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Derandomised Tensor Product Gap Amplification for Quantum Hamiltonians

Authors Thiago Bergamaschi , Tony Metger , Thomas Vidick , Tina Zhang

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum PCP conjecture
  • gap amplification
  • local Hamiltonians
  • tensor product amplification
  • expander random walks
  • quantum de Finetti theorems

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Abstract

The quantum PCP conjecture asks whether it is QMA-hard to distinguish between high- and low-energy Hamiltonians even when the gap between "high" and "low" energy is large (constant). A natural proof strategy is gap amplification: start from the fact that high- and low-energy Hamiltonians are hard to distinguish if the gap is small (inverse polynomial) [Alexei Y. Kitaev et al., 2002] and amplify the Hamiltonians to increase the energy gap while preserving hardness. Such a gap amplification procedure is at the heart of Dinur’s proof of the classical PCP theorem [Dinur, 2007]. In this work, following Dinur’s model, we introduce a new quantum gap amplification procedure for Hamiltonians which uses random walks on expander graphs to derandomise (subsample the terms of) the tensor product amplification of a Hamiltonian. Curiously, our analysis relies on a new technique inspired by quantum de Finetti theorems, which have previously been used to rule out certain approaches to the quantum PCP conjecture [Fernando G. S. L. Brandão and Aram Wettroth Harrow, 2013].

Cite As Get BibTex

Thiago Bergamaschi, Tony Metger, Thomas Vidick, and Tina Zhang. Derandomised Tensor Product Gap Amplification for Quantum Hamiltonians. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.15

Author Details

Thiago Bergamaschi
  • University of California, Berkeley, CA, USA
Tony Metger
  • ETH Zurich, Zurich, Switzerland
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
Thomas Vidick
  • Weizmann Institute of Science, Rehovot, Israel
  • EPFL, Lausanne, Switzerland
Tina Zhang
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Metger, Tony: acknowledges support from SNSF Grant No. 20QU-1_225171 and NCCR SwissMAP.
  • Vidick, Thomas: supported by AFOSR Grant No. FA9550-22-1-0391 and ERC Consolidator Grant VerNisQDevS (101086733).
  • Zhang, Tina: was partially supported by NSF CAREER grant number 2339948.

Acknowledgements

We thank Anurag Anshu for useful discussions and suggesting the DL-based construction of a streaming quantum PCP presented in Appendix A. We thank Dorit Aharonov, Niko Breuckmann, Sevag Gharibian, Louis Golowich, Anand Natarajan, Quynh The Nguyen, Nikhil Srivastava, and John Wright for helpful discussions. Part of this work was completed while the authors were visiting the Simons Institute for the Theory of Computing.

References

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