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NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes

Authors Louis Golowich , Tali Kaufman

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • NLTS Hamiltonian
  • Quantum PCP
  • Sum-of-squares lower bound
  • Quantum LDPC code

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Abstract

Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22).
In this work, we obtain improvements to both of these results using qLDPC codes of low rate:  
- Whereas Anshu et al. only obtained NLTS Hamiltonians from qLDPC codes of linear dimension, we show the stronger result that qLDPC codes of arbitrarily small positive dimension yield NLTS Hamiltonians. 
- The SoS lower bounds of Hopkins and Lin are only weakly explicit because they require running Gaussian elimination to find a nontrivial codeword, which takes polynomial time. We resolve this shortcoming by introducing a new method of planting a strongly explicit nontrivial codeword in linear-distance qLDPC codes, which in turn yields strongly explicit SoS lower bounds.  Our "planted" qLDPC codes may be of independent interest, as they provide a new way of ensuring a qLDPC code has positive dimension without resorting to parity check counting, and therefore provide more flexibility in the code construction.

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Louis Golowich and Tali Kaufman. NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 54:1-54:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.54

Author Details

Louis Golowich
  • Department of Computer Science, University of California at Berkeley, CA, USA
Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel

Funding

This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.
  • Golowich, Louis: Supported by a National Science Foundation Graduate Research Fellowship under Grant No. DGE 2146752, and in part by V. Guruswami’s Simons Investigator award and UC Noyce Initiative Award award.

Acknowledgements

We thank Max Hopkins for numerous helpful discussions, and for bringing the problem of strongly explicit SoS lower bounds to our attention. We also thank Venkat Guruswami for helping to improve the exposition.

References

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