Entanglement from Expansion: High Rank-Width in Deterministic Graphs

Abstract: Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on $n$ vertices with a provably maximum rank-width $Θ(n)$. Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than $Θ(\sqrt{n})$.

• The paper presents a new analytic framework using expansion properties and isoperimetric inequalities to establish linear rank-width lower bounds in bounded-degree graphs.
• It provides deterministic constructions that yield high entanglement in graph states, crucial for efficient quantum state preparation in quantum computing.
• The study compares rank-width with other width parameters, advancing our understanding of structural graph theory and its applications in complex network analysis.

High Rank-Width in Deterministic Graphs: Tight Lower Bounds From Expansion

Introduction and Background

The paper "Entanglement from Expansion: High Rank-Width in Deterministic Graphs" (2606.07110) addresses a fundamental open problem in structural graph theory and quantum information: establishing explicit, deterministic families of bounded-degree graphs with rank-width scaling linearly with the number of vertices. The motivation is twofold: (1) in quantum computing, the entanglement of a graph state is tightly controlled by the rank-width of its underlying graph, directly impacting the efficiency and complexity of quantum state preparation protocols; (2) in structural graph theory, despite extensive understanding of treewidth and related parameters, strong lower bounds for rank-width in explicit non-random families above the $\Theta(\sqrt{n})$ regime were lacking.

The authors resolve this barrier by developing general lower bound techniques leveraging graph expansion, edge-isoperimetric inequalities, and recent advances in analytic methods over product graphs and Boolean function influences. The approach yields not only optimal deterministic constructions but also a unified analytic framework applicable to a broad class of graphs, including hypercubes, products, Cayley graphs, Hamming graphs, tori, grids, and explicit Ramanujan expanders.

Rank-Width via Expansion and Isoperimetric Inequalities

The core technical innovation is a methodology for lower bounding rank-width in terms of graph expansion properties. Edge expansion $h(G)$ provides a lower bound on the number of edges crossing any balanced cut, and the presence of large induced or acyclic matchings in these cuts ensures high cut-rank in the corresponding adjacency submatrices. The rank-width $rw(G)$ and its variant mim-width $mimw(G)$ are then bounded below by the ability to find large induced matchings in every balanced cut. One of the central results states:

$$rw(G) \ge mimw(G) \ge \min_{p\in[1/3,\,1/2]} \frac{\mathscr{F}(p\,|V|)}{\chi'_S(G)}$$

where $\mathscr{F}$ encodes isoperimetric structure (e.g., expansion times set size), and $\chi'_S(G)$ is the strong chromatic index.

This immediately shows that for bounded-degree expanders (i.e., $h(G)=\Omega(1)$, degree $d=O(1)$), the rank-width is $\Theta(n)$. For standard grids and product graphs, improved analytic inequalities from [bollobas1991edge, DS25] and the greedy matching approach sharpen previous bounds from $h(G)$0 up to nearly linear, especially for high-dimensional product graphs.

Boolean Function Analysis and the Generalized KKL Approach

A substantial advance in the paper is the extension to analytic techniques based on Boolean influence. By analyzing functions on the vertex set of product graphs and partitioning cut-edges by coordinate, the generalized Kahn-Kalai-Linial (KKL) theorem for product spaces is invoked. For $h(G)$1, a $h(G)$2-fold product of a base graph $h(G)$3, it is shown:

$h(G)$4

where $h(G)$5 is the order of $h(G)$6, $h(G)$7 is the degree, $h(G)$8 the expansion, and $h(G)$9 the log-Sobolev constant.

For the hypercube, $rw(G)$0, it gives $rw(G)$1, which is almost tight up to polylogarithmic factors, closing the gap with treewidth—unifying discrete and analytic methodologies within product graphs.

Applications to Quantum State Preparation

The theoretical results have significant implications in quantum information theory. Graph states with large rank-width yield highly entangled quantum states; the construction of such states by deterministic means and in constant depth circuits on all-to-all architectures is crucial for quantum supremacy tests and robust entanglement certification. The explicitly constructed expander families (e.g., Ramanujan and near-Ramanujan graphs) provide deterministic graph states with maximum entanglement width $rw(G)$2, matching the previously best-known results for random graphs but now with provable deterministic certificates. The preparation of such states is achievable in constant depth on all-to-all qubit architectures and incurs only $rw(G)$3 depth overhead on physically realizable grid-like layouts [childs2019circuit].

The paper also establishes that for other families (Hamming graphs, grids, tori, Cayley and Johnson graphs), the entanglement width and rank-width can be made nearly linear, with strong lower bounds of the form $rw(G)$4 depending on precise expansion characteristics and graph degree.

Comparison to Other Width Parameters and Structural Implications

An extensive comparison is provided between rank-width and other classical width parameters: pathwidth, treewidth, clique-width, and mim-width. It is shown that the new lower bounds transfer to these parameters (which are polynomially related), and for key families like the hypercube and Johnson graphs, the results match or nearly match tight separator-based upper bounds. For instance, for the hypercube $rw(G)$5, the gap between rank-width and treewidth closes to a factor $rw(G)$6. This resolves several questions in the literature concerning extremal values of rank-width for explicit graph classes.

The use of spectral methods and the Expander Mixing Lemma further allows the authors to establish that explicit Ramanujan and near-Ramanujan graphs achieve $rw(G)$7 with bounded degree, providing for the first time deterministic polynomial-time constructible families with provably maximal rank-width.

Conclusion

This paper achieves a comprehensive characterization and construction of explicit deterministic families of bounded-degree graphs of high rank-width using expansion and analytic combinatorics, breaking the previous $rw(G)$8 ceiling into the linear regime. The analytic framework unifies combinatorial matching arguments, isoperimetric inequalities, and influence-based techniques, with applications spanning quantum state preparation, expansion analysis, and the theory of graph decompositions.

From a quantum information perspective, the results underpin the deterministic preparation of maximally entangled states in constant circuit depth, providing robust candidates for quantum advantage demonstrations. Structurally, the results open opportunities in fixed-parameter algorithms, decompositional methods, and graph minor theory for classes with high complexity.

Further directions include closing the remaining logarithmic gaps for specific product classes (e.g., the hypercube), deepening the connections to Boolean analysis and small-set expansion conjectures, and leveraging these bounds for complexity-theoretic lower bounds in restricted computational models.