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Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products

Authors Louis Golowich , Venkatesan Guruswami

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ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Quantum Error Correction
  • Quantum LDPC Code
  • Homological Product
  • Iterative Construction

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Abstract

The first linear-distance quantum LDPC codes were recently constructed by a line of breakthrough works (culminating in the result of Panteleev & Kalachev, 2021). All such constructions, even when allowing for almost-linear distance, are based on an operation called a balanced (or lifted) product, which is used in a one-shot manner to combine a pair of large classical codes possessing a group symmetry.
We present a new construction of almost-linear distance quantum LDPC codes that is iterative in nature. Our construction is based on a more basic and widely used product, namely the homological product (i.e. the tensor product of chain complexes).
Specifically, for every ε > 0, we obtain a family of [[N,N^{1-ε},N^{1-ε}]] (subsystem) quantum LDPC codes via repeated homological products of a constant-sized quantum locally testable code. Our key idea is to remove certain low-weight codewords using subsystem codes (while still maintaining constant stabilizer weight), in order to circumvent a particular obstruction that limited the distance of many prior homological product code constructions to at most Õ(√N).

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Louis Golowich and Venkatesan Guruswami. Quantum LDPC Codes of Almost Linear Distance via Iterated Homological Products. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 25:1-25:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.25

Author Details

Louis Golowich
  • UC Berkeley, CA, USA
Venkatesan Guruswami
  • UC Berkeley, CA, USA

Funding

Research supported in part by a ONR grant N00014-24-1-2491 and a UC Noyce initiative award. This work was done in part while the authors were attending the Spring 2024 programs at the Simons Institute for the Theory of Computing.
  • Golowich, Louis: Supported by a National Science Foundation Graduate Research Fellowship under Grant No. DGE 2146752.

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