2026-07-24T17:54:16Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19582 2024-03-06T11:04:23Z ddc:004 open_access

NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes Golowich, Louis Kaufman, Tali NLTS Hamiltonian Quantum PCP Sum-of-squares lower bound Quantum LDPC code Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22). In this work, we obtain improvements to both of these results using qLDPC codes of low rate: - Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes of linear dimension, we show the stronger result that qLDPC codes of arbitrarily small positive dimension yield NLTS Hamiltonians. - The SoS lower bounds of Hopkins and Lin are only weakly explicit because they require running Gaussian elimination to find a nontrivial codeword, which takes polynomial time. We resolve this shortcoming by introducing a new method of planting a strongly explicit nontrivial codeword in linear-distance qLDPC codes, which in turn yields strongly explicit SoS lower bounds. Our "planted" qLDPC codes may be of independent interest, as they provide a new way of ensuring a qLDPC code has positive dimension without resorting to parity check counting, and therefore provide more flexibility in the code construction. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Louis Golowich and Tali Kaufman 2024 Is Part Of LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2024.54 urn:nbn:de:0030-drops-195829 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.54 eng https://creativecommons.org/licenses/by/4.0/legalcode