  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.
