Finite Temperature Complexity Growth

• Complexity growth at finite temperature is defined as the temperature-dependent evolution of complexity measures in various systems, displaying non-monotonic behavior and sharp transitions.
• Rigorous diagnostics—including algebraic formulations, Krylov complexity, and holographic duals—reveal links between operator growth, chaos bounds, and simulation challenges.
• Universal resource bounds and temperature scaling laws connect computational, thermodynamic, and gravitational principles in many-body physics.

Complexity growth at finite temperature refers to the rich, system-dependent behavior of dynamical, computational, and operator-theoretic complexity measures as quantum or classical systems are heated. In strongly correlated many-body systems, statistical models, quantum circuits, and holographic duals, complexity does not generically increase monotonically or smoothly with temperature; rather, it often exhibits sharp features, nontrivial scaling laws, and transitions reflecting underlying physical structure, symmetry, or resource constraints. Diverse frameworks—including information theory, computational mechanics, circuit complexity, real-time operator evolution, and bulk gravitational duals—provide both rigorous and practical diagnostics of how complexity emerges and evolves in finite-temperature ensembles.

1. Algebraic and Quantum Computational Complexity at Finite Temperature

In algebraic formulations of finite-temperature quantum systems—such as those based on anti-commuting Schwinger sources and exterior algebras—the configuration space is encoded by polynomial rings deformed by parity and time-reversal gradings. The complexity of analyzing observables (e.g., partition functions, trace invariants) is dictated by the structure of Grobner bases computed within these rings. For fermionic systems at finite temperature, as shown in the construction using ideal quotients (I : J) and degree-reverse lexicographic orderings, the Grobner basis enables projection onto commutative polynomial rings suitable for polynomial-time quantum algorithms.

The complexity class of the associated computation—namely the evaluation of additive approximations to trace invariants—is BQP, as established by bounding the additive error via Chernoff techniques and analyzing the monomial reduction structure inherent in the Grobner basis algorithm. Although the underlying polynomial rings do not always faithfully represent the fundamental group associated with topological path weights (due to quantum deformation and loss of faithfulness at finite temperature), the resulting Braid group representations, constructed via intersections of primitive and maximal spectra, afford exact computability and encode complexity features unique to the thermal regime (Crompton, 2010).

2. Complexity Diagnostics in Classical and Quantum Spin Systems

Statistical complexity, as measured by the minimal memory required to model a stochastic process (the ε-machine statistical complexity), exhibits marked temperature dependence in paradigmatic models such as the 2D and 1D Ising ferromagnet. In the classical context, block entropy techniques and recursive pair substitution (NSRPS) algorithms applied to spin sequences reveal that the statistical complexity (as measured by the difference between single-site entropy and entropy rate, $c_\mu = 1 - h_\mu/h_\mu(1)$) peaks sharply near the critical temperature $T_c$, reflecting the emergence of long-range correlations at phase transitions. This peak remains robust upon finite-size scaling and is universal across algorithmic methods, including data-compression-based estimators (Melchert et al., 2012).

In quantum computational mechanics, the minimal quantum memory (von Neumann entropy over quantum causal states) for modeling a process, $C_q$, can behave non-monotonically with temperature. For the classical Ising spin chain, the classical complexity $C_\mu$ grows with $T$ until $T\to\infty$, while quantum complexity $C_q$ rises to a finite maximum at intermediate temperature before decaying to zero at high $T$ due to the possibility of compressing memory with nonorthogonal quantum states. This signals a quantum-classical divergence in defining operational structure, and demonstrates the crucial role of allowed processing resources in complexity growth (Suen et al., 2015).

