Circuit lower bounds for low-energy states of quantum code Hamiltonians

TL;DR

This paper proves circuit lower bounds for low-energy states of quantum code Hamiltonians, advancing understanding of the NLTS conjecture and quantum PCP by using entropic and indistinguishability techniques.

Contribution

It introduces new methods to establish super-constant circuit lower bounds for low-energy states of certain quantum error-correcting code Hamiltonians, including non-locally testable codes.

Findings

Super-constant circuit lower bounds for low-energy states of LDPC stabilizer codes.

Low-depth states cannot approximate ground energy in 2D lattice systems.

Techniques apply to codes with nearly linear rate or distance.

Abstract

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings, 2014 -- which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states -- identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.