  The recently-defined No Low-energy Sampleable States (NLSS) conjecture of
Gharibian and Le Gall [GL22] posits the existence of a family of local
Hamiltonians where all states of low-enough constant energy do not have
succinct representations allowing perfect sampling access. States that can be
prepared using only Clifford gates (i.e. stabilizer states) are an example of
sampleable states, so the NLSS conjecture implies the existence of local
Hamiltonians whose low-energy space contains no stabilizer states. We describe
families that exhibit this requisite property via a simple alteration to local
Hamiltonians corresponding to CSS codes. Our method can also be applied to the
recent NLTS Hamiltonians of Anshu, Breuckmann, and Nirkhe [ABN22], resulting in
a family of local Hamiltonians whose low-energy space contains neither
stabilizer states nor trivial states. We hope that our techniques will
eventually be helpful for constructing Hamiltonians which simultaneously
satisfy NLSS and NLTS.
