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 initiate the study of 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 $2 < d = o(n)$ when measured in a random $X$-$Y$-plane product basis. This implies anticoncentration for random constant-regular graph states. Second, in the regime $d = \Theta(n^c)$ 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.

翻译：图态是量子信息理论中的基本对象，因其简单的经典描述和丰富的纠缠结构而备受关注。它们也与IQP电路密切相关，后者在量子伪随机性和量子优势方面具有应用价值。对我们而言，图态是理解电路连通性、纠缠结构与计算复杂性之间关系的理想模型。在最坏情况下，正则图态的计算通用性会随其度数$d$呈现出严格二分现象[GDH+23]。本文首次研究了在随机乘积基下测量不同度数的随机图态的平均情况模拟复杂度，并给出了明确证据表明平均情况下存在类似的计算复杂性二分现象。具体而言，我们考虑随机$d$正则图态并证明了三个独立结果：首先，我们展示了两类深度为$d$的IQP电路族，并证明当在随机$X$-$Y$平面乘积基下测量时，对于任意满足$2 < d = o(n)$的度数均呈现反集中现象。这意味着随机常数正则图态具有反集中特性。其次，在$d = \Theta(n^c)$（其中$c \in (0,1)$）的区间内，我们证明随机$d$正则图态以高概率包含多项式规模网格图作为诱导子图。这表明它们是基于测量的计算的通用资源态。第三，在高阶度数区间（$d\sim n/2$），我们证明与哈尔随机态不同，随机图态因纠缠不足而无法被平凡地经典模拟。证明这三个结果需要不同的技术手段——分别采用基于Krawtchouck多项式的经典统计力学模型分析、利用交换方法的图论分析，以及随机邻接矩阵子矩阵的秩分析。