UC Berkeley

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 thesis describes a line of research culminating in a proof of the NLTS conjecture, first presentedby Anshu, Breuckmann, and Nirkhe. The construction is based on quantum error correction and the thesis elaborates on how error correction, local Hamiltonians, and low-depth quantum circuits are related.