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Documents authored by Paredes, Pedro

Document
DOI: 10.4230/LIPIcs.ITCS.2022.88
Explicit Abelian Lifts and Quantum LDPC Codes

Authors: Fernando Granha Jeronimo, Tushant Mittal, Ryan O'Donnell, Pedro Paredes, and Madhur Tulsiani

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
For an abelian group H acting on the set [𝓁], an (H,𝓁)-lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 copies, and each edge by a matching corresponding to the action of an element of H. Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O'Donnell [STOC 2021] achieving distance Ω̃(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ω(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019]. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ⩽ Sym(𝓁), constant degree d ≥ 3 and ε > 0, we construct explicit d-regular expander graphs G obtained from an (H,𝓁)-lift of a (suitable) base n-vertex expander G₀ with the following parameters: ii) λ(G) ≤ 2√{d-1} + ε, for any lift size 𝓁 ≤ 2^{n^{δ}} where δ = δ(d,ε), iii) λ(G) ≤ ε ⋅ d, for any lift size 𝓁 ≤ 2^{n^{δ₀}} for a fixed δ₀ > 0, when d ≥ d₀(ε), or iv) λ(G) ≤ Õ(√d), for lift size "exactly" 𝓁 = 2^{Θ(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.
Cite as

Fernando Granha Jeronimo, Tushant Mittal, Ryan O'Donnell, Pedro Paredes, and Madhur Tulsiani. Explicit Abelian Lifts and Quantum LDPC Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 88:1-88:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jeronimo_et_al:LIPIcs.ITCS.2022.88,
  author =	{Jeronimo, Fernando Granha and Mittal, Tushant and O'Donnell, Ryan and Paredes, Pedro and Tulsiani, Madhur},
  title =	{{Explicit Abelian Lifts and Quantum LDPC Codes}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{88:1--88:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.88},
  URN =		{urn:nbn:de:0030-drops-156846},
  doi =		{10.4230/LIPIcs.ITCS.2022.88},
  annote =	{Keywords: Graph lifts, expander graphs, quasi-cyclic LDPC codes, quantum LDPC codes}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.55
Spectrum Preserving Short Cycle Removal on Regular Graphs

Authors: Pedro Paredes

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given d ⩾ 3 and n, there exists an explicit distribution of d-regular Θ(n)-vertex graphs where with high probability its samples have girth Ω(log_{d-1} n) and are ε-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2√{d-1} + ε (excluding the single trivial eigenvalue of d). Then, for every constant d ⩾ 3 and ε > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on Θ(n)-vertices that is ε-near-Ramanujan and has girth Ω(√{log n}), based on the work of [Mohanty et al., 2020].
Cite as

Pedro Paredes. Spectrum Preserving Short Cycle Removal on Regular Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{paredes:LIPIcs.STACS.2021.55,
  author =	{Paredes, Pedro},
  title =	{{Spectrum Preserving Short Cycle Removal on Regular Graphs}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{55:1--55:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.55},
  URN =		{urn:nbn:de:0030-drops-137000},
  doi =		{10.4230/LIPIcs.STACS.2021.55},
  annote =	{Keywords: Ramanujan Graphs, High Girth Regular Graphs}
}
Document
DOI: 10.4230/LIPIcs.STACS.2020.50
The SDP Value for Random Two-Eigenvalue CSPs

Authors: Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
We precisely determine the SDP value (equivalently, quantum value) of large random instances of certain kinds of constraint satisfaction problems, "two-eigenvalue 2CSPs". We show this SDP value coincides with the spectral relaxation value, possibly indicating a computational threshold. Our analysis extends the previously resolved cases of random regular 2XOR and NAE-3SAT, and includes new cases such as random Sort₄ (equivalently, CHSH) and Forrelation CSPs. Our techniques include new generalizations of the nonbacktracking operator, the Ihara-Bass Formula, and the Friedman/Bordenave proof of Alon’s Conjecture.
Cite as

Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. The SDP Value for Random Two-Eigenvalue CSPs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 50:1-50:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mohanty_et_al:LIPIcs.STACS.2020.50,
  author =	{Mohanty, Sidhanth and O'Donnell, Ryan and Paredes, Pedro},
  title =	{{The SDP Value for Random Two-Eigenvalue CSPs}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{50:1--50:45},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.50},
  URN =		{urn:nbn:de:0030-drops-119110},
  doi =		{10.4230/LIPIcs.STACS.2020.50},
  annote =	{Keywords: Semidefinite programming, constraint satisfaction problems}
}
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