Random regular graph states are complex at almost any depth

TL;DR

This work studies average-case complexity of simulating random $d$-regular graph states, linking connectivity, entanglement, and computational hardness across three regimes: constant degree, intermediate degree, and high degree. It provides three main results: anticoncentration for depth-$d$ IQP circuits and constant-degree graphs (via a stat-mech/Krawtchouk framework), universality for MBQC by locating polynomial grid subgraphs in $d=\Theta(n^c)$ with $c\in(1/2,1)$ (using switching methods and second-moment analysis), and a nontrivial geometric-entanglement bound for high-degree graphs (via epsilon-nets and Markov-chain analysis). The findings suggest that structured randomness in graph states yields average-case hardness and universality features that challenge naive classical simulation and extend our understanding of when quantum resources enable computational advantage. Together, these results illuminate how graph connectivity shapes entanglement and simulability, with implications for MBQC, quantum advantage demonstrations, and future noisy/architectural analyses.

Abstract

Graph states are fundamental objects in the theory of quantum information due to their simple classical description and rich entanglement structure. They are also intimately related to IQP circuits, which have applications in quantum pseudorandomness and quantum advantage. For us, they are a toy model to understand the relation between circuit connectivity, entanglement structure and computational complexity. In the worst case, a strict dichotomy in the computational universality of such graph states appears as a function of the degree $d$ of a regular graph state [GDH+23]. In this paper, we study the average-case complexity of simulating random graph states of varying degree when measured in random product bases and give distinct evidence that a similar complexity-theoretic dichotomy exists in the average case. Specifically, we consider random $d$-regular graph states and prove three distinct results: First, we exhibit two families of IQP circuits of depth $d$ and show that they anticoncentrate for any $2 < d = o(n^{1/2})$ when measured in a random $X$-$Y$-plane product basis. This implies anticoncentration for random constant-regular graph states. Second, in the regime $d = Θ(n^c)$ with $c \in (0,1)$, we prove that random $d$-regular graph states contain polynomially large grid graphs as induced subgraphs with high probability. This implies that they are universal resource states for measurement-based computation. Third, in the regime of high degree ($d\sim n/2$), we show that random graph states are not sufficiently entangled to be trivially classically simulable, unlike Haar random states. Proving the three results requires different techniques -- the analysis of a classical statistical-mechanics model using Krawtchouck polynomials, graph theoretic analysis using the switching method, and analysis of the ranks of submatrices of random adjacency matrices, respectively.

Figures (2)

• Figure 1: Exact values of $\mathbb E_{G, \theta}[m_2(G, \theta)]$ (evaluated via \ref{['eq:anticoncentration random matching']}) for graph states drawn from (a) the random pairing ensemble $\mathcal{G}_p(d,n)$ and (b) the random matching ensemble $\mathcal{G}_m(d,n)$ for various values of $d$ as a function of $n$. The opaque (transparent) loosely dotted lines denote the asymptotic values of $m_2 = 2(3)$ of the average second moment. Solid (dashed) lines represent odd (even) values of $d$.
• Figure 2: For $n = 16,d = 3$, we show the dependence on $k,l$ of (left)$M_p(n,k,l,d)$ and (right)$2^{-n } \binom nk \binom{n-k}{k} |M_p(n,k,l,d)|$ for $k,l > 0$ and $k+l<n$. The red area corresponds to the region $1\le k \le l \le n -k-l$ in which we need to achieve a nontrivial bound.

Theorems & Definitions (68)

• Theorem 1: name=Anticoncentration of random pairing and matching graph states,label=thm:anticoncentration
• Corollary 2: name=Anticoncentration of random $d$-regular graph states,label=cor:anticoncentration
• Theorem 3: name=Induced grid graphs,label=thm:grid_subgraph
• Corollary 4: name=Sparsified induced grid graphs,label=thm:grid_subgraph_sparsified
• Theorem 5: name=Geometric entanglement upper bound for graph states,label=thm:geom_ent_upper
• Theorem 6: name=Geometric entanglement lower bound for stabilizer states,label=thm:geom_ent_lower_stab
• Corollary 7: name=Geometric entanglement lower bound for graph states,label=cor:geom_ent_lower_graph
• Definition 8: uniformly random $d$-regular graphs
• Definition 9: Matching model of $d$-regular multigraphs
• Definition 10: Pairing model of $d$-regular multigraphs