Quantum Percolation Models

• Quantum Percolation Models are theoretical frameworks that integrate quantum mechanics with classical percolation theory to study electron localization and quantum transport in disordered systems.
• They utilize tight-binding Hamiltonians, quantum walks, and entropy diagnostics to analyze localization–delocalization transitions and multifractal eigenstate behavior.
• Insights from QPM advance experimental designs and quantum computing protocols by setting percolation thresholds and optimizing entanglement distribution in complex networks.

Quantum Percolation Models (QPM) represent a class of theoretical frameworks that generalize classical percolation theory to quantum systems. In these models, disorder is introduced via random dilution (of sites or bonds) on a lattice, but the dynamics of the system are governed by quantum mechanics—most prominently, by quantum interference effects, localization phenomena, and transport properties that fundamentally differ from their classical counterparts. QPM are instrumental in understanding electron localization, quantum walks, entanglement propagation, and quantum phase transitions in disordered media.

1. Fundamental Concepts and Model Definitions

A prototypical quantum percolation model is constructed by considering a tight-binding Hamiltonian on a regular lattice, where a fraction $p$ of the sites (or bonds) are “accessible” (i.e., part of the so-called $A$-sites). In the limit of infinite on-site energy difference between accessible ($A$) and inaccessible ($B$) sites, the Hamiltonian reduces to

$$H_{AA} = -t \sum_{\langle ij \rangle \in A} \left( c_i^\dagger c_j + \mathrm{H.c.} \right)$$

where $t$ is the hopping amplitude, $c_i^\dagger$ creates a particle at site $i$, and the sum runs over nearest-neighbor $A$-sites (0911.5531).

Disorder is introduced by randomly removing sites or bonds: in site percolation, only certain lattice sites remain “active” for quantum transport; in bond percolation, only certain nearest-neighbor links permit hopping. The disorder is typically “quenched,” i.e., fixed for each realization. Alternatively, dynamic (annealed) disorder can be introduced by allowing the percolation configuration to change in time (Kollár et al., 2014).

In the context of wave transport or quantum information, the state evolution may be modeled using either continuous-time or discrete-time (coined) quantum walks (1006.1283).

2. Localization–Delocalization Transitions, Phase Diagrams, and Critical Exponents

A central feature of QPM is the existence of localization–delocalization transitions (LDTs), often reflected in a quantum percolation threshold $p_q < 1$—markedly different from the classical percolation threshold $p_c$. For $p < p_q$, all quantum states are exponentially localized due to interference; for $p > p_q$, a subset of eigenstates becomes extended.

Key diagnostics and methodologies:

• Von Neumann entropy as a localization probe: The site-averaged local entropy,

$$E_{vi}^\alpha = -z_i \log_2 z_i - (1-z_i) \log_2 (1-z_i), \quad z_i = |\psi_i^\alpha|^2$$

is sensitive to the spatial extension of the eigenstate $|\alpha\rangle$: $\langle E_v^\alpha \rangle \sim 1$ for delocalized states and $\sim 0$ for localized states (0911.5531).

• The localization–delocalization transition can be identified by a maximum in the derivative $\frac{d\langle E_v^\alpha\rangle}{dp}$ at $p_{max}$, and $p_q$ is obtained by extrapolating $p_{max}$ to $L \rightarrow \infty$.
• The phase diagram in the $(\varepsilon, p)$-plane generally reveals a non-monotonic $p_q(\varepsilon)$, with the minimal $p_q \approx 0.665$ found for 2D square lattices (0911.5531).
• Transmission coefficients and their scaling: The functional dependence of the transmission $T(L)$ on system size $L$ distinguishes three regimes (Dillon et al., 2013, Thomas et al., 2016):
  • Delocalized: $T=a \exp(-bL)+c$, $c>0$ (finite transmission as $L\to\infty$)
  • Power-law localized: $T=aL^{-b}$
  • Exponentially localized: $T=a \exp(-L/\ell)$
• Critical exponents ($\nu$): Finite-size scaling of localization indicators yields non-universal (energy-dependent) $\nu$, typically $2.6\leq\nu\leq 3.2$ in 2D, with all values $\nu \geq 2/D$ ($D=2$) (0911.5531). In 3D, QPM belong to the Anderson universality class with $\nu \approx 1.62$ (Ujfalusi et al., 2014).
• Multifractality: At the quantum metal–insulator transition, eigenstates exhibit multifractal scaling with generalized dimensions $D_q$ and singularity spectra $f(\alpha)$. These quantities are universal across the mobility edge, but exhibit anomalous $p$-dependent behavior close to the classical threshold (Ujfalusi et al., 2014).

3. Quantum Transport, Walks, and Scaling Regimes

Quantum percolation models yield rich transport phenomena that sharply contrast classical behavior:

• Quantum walks on percolation lattices exhibit a crossover between ballistic, diffusive, and localized transport (1006.1283, Kollár et al., 2014). In 1D, even isolated gaps can block motion unless quantum tunneling (modeled by a biased coin) is introduced. In 2D, the time-averaged displacement scales as $\langle r \rangle \sim t^{\alpha(p)}$, with $\alpha$ interpolating continuously between localized ($\alpha=0$), classical diffusive ($\alpha=0.5$), and quantum ballistic ($\alpha=1$) regimes.
• Dynamic percolation (i.e., time-dependent disorder) transforms the closed (unitary) quantum dynamics into a “random unitary map,” characterized by “attractor spaces” of steady-state operators. This framework can capture decoherence and persistent localization effects, especially for specific coin operators (e.g., Grover coin in 2D) (Kollár et al., 2014).
• Experimental realizations using photonic chips (via femtosecond laser writing) establish quantum percolation thresholds higher than classical counterparts (e.g., $\sim80\%$ compared to $63\%$ for the hexagonal lattice), reflecting the necessity of denser connectivity to maintain extended quantum transport (Feng et al., 2020).
• Conductivity and scaling: In the delocalized regime, conductivity $\sigma \propto p/(1-p)$ follows Drude-like expectations at high $p$, but strong disorder induces a crossover to Gaussian decay of the current autocorrelation, and the mean free path can be estimated as $\lambda(p) = p^2/(1-p)$ (Schmidtke et al., 2014).

