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DOI: 10.4230/LIPIcs.CCC.2024.21
Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure

Authors: Adam Bouland, Bill Fefferman, Soumik Ghosh, Tony Metger, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
Given a local Hamiltonian, how difficult is it to determine the entanglement structure of its ground state? We show that this problem is computationally intractable even if one is only trying to decide if the ground state is volume-law vs near area-law entangled. We prove this by constructing strong forms of pseudoentanglement in a public-key setting, where the circuits used to prepare the states are public knowledge. In particular, we construct two families of quantum circuits which produce volume-law vs near area-law entangled states, but nonetheless the classical descriptions of the circuits are indistinguishable under the Learning with Errors (LWE) assumption. Indistinguishability of the circuits then allows us to translate our construction to Hamiltonians. Our work opens new directions in Hamiltonian complexity, for example whether it is difficult to learn certain phases of matter.
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Adam Bouland, Bill Fefferman, Soumik Ghosh, Tony Metger, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2024.21,
  author =	{Bouland, Adam and Fefferman, Bill and Ghosh, Soumik and Metger, Tony and Vazirani, Umesh and Zhang, Chenyi and Zhou, Zixin},
  title =	{{Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{21:1--21:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.21},
  URN =		{urn:nbn:de:0030-drops-204175},
  doi =		{10.4230/LIPIcs.CCC.2024.21},
  annote =	{Keywords: Quantum computing, Quantum complexity theory, entanglement}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.2
Quantum Pseudoentanglement

Authors: Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement", a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states - first defined by Ji, Liu and Song - which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to log n across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.
Cite as

Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Quantum Pseudoentanglement. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aaronson_et_al:LIPIcs.ITCS.2024.2,
  author =	{Aaronson, Scott and Bouland, Adam and Fefferman, Bill and Ghosh, Soumik and Vazirani, Umesh and Zhang, Chenyi and Zhou, Zixin},
  title =	{{Quantum Pseudoentanglement}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.2},
  URN =		{urn:nbn:de:0030-drops-195300},
  doi =		{10.4230/LIPIcs.ITCS.2024.2},
  annote =	{Keywords: Quantum computing, Quantum complexity theory, entanglement}
}
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