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Quantum Information Processing

Explosiveness in percolating quantum networks

Michael Siomau

Published Apr 6, 2026• 0 citations
Publisher

Abstract

A sudden change in size, number, or extent is rare, yet highly desired effect in theoretical models. The abrupt emergence of large-scale connectivity in a percolating network near the transition point gained the name explosive percolation. Classical network science lacks a simple recipe for explosiveness. Quantum networks governed by the laws of quantum mechanics intrinsically possess such an explosive component due to quantum superposition and measurements, which may be employed to achieve macroscopic explosiveness. Any percolating quantum network may exhibit explosive properties due to different state preparation and observation strategies near the transition point.

DOI
10.1007/s11128-026-05152-x

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A sudden change in size, number, or extent is rare, yet highly desired effect in theoretical models. The abrupt emergence of large-scale connectivity in a percolating network near the transition point gained the name explosive percolation. Classical network science lacks a simple recipe for explosiveness. Quantum networks governed by the laws of quantum mechanics intrinsically possess such an explosive component due to quantum superposition and measurements, which may be employed to achieve macroscopic explosiveness. Any percolating quantum network may exhibit explosive properties due to different state preparation and observation strategies near the transition point.
The macroscopic manifestation of quantum coherence in complex many-body systems remains one of the most profound and challenging frontiers in modern physics. While traditional percolation theory successfully describes classical connectivity phase transitions in disordered media, and quantum phase transitions typically focus on zero-temperature Hamiltonian parameter variations, the interplay between network topology and quantum coherence propagation lacks a unifying macroscopic framework. In this paper, we propose the theory of Quantum Coherence Percolation Threshold (QCPT), introducing the concept that quantum coherence can undergo a sudden, non-analytic macroscopic percolation transition across complex networks. We formally define the coherence connection probability, denoted as p_C, and demonstrate that when p_C exceeds a critical threshold p_C*, a global coherent cluster abruptly emerges, characterized by a non-zero macroscopic global coherence measure C_global > 0. By formulating the density matrix evolution, local coherence sharing, and topological constraints over complex graph structures, we establish the analytical foundation for this network-mediated quantum phase transition. This framework bridges quantum information theory, statistical mechanics, and network science, offering a foundational blueprint for understanding macroscopic quantum states in complex environments.
Quantum communication demands efficient distribution of quantum entanglement across a network of connected partners. The search for efficient strategies for the entanglement distribution may be based on percolation theory, which describes evolution of network connectivity with respect to some network parameters. In this framework, the probability to establish perfect entanglement between two remote partners decays exponentially with the distance between them before the percolation transition point, which unambiguously defines percolation properties of any classical network or lattice. Here we introduce quantum networks created with local operations and classical communication, which exhibit non-classical percolation transition points leading to striking communication advantages over those offered by the corresponding classical networks. We show, in particular, how to establish perfect entanglement between any two nodes in the simplest possible network-the 1D chain-using imperfectly entangled pairs of qubits.
Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.
Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [], we demonstrated that the correct critical behaviors of EP in infinite dimensions (complete graph) can be accurately extracted using the event-based method, with finite-size scaling behaviors still described by the standard finite-size scaling theory. We perform an extensive simulation of EPs on hypercubic lattices ranging from dimensions d = 2 to 6, and find that the critical behaviors consistently obey the standard finite-size scaling theory. Consequently, we obtain a high-precision determination of the percolation thresholds and critical exponents, revealing that EPs governed by the product and sum rules belong to different universality classes. Remarkably, despite the mean of the dynamic pseudocritical point T L deviating from the infinite-lattice criticality by a distance determined by the d -dependent correlation-length exponent, T L follows a normal (Gaussian) distribution across all dimensions, with a standard deviation proportional to 1 / V , where V denotes the system volume. A theoretical argument associated with the central-limit theorem is further proposed to understand the probability distribution of T L . These findings offer a comprehensive understanding of critical behaviors in EPs across various dimensions, revealing a different dimension-dependence compared to standard bond percolation. Published by the American Physical Society 2024
Abstract Quantum communication demands efficient distribution of quantum entanglement across a network of connected partners. The search for efficient strategies for the entanglement distribution may be based on percolation theory, which describes evolution of network connectivity with respect to some network parameters. In this framework, the probability to establish perfect entanglement between two remote partners decays exponentially with the distance between them before the percolation transition point, which unambiguously defines percolation properties of any classical network or lattice. Here we introduce quantum networks created with local operations and classical communication, which exhibit non-classical percolation transition points leading to striking communication advantages over those offered by the corresponding classical networks. We show, in particular, how to establish perfect entanglement between any two nodes in the simplest possible network—the 1D chain—using imperfectly entangled pairs of qubits.