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Recent constructions of the first asymptotically good quantum LDPC (qLDPC)codes led to two breakthroughs in complexity theory: the NLTS (No Low-EnergyTrivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicitlower 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 qLDPCcodes of low rate:  - Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes oflinear dimension, we show the stronger result that qLDPC codes of arbitrarilysmall positive dimension yield NLTS Hamiltonians.  - The SoS lower bounds of Hopkins and Lin are only weakly explicit becausethey require running Gaussian elimination to find a nontrivial codeword, whichtakes polynomial time. We resolve this shortcoming by introducing a new methodof planting a strongly explicit nontrivial codeword in linear-distance qLDPCcodes, which in turn yields strongly explicit SoS lower bounds.  Our "planted" qLDPC codes may be of independent interest, as they provide anew way of ensuring a qLDPC code has positive dimension without resorting toparity check counting, and therefore provide more flexibility in the codeconstruction.

NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes