[ { "id": "https://authors.library.caltech.edu/records/f9txd-6jq82", "eprint_id": 43591, "eprint_status": "archive", "datestamp": "2023-08-19 14:18:46", "lastmod": "2023-10-25 23:41:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Arad-Itai", "name": { "family": "Arad", "given": "Itai" } }, { "id": "Kitaev-A", "name": { "family": "Kitaev", "given": "Alexei" } }, { "id": "Landau-Zeph", "name": { "family": "Landau", "given": "Zeph" } }, { "id": "Vazirani-Umesh-V", "name": { "family": "Vazirani", "given": "Umesh" } } ] }, "title": "An area law and sub-exponential algorithm for 1D systems", "ispublished": "unpub", "full_text_status": "public", "note": "We are grateful to Dorit Aharonov, Fernando Brandao, and Matt Hastings for inspiring discussions\nabout the above and related topics.\n\n
Submitted - 1301.1162v1.pdf
", "abstract": "We give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result [1]. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap \u03b5 > 0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by O(log^3 d/\u03b5 ). Our approach uses the framework of Refs. [2, 3] to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the 1/\u03b5 dependence of the bound in Ref. [3]. To achieve that, we establish a new, \"random-walk like\", bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: ER(H^\u2113) \u2264 (\u2113d)O(\u221a\u2113). Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension B = \u03b5 ^O(log^(3/4) n/\u03b5^(1/4)). Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1=poly(n) approximation of the ground energy with a subexponential running time T \u2264 exp (\u03b5O(log^(3/4) n/\u03b5^(1/4))).", "date": "2014-01-30", "date_type": "published", "id_number": "CaltechAUTHORS:20140130-142058060", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140130-142058060", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "local_group": { "items": [ { "id": "IQIM" } ] }, "doi": "10.48550/arXiv.1301.1162v1", "primary_object": { "basename": "1301.1162v1.pdf", "url": "https://authors.library.caltech.edu/records/f9txd-6jq82/files/1301.1162v1.pdf" }, "resource_type": "monograph", "pub_year": "2014", "author_list": "Arad, Itai; Kitaev, Alexei; et el." }, { "id": "https://authors.library.caltech.edu/records/tf8r6-qs172", "eprint_id": 32421, "eprint_status": "archive", "datestamp": "2023-08-19 10:37:37", "lastmod": "2023-10-17 23:48:25", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Kitaev-A", "name": { "family": "Kitaev", "given": "Alexei" } }, { "id": "Wang-Zhenghan", "name": { "family": "Wang", "given": "Zhenghan" }, "orcid": "0000-0002-5253-6400" } ] }, "title": "Solutions to generalized Yang-Baxter equations via ribbon fusion categories", "ispublished": "unpub", "full_text_status": "public", "note": "The second author is partially supported by NSF DMS 1108736 and would like to thank E.\nRowell for observing (3) of Thm. 2.5, S. Hong for helping on 6j symbols, and R. Chen for\nnumerically testing the solutions.\n\nSubmitted - 1203.1063v2.pdf
", "abstract": "Inspired by quantum information theory, we look for representations of the braid groups B_n on V^(\u2297(n+m\u22122)) for some fixed vector space V such\nthat each braid generator \u03c3_i, i = 1, ..., n\u22121, acts on m consecutive tensor factors\nfrom i through i +m\u22121. The braid relation for m = 2 is essentially the Yang-Baxter equation, and the cases for m > 2 are called generalized Yang-Baxter\nequations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given\nfrom the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and\nSU(3)_3. The solution from the Jones-Kauffman theory at a 6th root of unity,\nwhich is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.", "date": "2012-07-19", "date_type": "published", "id_number": "CaltechAUTHORS:20120713-102318475", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120713-102318475", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1108736" } ] }, "local_group": { "items": [ { "id": "IQIM" } ] }, "doi": "10.48550/arXiv.1203.1063", "primary_object": { "basename": "1203.1063v2.pdf", "url": "https://authors.library.caltech.edu/records/tf8r6-qs172/files/1203.1063v2.pdf" }, "resource_type": "monograph", "pub_year": "2012", "author_list": "Kitaev, Alexei and Wang, Zhenghan" } ]