Lower bounds on the complexity of quantum proofs


Chinmay Nirkhe

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2022-184
August 9, 2022

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 presented by 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.

Advisor: Umesh Vazirani

Author Comments: This thesis was updated for minor formatting and to reflect helpful comments and suggestions. The technical content in this version is correct. The new version is available at: https://www2.eecs.berkeley.edu/Pubs/TechRpts/2022/EECS-2022-236.pdf