License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.6
URN: urn:nbn:de:0030-drops-156023
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15602/
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Anshu, Anurag ; Nirkhe, Chinmay

Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians

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Abstract

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings [Freedman and Hastings, 2014] - which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states - identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes.
For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.

BibTeX - Entry

@InProceedings{anshu_et_al:LIPIcs.ITCS.2022.6,
  author =	{Anshu, Anurag and Nirkhe, Chinmay},
  title =	{{Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{6:1--6:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15602},
  URN =		{urn:nbn:de:0030-drops-156023},
  doi =		{10.4230/LIPIcs.ITCS.2022.6},
  annote =	{Keywords: quantum pcps, local hamiltonians, error-correcting codes}
}

Keywords: quantum pcps, local hamiltonians, error-correcting codes
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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