# Local Hamiltonians with no low-energy stabilizer states

## TL;DR

This paper constructs local Hamiltonians with low-energy spaces that exclude stabilizer states and trivial states, advancing understanding of the NLSS and NLTS conjectures in quantum complexity.

## Contribution

It introduces a simple modification to CSS code Hamiltonians to eliminate stabilizer states from low-energy spaces, extending to NLTS Hamiltonians.

## Key findings

- Constructed Hamiltonians with no low-energy stabilizer states.
- Extended techniques to NLTS Hamiltonians, excluding trivial states.
- Potential pathway for Hamiltonians satisfying both NLSS and NLTS.

## Abstract

The recently-defined No Low-energy Sampleable States (NLSS) conjecture of Gharibian and Le Gall [GL22] posits the existence of a family of local Hamiltonians where all states of low-enough constant energy do not have succinct representations allowing perfect sampling access. States that can be prepared using only Clifford gates (i.e. stabilizer states) are an example of sampleable states, so the NLSS conjecture implies the existence of local Hamiltonians whose low-energy space contains no stabilizer states. We describe families that exhibit this requisite property via a simple alteration to local Hamiltonians corresponding to CSS codes. Our method can also be applied to the recent NLTS Hamiltonians of Anshu, Breuckmann, and Nirkhe [ABN22], resulting in a family of local Hamiltonians whose low-energy space contains neither stabilizer states nor trivial states. We hope that our techniques will eventually be helpful for constructing Hamiltonians which simultaneously satisfy NLSS and NLTS.