Local Hamiltonians with no low-energy stabilizer states

• URL: http://arxiv.org/abs/2302.14755v1
• Date: Tue, 28 Feb 2023 16:55:05 GMT
• Title: Local Hamiltonians with no low-energy stabilizer states
• Authors: Nolan J. Coble, Matthew Coudron, Jon Nelson, Seyed Sajjad Nezhadi
• Abstract summary: A family of local Hamiltonians with low-enough constant energy do not have succinct representations allowing perfect sampling access. 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.
• Score: 0.4588028371034407
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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