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DOI: 10.4230/LIPIcs.TQC.2025.12

The Rotation-Invariant Hamiltonian Problem Is QMA_EXP-Complete

Authors: Jon Nelson and Daniel Gottesman

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract

In this work we study a variant of the local Hamiltonian problem where we restrict to Hamiltonians that live on a lattice and are invariant under translations and rotations of the lattice. In the one-dimensional case this problem is known to be QMA_EXP-complete. On the other hand, if we fix the lattice length then in the high-dimensional limit the ground state becomes unentangled due to arguments from mean-field theory. We take steps towards understanding this complexity spectrum by studying a problem that is intermediate between these two extremes. Namely, we consider the regime where the lattice dimension is arbitrary but fixed and the lattice length is scaled. We prove that this rotation-invariant Hamiltonian problem is QMA_EXP-complete answering an open question of [Gottesman and Irani, 2013]. This characterizes a broad parameter range in which these rotation-invariant Hamiltonians have high computational complexity.

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Jon Nelson and Daniel Gottesman. The Rotation-Invariant Hamiltonian Problem Is QMA_EXP-Complete. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{nelson_et_al:LIPIcs.TQC.2025.12,
  author =	{Nelson, Jon and Gottesman, Daniel},
  title =	{{The Rotation-Invariant Hamiltonian Problem Is QMA\underlineEXP-Complete}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.12},
  URN =		{urn:nbn:de:0030-drops-240615},
  doi =		{10.4230/LIPIcs.TQC.2025.12},
  annote =	{Keywords: Hamiltonian complexity, Local Hamiltonian problem, Monogamy of entanglement}
}

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DOI: 10.4230/LIPIcs.TQC.2023.14

Local Hamiltonians with No Low-Energy Stabilizer States

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

Published in: LIPIcs, Volume 266, 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

Abstract

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.

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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)

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@InProceedings{coble_et_al:LIPIcs.TQC.2023.14,
  author =	{Coble, Nolan J. and Coudron, Matthew and Nelson, Jon and Nezhadi, Seyed Sajjad},
  title =	{{Local Hamiltonians with No Low-Energy Stabilizer States}},
  booktitle =	{18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-283-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{266},
  editor =	{Fawzi, Omar and Walter, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2023.14},
  URN =		{urn:nbn:de:0030-drops-183243},
  doi =		{10.4230/LIPIcs.TQC.2023.14},
  annote =	{Keywords: Hamiltonian complexity, Stabilizer codes, Low-energy states}
}

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The Rotation-Invariant Hamiltonian Problem Is QMA_EXP-Complete

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Local Hamiltonians with No Low-Energy Stabilizer States

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