  In an important recent development, Anshu, Breuckmann, and Nirkhe [ABN22]
resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture
by Freedman and Hastings. The conjecture postulated the existence of
linear-size local Hamiltonians on n qubit systems for which no near-ground
state can be prepared by a shallow (sublogarithmic depth) circuit. The
construction in [ABN22] is based on recently developed good quantum codes.
Earlier results in this direction included the constructions of the so-called
Combinatorial NLTS -- a weaker version of NLTS -- where a state is defined to
have low energy if it violates at most a vanishing fraction of the Hamiltonian
terms [AB22]. These constructions were also based on codes.
  In this paper we provide a "non-code" construction of a class of Hamiltonians
satisfying the Combinatorial NLTS. The construction is inspired by one in
[AB22], but our proof uses the complex solution space geometry of random K-SAT
instead of properties of codes. Specifically, it is known that above a certain
clause-to-variables density the set of satisfying assignments of random K-SAT
exhibits an overlap gap property, which implies that it can be partitioned into
exponentially many clusters each constituting at most an exponentially small
fraction of the total set of satisfying solutions. We establish a certain
robust version of this clustering property for the space of near-satisfying
assignments and show that for our constructed Hamiltonians every combinatorial
near-ground state induces a near-uniform distribution supported by this set.
Standard arguments then are used to show that such distributions cannot be
prepared by quantum circuits with depth o(log n). Since the clustering property
is exhibited by many random structures, including proper coloring and maximum
cut, we anticipate that our approach is extendable to these models as well.
