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  • Open Access

Random Regular Graph States Are Complex at Almost Any Depth

Soumik Ghosh1, Dominik Hangleiter2,*, and Jonas Helsen3

  • *Contact author: mail@dhangleiter.eu
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PRX Quantum 6, 040344 – Published 24 November, 2025

DOI: https://doi.org/10.1103/52xz-3hpc

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 instantaneous quantum polynomial-time (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 [Ghosh et al., Phys. Rev. Lett. 131, 030601 (2023)]. 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 show two families of IQP circuits of depth d and show that they anticoncentrate for any 2<d=o(n1/2) 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(nc) with c∈(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∼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.

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Popular Summary

Graph states play a fundamental role in many areas of quantum physics, including in quantum computation, complexity of physical systems studies, and quantum advantage experiments. In this work, we study how typical the computational properties of graph states are for the special ensemble of “d-regular graph states.” These states have a special symmetry in the underlying graph—each vertex is connected to d other vertices.

We show that for small values of d, d-regular graph states have the so-called anticoncentration property, which is a key ingredient for quantum advantage. For larger values of d, they are even more powerful—they can be used to simulate any quantum computation. For even larger values of d, we show that their entanglement structure is very different from that of generic quantum states. This underscores that they are special and that we need to find new techniques to analyze them.

Our work takes one of the first steps in characterizing the typical-case behavior of states with a significant amount of structure, which can eventually help us understand the behavior of specific algorithms. Since constant-regularity graph states have many desirable properties with regard to error detection and fault tolerance, this also strongly motivates the use of random regular graph states in the next generation of quantum advantage experiments.

From a different perspective, our work shows that, in a certain regime, typical d-regular graph states are resources for quantum computation. This property has deep connections to the interplay between computational and physical properties in phases of matter.

Article Text

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PRX Quantum 6, 040344– Published 24 November, 2025
  • Received 14 January 2025
  • Revised 13 June 2025
  • Accepted 22 October 2025
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