Title: Explicit Abelian Lifts and Quantum LDPC Codes

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. more »« less

Jeronimo, Fernando Granha; Quintana, Dylan; Srivastava, Shashank; Tulsiani, Madhur(
, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS))

null
(Ed.)

The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance 1/2-ε and rate Ω(ε 2 ) (where an upper bound of O(ε 2 log(1/ε)) is known). Ta-Shma [STOC 2017] gave an explicit construction of ε-balanced binary codes, where any two distinct codewords are at a distance between 1/2-ε/2 and 1/2+ε/2, achieving a near optimal rate of Ω(ε 2+β ), where β→ 0 as ε→ 0. We develop unique and list decoding algorithms for (a slight modification of) the family of codes constructed by Ta-Shma, in the adversarial error model. We prove the following results for ε-balanced codes with block length N and rate Ω(ε 2+β ) in this family: -For all , there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time N Oε,β(1) . -For any fixed constant β independent of ε, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time (log(1/ε)) O(1) ·N Oβ(1) . -For any , there are explicit ε-balanced codes with rate Ω(ε 2+β ) which can be list decoded up to error 1/2-ε ' in time N Oε,ε' ,β(1), where ε ' ,β→ 0 as ε→ 0. The starting point of our algorithms is the framework for list decoding direct-sum codes develop in Alev et al. [SODA 2020], which uses the Sum-of-Squares SDP hierarchy. The rates obtained there were quasipolynomial in ε. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of Ta-Shma's construction.

Blocki, Jeremiah; Cinkoske, Mike; Lee, Seunghoon; Son, Jin Young(
, Leibniz international proceedings in informatics)

Berenbrink, Petra and
(Ed.)

A directed acyclic graph G = (V,E) is said to be (e,d)-depth robust if for every subset S ⊆ V of |S| ≤ e nodes the graph G-S still contains a directed path of length d. If the graph is (e,d)-depth-robust for any e,d such that e+d ≤ (1-ε)|V| then the graph is said to be ε-extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents running in time polylog|V| which takes as input a node v ∈ V and returns the set of v’s parents. We give the first explicit construction of locally navigable ε-extreme depth-robust graphs with indegree O(log |V|). Previous constructions of ε-extreme depth-robust graphs either had indegree ω̃(log² |V|) or were not explicit.

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space.

Goldreich, Oded; Wigderson, Avi(
, Electronic colloquium on computational complexity)

null
(Ed.)

A graph G is called {\em self-ordered} (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G.
We say that G=(VE) is {\em robustly self-ordered}if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation :VV is proportional to the number of non-fixed-points of .
In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs.
We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph,
it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph).
We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph,
on {\em pairs} of vertices of the original graph, is expanding. In this case the original graph is
(not only robustly self-ordered but) also expanding.
The second construction proceeds in three steps: It boosts the mere existence of robustly self-ordered graphs,
which provides explicit graphs of sublogarithmic size, to an efficient construction of polynomial-size graphs,
and then, repeating it again, to exponential-size(robustly self-ordered) graphs that are locally constructible. This construction can yield robustly self-ordered graphs that are either expanders or highly disconnected,
having logarithmic size connected components.
We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters.
We again demonstrate that such graphs (of linear degree)exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors with very weak parameters but with some additional natural features. We actually show two reductions, one simpler than the other but yielding a less efficient construction when combined with the known constructions of extractors.
We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems
on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs.
One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.
Changes to previous version:
We retract the claims made in our initial posting regarding the construction of non-malleable two-source extractors
(which are quasi-orthogonal) as well as the claims about the construction of relocation-detecting codes (see Theorems 1.5 and 1.6 in the original version). The source of trouble is a fundamental flaw in the proof of Lemma 9.7 (in the original version), which may as well be wrong.
Hence, the original Section 9 was omitted, except that the original Section 9.3 was retained as a new Section 8.3.
The original Section 8 appears as Section 8.0 and 8.1, and Section 8.2 is new.

Abboud, Amir; Cohen-Addad, Vincent; Klein, Philip N.(
, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing)

null
(Ed.)

The Sparsest Cut is a fundamental optimization problem that have been extensively studied. For planar inputs the problem is in P and can be solved in Õ(n 3 ) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90’s and can only achieve O(log n)-approximation in Õ(n) time or 3.5-approximation in Õ(n 2 ) time [Rao, STOC92]. Our main result is an Ω(n 2−ε ) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approxima- tions are inevitable in the near-linear time regime. To complement the lower bound, we provide a 3.3-approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. We also show that our lower bound is not far from optimal by observing an exact algorithm with running time Õ(n 5/2 ) improving upon the Õ(n 3 ) algorithm of Park and Phillips [STOC93]. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Building on our construction we prove near-quadratic lower bounds under SETH for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGET model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.

Jeronimo, Fernando Granha. Explicit Abelian Lifts and Quantum LDPC Codes. Retrieved from https://par.nsf.gov/biblio/10339909. Leibniz international proceedings in informatics 215. Web. doi:10.4230/LIPIcs.ITCS.2022.88.

Jeronimo, Fernando Granha. Explicit Abelian Lifts and Quantum LDPC Codes. Leibniz international proceedings in informatics, 215 (). Retrieved from https://par.nsf.gov/biblio/10339909. https://doi.org/10.4230/LIPIcs.ITCS.2022.88

@article{osti_10339909,
place = {Country unknown/Code not available},
title = {Explicit Abelian Lifts and Quantum LDPC Codes},
url = {https://par.nsf.gov/biblio/10339909},
DOI = {10.4230/LIPIcs.ITCS.2022.88},
abstractNote = {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.},
journal = {Leibniz international proceedings in informatics},
volume = {215},
author = {Jeronimo, Fernando Granha},
editor = {Braverman, Mark}
}

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