In quantum information theory, the no low-energy trivial state (NLTS) conjecture is a precursor to a quantum PCP theorem (qPCP) and posits the existence of families of Hamiltonians with all low-energy states of non-trivial complexity.
NLTS is a consequence of one aspect of qPCP problems – the inability to certify an approximation of local Hamiltonians via NP completeness.
[5] On a high level, it is one property of the non-Newtonian complexity of quantum computation.
[5] NLTS and qPCP conjectures posit the near-infinite complexity involved in predicting the outcome of quantum systems with many interacting states.
Kliesch states the following as a definition for local Hamiltonians in the context of NLTS:[2] Let I ⊂ N be an index set.
[2] Kliesch defines the NLTS property thus:[2] Let I be an infinite set of system sizes.
[1] The qPCP conjecture is a quantum analogue of the classical PCP theorem.
[7] In layman's terms, classical PCP describes the near-infinite complexity involved in predicting the outcome of a system with many resolving states, such as a water bath full of hundreds of magnets.
[6] qPCP increases the complexity by trying to solve PCP for quantum states.