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We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>&amp;#x221D;<\/mml:mo><mml:mn>1<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:mi>log<\/mml:mi><mml:mo>&amp;#x2061;<\/mml:mo><mml:mi>N<\/mml:mi><\/mml:math> can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.<\/jats:p>","DOI":"10.22331\/q-2024-07-10-1401","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T14:05:28Z","timestamp":1720620328000},"page":"1401","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Matrix product state approximations to quantum states of low energy variance"],"prefix":"10.22331","volume":"8","author":[{"given":"Kshiti Sneh","family":"Rai","sequence":"first","affiliation":[{"name":"Instituut-Lorentz, Niels Bohrweg 2, Leiden, NL-2333 CA, The Netherlands"},{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Stra\u00dfe 1, D-85748 Garching, Germany"}]},{"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Stra\u00dfe 1, D-85748 Garching, Germany"}]},{"given":"\u00c1lvaro M.","family":"Alhambra","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica Te\u00f3rica UAM\/CSIC, C\/ Nicol\u00e1s Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain"},{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Stra\u00dfe 1, D-85748 Garching, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2024,7,10]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac. Criticality, the area law, and the computational power of projected entangled pair states. Physical Review Letters, 96 (22), jun 2006. 10.1103\/physrevlett.96.220601. URL https:\/\/doi.org\/10.1103.","DOI":"10.1103\/physrevlett.96.220601"},{"key":"1","doi-asserted-by":"publisher","unstructured":"J. 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