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Explicit Lossless Vertex Expanders

Jun-Ting Hsieh, Alexander Lubotzky, Sidhanth Mohanty, Assaf Reiner, Rachel Yun Zhang

TL;DR

This work resolves the long-standing open problem of explicit constant-degree lossless vertex expanders by constructing an explicit infinite family of $d$-regular graphs where every small subset has at least $(1-\varepsilon)$ times its expected neighbor count, with $\varepsilon\to0$ as $d\to\infty$. The authors achieve two-sided lossless expansion by composing a constant-sized gadget with a base graph derived from Ramanujan Cayley cubical complexes through a tripartite line product, and they encode incidence via the Hadamard code to enforce symmetry and control neighborhood structure. A key technical contribution is a two-part analysis: a left-to-middle bound using small-set subcube density and Loomis–Whitney-type entropy arguments, and a middle-to-right bound using a collision graph and skeleton expansion, together yielding $(1-\varepsilon)$-expansion on both sides. The results yield new families of quantum LDPC codes with linear-time decoding under a free group action and demonstrate the power of high-dimensional expanders for lifting local properties to large combinatorial objects, with potential extensions to ultra-lossless edge expanders and other high-dimensional amplification frameworks.

Abstract

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where every small set $S$ of vertices has $(1-\varepsilon)d|S|$ neighbors (which implies $(1-2\varepsilon)d|S|$ unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

Explicit Lossless Vertex Expanders

TL;DR

This work resolves the long-standing open problem of explicit constant-degree lossless vertex expanders by constructing an explicit infinite family of -regular graphs where every small subset has at least times its expected neighbor count, with as . The authors achieve two-sided lossless expansion by composing a constant-sized gadget with a base graph derived from Ramanujan Cayley cubical complexes through a tripartite line product, and they encode incidence via the Hadamard code to enforce symmetry and control neighborhood structure. A key technical contribution is a two-part analysis: a left-to-middle bound using small-set subcube density and Loomis–Whitney-type entropy arguments, and a middle-to-right bound using a collision graph and skeleton expansion, together yielding -expansion on both sides. The results yield new families of quantum LDPC codes with linear-time decoding under a free group action and demonstrate the power of high-dimensional expanders for lifting local properties to large combinatorial objects, with potential extensions to ultra-lossless edge expanders and other high-dimensional amplification frameworks.

Abstract

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any and sufficiently large , we give an explicit construction of an infinite family of -regular graphs where every small set of vertices has neighbors (which implies unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

Paper Structure

This paper contains 16 sections, 19 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. History of vertex expanders
  3. Cubical complexes
  4. Our construction of lossless expanders
  5. Overview of the analysis
  6. Discussion and future directions
  7. Construction of lossless vertex expanders
  8. Base and gadget graph constructions
  9. Proof of Theorem \ref{['thm:main']}
  10. Cubical complexes and coded incidence graphs
  11. Proof of Lemma \ref{['lem:base-graph']}: structured bipartite graph construction
  12. Small-set subcube density in cubical complexes
  13. Ramanujan cubical complexes
  14. LPS Ramanujan graphs
  15. Construction of Ramanujan Cayley cubical complexes
  16. ...and 1 more sections

Key Result

Theorem 1

For every $\varepsilon > 0$, there exists a sufficiently large integer $d_0$ such that for every integer $d\geqslant d_0$, there is an explicit (deterministic polynomial-time constructible) infinite family of $d$-regular graphs $G$ that are $(1-\varepsilon)$-vertex expanders.

Figures (3)

  • Figure 1: A $3$-dimensional (decorated) cubical complex $X = \mathrm{Cay}(\Gamma; (A_1,A_2,A_3))$, where the vertex set $X(0) = \Gamma \times \mathbb F_2^3$. An element $g\in \Gamma$ and $a_1 \in A_1$, $a_2\in A_2$, $a_3\in A_3$ uniquely specify a face (or cube) $f \in X(3)$, as depicted in the figure. Note that by the properties of $A_1,A_2,A_3$, there exist unique $a_1' \in A_1$, $a_2' \in A_2$ and $a_3' \in A_3$ such that $a_1a_2a_3 = a_2' a_3' a_1'$. The vertex-face incidence graph we need for our base graph construction will be restricted to a linear code $\mathcal{C} \subseteq \mathbb F_2^k$ of large distance --- the bipartite graph between $X(k)$ and $\Gamma \times \mathcal{C} \subseteq X(0)$ where edges indicate containment. Here, a code $\{000, 011, 110, 101\}$ is highlighted.
  • Figure 2: The tripartite line product between a base graph $G$ and gadget graph $H$. In this figure, only the edges from the copy of $H$ placed at the red vertex in $M$ are drawn.
  • Figure 3: The two bipartite base graphs $G_L, G_R$ have the structure that $M$ has $k$ parts, and for $u\in M$ and $v, w \in M$ from a different part, the common neighborhoods $N_{G_R}(u) \cap N_{G_R}(v)$ and $N_{G_R}(u) \cap N_{G_R}(w) \subseteq R$ are disjoint, each corresponding to a special set in $[D_R]$, i.e., $N_{G_R}(u) \cap N_{G_R}(v) = \mathrm{Nbr}_u(Q_i)$ for some special set $Q_i \subseteq [D_R]$. \ref{['fig:special-sets']} shows an example of $\mathsf{RED}(S)$, a subgraph of $G_R$. The middle-to-right analysis involves upper bounding the collisions of the red edges on the right. Here, $u$ has collisions with $v$ and $w$, represented as edges in the collision graph $C$ in \ref{['fig:collision']}. We will show that this cannot happen too often by upper bounding $e(C)$.

Theorems & Definitions (53)

  • Theorem 1: Constant-degree lossless expanders
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4: Tripartite line product
  • Definition 2.5: Structured bipartite graph
  • Definition 2.6: Small-set $j$-neighbor expansion
  • Definition 2.7: Small-set skeleton expansion
  • ...and 43 more