Random regular graph states are complex at almost any depth

2024-12-09

arXiv.org

AI Summary

This work investigates the average-case quantum simulation complexity of random $d$-regular graph states, aiming to uncover phase-transition behavior and a potential worst-case-like complexity dichotomy as a function of degree $d$. Employing Krawtchouk polynomial analysis, graph-theoretic switching techniques, and rank analysis of random submatrices of adjacency matrices, we establish the first rigorous characterization: for constant $d$, graph states exhibit anti-clustering and are efficiently classically simulable; for $d = Theta(log n)$, they contain large grid minors with high probability, rendering them universal for measurement-based quantum computation (MBQC); for $d = Omega(n^varepsilon)$, they lose the efficient simulability property of Haar-random states, exhibiting nontrivial quantum hardness. Our results establish a degree-dependent computational complexity dichotomy in the average case, providing a new paradigm for assessing quantum advantage in random graph states.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Security and Privacy: Large-scale security measurements

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 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 ($dsim n/2$), 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.

Problem

• Analyzing computational complexity of simulating random regular graph states
• Studying entanglement structure and universality in quantum graph states
• Investigating classical simulability versus quantum advantage in different degree regimes

Innovation

• Anticoncentration proof for IQP circuits using Krawtchouck polynomials
• Grid graph embedding in random regular graphs via switching method
• Adjacency matrix analysis showing nontrivial entanglement in high-degree regimes