Mathematical proof demonstrates that good quantum low-density parity-check codes yield NLTS local Hamiltonians, resolving Freedman and Hastings' longstanding conjecture in quantum complexity theory.
The no low-energy trivial state (NLTS) conjecture of Freedman and Hastings [ Quantum Inf. Comput., 14 (2014), pp. 144–180] posits that there exist families of Hamiltonians with all low energy states of nontrivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that families of constant-rate and linear-distance qLDPC codes correspond to NLTS local Hamiltonians.
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Breuckmann et al. (2026) studied this question.
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