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Local Hamiltonians with No Low-Energy Stabilizer States

Authors Nolan J. Coble, Matthew Coudron, Jon Nelson, Seyed Sajjad Nezhadi

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Author Details

Nolan J. Coble
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA
Matthew Coudron
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA
  • National Institute of Standards and Technology, Gaithersburg, MD, USA
Jon Nelson
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA
Seyed Sajjad Nezhadi
  • Joint Center for Quantum Information and Computer Science (QuICS), Department of Computer Science, University of Maryland, College Park, MD, USA


This paper is a contribution of NIST, an agency of the US government, and is not subject to US copyright. We thank Alexander Barg and Chinmay Nirkhe for helpful discussions.

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Nolan J. Coble, Matthew Coudron, Jon Nelson, and Seyed Sajjad Nezhadi. Local Hamiltonians with No Low-Energy Stabilizer States. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 14:1-14:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The recently-defined No Low-energy Sampleable States (NLSS) conjecture of Gharibian and Le Gall [Sevag Gharibian and François {Le Gall}, 2022] 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 [Anshu et al., 2022], 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Hamiltonian complexity
  • Stabilizer codes
  • Low-energy states


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