4. Effects of Quantum Entanglement, Correlated Disorder, and Modified Percolation Schemes

Quantum percolation theory extends naturally to quantum information science, entanglement transport, and the paper of correlated disorder:

• Entanglement percolation: By converting imperfect local entanglement links probabilistically into perfect (singlet) links (with probability $p = 2\lambda_2$ for $$|\phi\rangle = \sqrt{\lambda_1}|00\rangle + \sqrt{\lambda_2}|11\rangle$$), protocols using LOCC and entanglement swapping can substantially lower the percolation threshold, e.g., to $p_t=1/2$ for a 1D chain (compared to $p=1$ classically) (Siomau, 2016, Siomau, 2019, Girolamo et al., 24 Feb 2025). This demonstrates striking quantum advantages for network connectivity and distributed quantum communication.
• Correlated disorder: When the underlying structure is defined by a classical system at criticality—such as a 2D Ising model at $T_c$—the resulting disorder is scale-free. The QPM then exhibits a “correlation-induced” localization-delocalization transition, with critical energy levels and suppressed density of compact localized states compared to the uncorrelated case (Tomasi et al., 2022).
• Hybrid classical–quantum percolation: Networks combining classical communication channels (probability $p$ open) and quantum filtering for entangled links (probability $p_e$) allow for engineered control of the percolation threshold at $p_c = 1 - p_e$, with either continuous or discontinuous transitions depending on protocol timing (Siomau, 2019).
• Interacting many-body quantum systems: In QPM realized by interacting fermionic quantum circuits, the Krylov complexity exhibits a novel phase transition—the Krylov complexity phase transition (KCPT)—whose position coincides with the classical percolation threshold for free systems, but in interacting systems is generically separated and can occur at lower occupancy (Griffiths-type shift); this is governed by different scaling exponents (Xia et al., 29 Jul 2025).

5. Advanced Topics: Universality, Lattice Geometry, Non-Standard Percolation, and Simulation Techniques

• Universality classes: The localization–delocalization transitions in QPM generally belong to the same universality class as the Anderson model for analogous dimensions (e.g., 3D QPM has $\nu \approx 1.62$) (Ujfalusi et al., 2014, Oliveira et al., 6 Sep 2024). Both site and bond percolation problems on complex lattices (e.g., 2D/3D Lieb lattices) display identical critical exponents and scaling functions, regardless of microscopic details (Oliveira et al., 6 Sep 2024).
• Lattice geometry: The critical threshold $p_q$ decreases as the average coordination number increases, as observed across simple cubic, Lieb, and perovskite lattices (Oliveira et al., 6 Sep 2024). Quasicrystalline lattices exhibit enhanced localization due to their aperiodicity, with percolation probability dropping more rapidly under disorder compared to periodic lattices (Chawla et al., 2019).
• Non-standard percolation in quantum computing: In measurement-based photonic quantum computing, “effective” percolation rules—arising from photon loss, destructive measurement, and graph fusion—are simulated via modified Newman–Ziff algorithms. The critical photon loss tolerance (percolation threshold) determines whether the resulting graph state can support universal computation (Löbl et al., 2023).
• Experimental protocols: Observing QPM predictions is feasible with current quantum hardware by constructing accessible Krylov subspaces (via time-evolved operators), performing quantum Gram–Schmidt orthogonalization, and measuring overlaps to extract complexity or transport characteristics (Xia et al., 29 Jul 2025).

6. Open Problems, Future Directions, and Synthesis

Numerical and analytical investigations of QPM continue to raise important questions:

• The exact nature (continuous versus discontinuous, Kosterlitz-Thouless-type, etc.) of localization transitions and the role of intermediate, power-law localized regimes require more extensive simulations for larger systems and alternative scaling analyses (Dillon et al., 2013, Thomas et al., 2016).
• The dependence of universality classes on lattice geometry or the inclusion of dynamic and correlated disorder remains an active area, particularly for hybrid or engineered networks with both classical and quantum resources (Siomau, 2019, Tomasi et al., 2022).
• Quantum information science motivates further exploration of percolation algorithms—especially physics-informed heuristics—to optimize network connectivity, resource consumption, and entanglement propagation in realistic, noisy architectures (Girolamo et al., 24 Feb 2025).
• Direct experimental observation of QPM transitions beyond ballistic-diffusive-localized crossovers promises to clarify the empirical values of quantum thresholds and the influence of disorder or interaction-induced complexity transitions (Feng et al., 2020, Xia et al., 29 Jul 2025).

Overall, quantum percolation models elucidate the interplay of geometry, disorder, and quantum coherence in a range of settings from condensed matter physics to quantum information. Their paper informs our understanding of localization physics, quantum algorithms, network engineering, and quantum phase transitions distinct from classical paradigms.