The Physics of (good) LDPC Codes I. Gauging and dualities

Low-depth parity check (LDPC) codes are a paradigm of error correction that allow for spatially non-local interactions between (qu)bits, while still enforcing that each (qu)bit interacts only with finitely many others. On expander graphs, they can give rise to ``good codes'' that combine a finite encoding rate with an optimal scaling of the code distance, which governs the code's robustness against noise. Such codes have garnered much recent attention due to two breakthrough developments: the construction of good quantum LDPC codes and good locally testable classical LDPC codes, using similar methods. Here we explore these developments from a physics lens, establishing connections between LDPC codes and ordered phases of matter defined for systems with non-local interactions and on non-Euclidean geometries. We generalize the physical notions of Kramers-Wannier (KW) dualities and gauge theories to this context, using the notion of chain complexes as an organizing principle. We discuss gauge theories based on generic classical LDPC codes and make a distinction between two classes, based on whether their excitations are point-like or extended. For the former, we describe KW dualities, analogous to the 1D Ising model and describe the role played by ``boundary conditions''. For the latter we generalize Wegner's duality to obtain generic quantum LDPC codes within the deconfined phase of a Z_2 gauge theory. We show that all known examples of good quantum LDPC codes are obtained by gauging locally testable classical codes. We also construct cluster Hamiltonians from arbitrary classical codes, related to the Higgs phase of the gauge theory, and formulate generalizations of the Kennedy-Tasaki duality transformation. We use the chain complex language to discuss edge modes and non-local order parameters for these models, initiating the study of SPT phases in non-Euclidean geometries

High Dimensional Expanders and Property Testing

We show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable. Using this connection, we derive several testability results

Single-shot decoding of good quantum LDPC codes

Quantum Tanner codes constitute a family of quantum low-density parity-check (LDPC) codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Z\'emor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.Comment: 35 pages, 3 figure

Expanders with Symmetry: Constructions and Applications

Expanders are sparse yet well-connected graphs with numerous theoretical and practical uses. Symmetry is a valuable structure for expanders as it enables efficient algorithms and a richer set of applications. This thesis studies expanders with symmetry, giving new constructions and applications. We extend expander construction techniques to work with symmetry and give explicit constructions of expanders with varying quality of expansion and symmetries of various groups. In particular, we construct graphs with large Abelian group symmetries via the technique of \textit{graph lifts}. We also give a generic amplification procedure that converts a weak expander to an almost optimal one while preserving symmetries. This procedure is obtained by generalizing prior amplification techniques that work for Cayley graphs over Abelian groups to Cayley graphs over any finite group. In particular, we obtain almost-Ramanujan expanders over every non-abelian finite simple group. We then explore the utility of having both symmetry and expansion simultaneously. We obtain explicit quantum LDPC codes of almost linear distance and \textit{good} classical quasi-cyclic codes with varying circulant sizes using prior results and our constructions of graphs with Abelian symmetries. We show how our generic amplification machinery boosts various structured expander-like objects: \textit{quantum expanders}, \textit{dimension expanders}, and \textit{monotone expanders}. Finally, we prove a structural result about expanding Cayley graphs, showing that they satisfy a \enquote{degree-2} variant of the \textit{expander mixing lemma}. As an application of this, we give a randomness-efficient query algorithm for \textit{homomorphism testing} of unitary-valued functions on finite groups and a derandomized version of the celebrated Babai--Nikolov--Pyber (BNP) lemma

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D

Explicit Abelian Lifts and Quantum LDPC Codes

For an abelian group H acting on the set [?], an (H,?)-lift of a graph G? is a graph obtained by replacing each vertex by ? 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\u27Donnell [STOC 2021] achieving distance ??(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance ?(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(?), constant degree d ? 3 and ? > 0, we construct explicit d-regular expander graphs G obtained from an (H,?)-lift of a (suitable) base n-vertex expander G? with the following parameters: ii) ?(G) ? 2?{d-1} + ?, for any lift size ? ? 2^{n^{?}} where ? = ?(d,?), iii) ?(G) ? ? ? d, for any lift size ? ? 2^{n^{??}} for a fixed ?? > 0, when d ? d?(?), or iv) ?(G) ? O?(?d), for lift size "exactly" ? = 2^{?(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\u27Donnell 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