Complexity Phase Transition

• Complexity phase transition is a sharp, nonanalytic change in complexity measures—such as computational cost, circuit, and entanglement complexity—that delineates distinct regimes in system behavior.
• It is characterized by discontinuities or divergences in various markers across models ranging from classical constraint satisfaction and Minesweeper to quantum many-body and holographic systems.
• Studying these transitions aids in optimizing simulation strategies, enhancing algorithmic performance, and deepening insights into quantum information and computational physics.

A complexity phase transition is a sharp, nonanalytic change in a complexity-based order parameter—such as computational cost, circuit/gate complexity, Krylov complexity, or related structural measures—that occurs as a control parameter is varied in a statistical or quantum system. Unlike conventional thermodynamic or local order-parameter transitions, complexity phase transitions demarcate regions in parameter space with fundamentally different scaling or qualitative behavior of algorithmic, geometric, or information-theoretic complexity. These phenomena occur in classical combinatorial optimization, quantum many-body physics, holographic duality, and network science, and are detected using rigorous markers such as scaling nonanalyticities, discontinuities, or sudden divergences in properly normalized complexity measures.

1. Definitions and Representative Complexity Markers

A range of quantitative diagnostic functions—“complexity markers”—have been used to witness complexity phase transitions:

• State-space/algorithmic complexity: Typical or worst-case running time for a given algorithm, e.g., in random SAT, Minesweeper, or genome assembly (Schawe et al., 2017, Dempsey et al., 2020, Fernandez et al., 2022).
• Quantum circuit complexity: Minimal gate count in a specified gate set preparing a given quantum state or unitary, as defined by Nielsen’s geometric approach, Fubini-Study metric, or Krylov/Lanczos spread (Liu et al., 2019, Huang, 2021, Roca-Jerat et al., 2023, Grabarits et al., 15 Oct 2025).
• Entanglement/structural complexity: Measures such as the entanglement width (ew), stabilizer Rényi (magic) entropy 𝓜₂, or fractal dimension D₂ quantify the classical simulatability and resource scaling of quantum states (Ghosh et al., 2022, Santra et al., 14 May 2025).
• Holographic complexity growth: Holographic dual functionals (complexity=volume/action/general observable) in AdS/CFT, acting as order parameters across deconfinement or confinement transitions (Yang et al., 2023, Zhang, 2017, Ghodrati, 2018).
• Krylov complexity: The spreading of a state or density matrix in a Krylov/Lanczos basis, connecting complexity to dynamical/information-theoretic features and serving as an order parameter in both closed and open systems (Grabarits et al., 15 Oct 2025, Teh et al., 26 Oct 2025, Xia et al., 29 Jul 2025).
• Network/graph entropy: Variational entropy density, e.g., for edge/triangle densities in random graphs, showing distinct multipartite and disordered phases (Radin et al., 2013, Torres et al., 2015).

These markers exhibit singularities (jumps, kinks, cusp, divergence) or abrupt regime changes at critical values of relevant physical or control parameters (e.g., disorder strength, coupling, measurement rate, density, etc.).

2. Paradigmatic Examples in Classical and Quantum Systems

Multiple canonical models display complexity phase transitions, each with domain-specific order parameters and critical behaviors:

• Random SAT and constraint satisfaction: Easy–hard transitions for polynomial-time algorithms occur at clause densities well below the satisfiability threshold; transitions are algorithm specific and not always linked with simple graph structural thresholds. For LP-based solvers in random 3-SAT, the transition is observed at α_c ≈ 2.36, far below the SAT-UNSAT threshold of α_s ≈ 4.26 (Schawe et al., 2017).
• Minesweeper: The probability that logical inference algorithms can deduce all mine positions drops abruptly—at a density ρ_c ≈ 0.27—accompanied by a peak in the "hardness" (minimal unsatisfiable core size), and polynomial-time heuristics fail as the phase boundary is crossed (Dempsey et al., 2020).
• Quantum graph states: For k-regular graph states, a sharp transition from polynomial-time simulability (low complexity) to #P-hardness (high complexity) occurs precisely as entanglement width rises from constant (ew=O(1)) to logarithmic (ew=Ω(log n)) at k=3, then back to easy at k = n–3 (Ghosh et al., 2022).
• Genome assembly (SCS problem): A first-order-like complexity transition is observed in the success probability of polynomial heuristics, tied to read density, with the hard phase corresponding to exponentially small success rates and true NP-hard scaling (Fernandez et al., 2022).

3. Complexity Transitions in Quantum Circuits and Many-Body Systems

Quantum many-body dynamics offer a wide range of complexity phase transitions, both equilibrium and dynamical:

• Kitaev Chain and Topological QPTs: The circuit complexity (Nielsen's approach) of ground states exhibits a log-singularity in its parameter derivative at critical points (e.g., μ_t = ±1), marking the topological phase boundary; the optimal real-space generator structure changes from local (within-phase) to nonlocal (between-phase) (Liu et al., 2019).
• Bose-Hubbard Model: Exact (operator-norm) circuit complexity diverges as a pole at the superfluid–Mott-insulator critical coupling (g_c=1), sharply distinguishing the quantum phase transition and functioning as a nonlocal order parameter (Huang, 2021).
• Krylov Complexity and Kibble-Zurek Scaling: All cumulants of Krylov complexity scale with the same Kibble–Zurek exponents as defect density when a system is slowly ramped through a continuous quantum phase transition (example: transverse-field Ising chain, κ_q ∼ τ_Q^{−ν(d−D)/(1+zν)}) (Grabarits et al., 15 Oct 2025).
• Open-System (Mixed-State) Transitions: Under decoherence, the Krylov complexity of a density matrix can display a singular area-to-volume-law scaling change at a critical decoherence time, sharp for infinite-range noise and absent for strictly local dephasing (Teh et al., 26 Oct 2025).
• Monitored Random Circuits: Complexity transitions arise in hybrid unitary–measurement circuits at a critical measurement rate p_c=½. Below p_c, circuit complexity grows linearly in time up to exponential times and saturates at exp(Ω(n)); above, it never exceeds poly(n). This transition is underpinned by a percolation mapping via open-edge connectivity (Suzuki et al., 2023).
• Chaotic vs. Integrable Regimes: In models like the power-law random banded matrix, RP, or hybrid SYK+Ising, the locations of complexity phase transitions differ depending on the marker (e.g., fractal dimension D₂, von Neumann entropy, stabilizer Rényi entropy), reflecting distinct classical simulatability barriers. In intermediate ("fractal") regimes, high entanglement and high "magic" can persist even as wavefunction support shrinks, complicating criteria for classical tractability (Santra et al., 14 May 2025).

