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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, Zixin Zhou

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ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Quantum computing
  • Quantum complexity theory
  • entanglement

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.CCC.2024.21

Author Details

Adam Bouland
  • Department of Computer Science, Stanford University, CA, USA
Bill Fefferman
  • Department of Computer Science, University of Chicago, IL, USA
Soumik Ghosh
  • Department of Computer Science, University of Chicago, IL, USA
Tony Metger
  • ETH Zürich, Switzerland
Umesh Vazirani
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
Chenyi Zhang
  • Department of Computer Science, Stanford University, CA, USA
Zixin Zhou
  • Department of Computer Science, Stanford University, CA, USA

Funding

B.F. and S.G. acknowledge support from AFOSR (FA9550-21-1-0008). This material is based upon work partially supported by the National Science Foundation under Grant CCF-2044923 (CAREER) and by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers (Q-NEXT). This research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. A.B., B.F., C.Z., and Z.Z. were supported in part by the DOE QuantISED grant DE-SC0020360. A.B. and C.Z. were supported in part by the U.S. DOE Office of Science under Award Number DE-SC0020266. A.B. was supported in part by the AFOSR under grant FA9550-21-1-0392. C.Z. was supported in part by the Shoucheng Zhang graduate fellowship. T.M. acknowledges support from SNSF Grant No. 200021_188541, the ETH Zurich Quantum Center, and an ETH Doc.Mobility Fellowship. U.V. was supported in part by DOE NQISRC QSA grant FP00010905, NSF QLCI Grant No. 2016245, and MURI Grant FA9550-18-1-0161.

Acknowledgements

We thank Rotem Arnon-Friedman, Jordan Docter, Tudor Giurgica-Tiron, Andru Gheorghiu, Hsin-Yuan Huang, Vinod Vaikuntanathan, and Thomas Vidick for helpful discussions. We thank the Simons Institute for the Theory of Computing, where some of this work was conducted.

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