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.

翻译：最近，首个渐近好量子LDPC（qLDPC）码的构造带来了复杂性理论中的两大突破：NLTS（无低能平凡态）定理（Anshu、Breuckmann和Nirkhe，STOC'23），以及针对和积（SoS）层级线性级数的显式下界（Hopkins和Lin，FOCS'22）。本文利用低速率qLDPC码对这两项结果进行了改进：- 虽然Anshu等人仅从线性维度的qLDPC码获得了NLTS哈密顿量，但我们证明了更强的结果：任意小正维度的qLDPC码均可导出NLTS哈密顿量。- Hopkins和Lin的SoS下界仅为弱显式，因其需运行高斯消元法（耗时多项式时间）才能找到非平凡码字。我们通过引入一种新方法解决了这一缺陷：在线性距离qLDPC码中植入强显式非平凡码字，进而获得强显式SoS下界。我们的“植入式”qLDPC码可能具有独立价值，因为它提供了一种无需依赖奇偶校验计数即可确保qLDPC码具有正维度的新途径，从而为码构造提供了更多灵活性。