4. Holographic and Network-Structural Complexity Transitions

Holography and network theory offer further manifestations:

• Holographic Complexity as Order Parameter: In AdS/CFT, several complexity functionals (volume, action, generalized observables) display nonanalyticities—finite jumps, cusps, or inflections—precisely at first-order, second-order, or crossover transitions in the dual field theory. These behaviors have been systematically verified in confinement–deconfinement and QCD-like settings, and complexity growth rates typically match the singularities seen in entropy, speed of sound, and other thermodynamic observables (Yang et al., 2023, Zhang, 2017, Ghodrati, 2018).
• Network and Topological Phase Transitions: In Ising models on complex networks, the emergence of a phase transition is triggered by the fractal dimension D_F crossing 1, with new collective (loop) excitations stabilizing long-range order in otherwise 1D or weakly connected systems (Torres et al., 2015).
• Entropy-Based Graph Transitions: In random graph ensembles controlled by edge and triangle densities, transitions between multipartite, heterogeneous, and disordered phases correspond to nonanalytic changes (loss of real analyticity) in the entropy density as a function of these parameters (Radin et al., 2013).

5. Extraction of Critical Points and Scaling Behavior

Complexity transitions are characterized via finite-size scaling, discontinuities or kinks in complexity markers, or by divergence of operational quantities:

System/Class	Control Parameter	Transition Location	Order Parameter(s) / Marker(s)
Random SAT (LP)	Clause density α	Algorithm-dependent α_c < α_s	Integrality probability, solver cost
Monitored circuits	Measurement rate p	p_c = ½	Circuit complexity (min. gate number)
Graph states	Regularity k	k=3, k=n–3	Entanglement width, simulation runtime
Krylov Complexity	Quench time, coupling	τ_quench, τ_decoh, p, ...	Cumulants κ_q, spread C(t), scaling exponents
Holographic QCD	Temperature, Wilson line	T_c, Φ_c	Complexity growth rate (CA/CV/any)

At these transitions, system size scalings change sharply (e.g., area → volume law), critical exponents can often be extracted from data collapse (cf. (Grabarits et al., 15 Oct 2025)), and diverging derivatives or discontinuities are visible in complexity or its derivatives (Roca-Jerat et al., 2023).

6. Role of Symmetry, Disorder, and Structural Constraints

Critical behavior of complexity markers is often more sensitive to symmetries or system structure than traditional thermodynamic quantities:

• Symmetry dependence: Magic entropy 𝓜₂ responds to fermion parity and time-reversal in the SYK+Ising model, yielding different Haar values depending on symmetry class (Santra et al., 14 May 2025).
• Localization and fractal regimes: Transitions in fractal dimension (D₂), entanglement entropy (S_{vN}), and "magic" can occur at distinct critical parameters, leading to a rich taxonomy of regimes aligned neither with standard localization nor ergodicity alone.
• Algorithmic specificity: In classical problems such as SAT and genome assembly, where average-case complexity transitions are detected, the critical value depends strongly on the algorithmic strategy (e.g., LP, message-passing, Markov-chain heuristics), rather than being a property of the solution space alone (Schawe et al., 2017, Fernandez et al., 2022).

7. Impact and Implications for Simulatability and Quantum Information

The discovery and quantitative characterization of complexity phase transitions inform both fundamental physics and computational practice:

• Classical simulation boundaries: Entanglement width, magic entropy, and fractal dimension delimit regions where tensor network methods, stabilizer-based simulators, or variational Ansätze can operate efficiently, versus when exponential resources are required (Ghosh et al., 2022, Santra et al., 14 May 2025).
• Quantum information: Sharp transitions in complexity growth under measurement, decoherence, or interaction define the boundary between phases robust to error and information loss, and those exhibiting complexity-proliferation or "magic" condensation (Teh et al., 26 Oct 2025, Suzuki et al., 2023).
• Holography and black holes: Complexity provides a nonlocal diagnostic in gauge/gravity correspondence, acting as a more sensitive probe than local energy or entropy for certain quantum-gravitational transitions (Jiang et al., 2023, Zhang, 2017, Ghodrati, 2018).
• Relevance for experimental quantum simulation: Protocols for direct measurement of Krylov complexity and its scaling exponents on current quantum hardware have been devised, making complexity transitions experimentally accessible in quantum simulators (Xia et al., 29 Jul 2025).

In conclusion, complexity phase transitions stand as robust, multifaceted indicators of qualitative changes in computational, dynamical, and physical character of high-dimensional systems—bridging statistical mechanics, quantum many-body theory, information theory, and computational complexity. The marker used, the domain, and the application inform the critical value and the physical or computational interpretation, requiring an overview of measures to fully delineate the boundaries of tractability and emergent order.