The quantum PCP conjecture is one of the central open questions in quantum complexity theory. It asserts that calculating even a rough approximation to the ground energy of a local Hamiltonian is intractable even for quantum devices. The widely believed separation between the complexity classes NP and QMA necessitates that polynomial length classical proofs do not exist for calculating the ground energy. This further implies that low-energy states of local Hamiltonians cannot be described by constant-depth quantum circuits.
The "No low-energy trivial states (NLTS)" conjecture by Freedman and Hastings posited the existence of such Hamiltonians. This talk will describe a line of research culminating in proofs of the NLTS conjecture by Anshu, Breuckmann, and Nirkhe. The construction is based on quantum error correction and in the talk, I will elaborate on how error correction, local Hamiltonians, and low-depth quantum circuits are related.