \eps>0, the entanglement entropy of the groundstate with respect to any cut in the chain is upper bounded by $O{\frac{\log^3d}{\eps}}$. Our approach uses the framework Arad et al to construct aChebyshev-based AGSP (Approximate Ground Space Projection) with favorablefactors. However, our construction uses the Hamiltonian directly, instead ofusing the Detectability lemma, which allows us to work with general(frustrated) Hamiltonians, as well as slightly improving the $1/\eps$dependence of the bound in Arad et al. To achieve that, we establish a new,"random-walk like", bound on the entanglement rank of an arbitrary power of a1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\elld)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that theground state is well approximated by a matrix product state with a sublinearbond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction withknown dynamical programing algorithms, yields an algorithm for a 1/\poly(n)approximation of the ground energy with a subexponential running time T\le\exp(e^{O(\log^{3/4}n/\eps^{1/4})}).

An area law and sub-exponential algorithm for 1D systems