3. Operator and Holographic Complexity: Growth Regimes and Universal Bounds

Operator-theoretic complexity measures, especially Krylov complexity, leverage the action of the Liouvillian on local operators to construct a Krylov basis and Lanczos coefficients $\{b_n\}$. At infinite temperature and in random matrix theory, $b_n$ rapidly saturates to a plateau and Krylov complexity grows linearly ($K(t) \sim t$), while in chaotic local systems, an initial regime of exponentially growing complexity ($K(t) \sim \exp(2\alpha t)$, $\alpha$ the slope of $b_n$) yields to linear growth after the scrambling time. At low temperature, $b_n$ exhibits a transient linear-in-$n$ regime with slope saturating the chaos bound ($\alpha = \pi k_BT$), resulting in exponential complexity growth up to $t_* \sim \beta \log \beta$, before crossover to linear growth at the RMT plateau value (Tang, 2023, Tan et al., 19 Jan 2024).

Temperature-dependent bounds have been established for the slope of the Lanczos coefficients and thus for operator growth—a universal upper bound of $\alpha \leq \pi k_B T$ is found in both GOE and random spin chains when the Wightman inner product is used. The precise temperature scaling of the operator growth rate provides both a diagnostic of quantum chaos and a bridge to spectral ergodicity, as encoded by relationships between $\alpha(T)$ and the spectrum form factor’s Thouless time (Tan et al., 19 Jan 2024).

Holographic complexity proposals, notably the complexity=action (CA) and complexity=volume (CV) conjectures, further illuminate complexity growth after finite-temperature quenches, in black hole spacetimes, and for holographic superconductors. The CA proposal yields linear complexity growth saturating at $2E/\pi$ (with $E$ the final state’s energy) respecting Lloyd’s bound in equilibrium, and shows that the complexity growth rate may depend more strongly on temperature than entropy. CV complexity can display unbounded late-time linear growth, with temperature enhancing the linear growth coefficient, occasionally leading to violations of the Lloyd bound, particularly at early times or in the presence of a local quench (Tanhayi et al., 2018, Ageev, 2019, Ghodrati, 2017, Yang et al., 2019).

4. Complexity in Quantum Simulations and Many-Body Algorithms

The growth of computational complexity with temperature also appears in classical simulations of quantum many-body systems. In auxiliary-field quantum Monte Carlo (AFMC) simulations at finite temperature, algorithms that systematically exclude negligibly-occupied single-particle states dramatically reduce computational cost. Key steps include block decomposition (QR, QDR), truncation to a subspace of “occupied” states (order $N$ versus full $N_s$), and efficient implementation of number-projected observables, reducing scaling of critical operations from $O(N_s^3)$ to $O(N_s N_\text{occ}^2)$ or better. This enables ab initio simulation of large-scale systems such as the unitary Fermi gas across temperature regimes (Gilbreth et al., 2019).

In algorithms based on matrix product states (MPS), the sample complexity required to represent or sample thermal states undergoes a sharp crossover: at high temperature, normalized partition function fluctuations (and thus the required number of samples) scale linearly with system size; at low temperature, quadratic scaling sets in, with the crossover controlled by the thermal Renyi-2 entropy and underlying spectral gaps. This identifies a quantitative boundary between regimes of efficient and demanding simulation as a function of temperature (Iwaki et al., 15 Mar 2024).

5. Geometric and Quantum Circuit Complexity of Thermal States

For quantum systems with well-defined symmetry structures (e.g., Bose-Einstein condensates with $SU(1,1)$ symmetry), geometric measures of complexity for finite-temperature density matrices can be constructed via generalizations of the Fubini-Study metric (Bures, Sjöqvist) and by the circuit complexity of Nielsen’s purification approach. These metrics quantify the minimal geodesic length between reference and target mixed states on the appropriate statistical manifold. Analytical and numerical evaluations reveal characteristic logarithmic scaling of complexity with temperature, with both quantum coherence and classical probability contributions, and notable equivalence between optimal circuit (Nielsen) and Bures measures after gauge fixing (Wang, 18 Mar 2025).

6. Thermodynamic and Universal Resource Bounds on Complexity Growth

Complexity-windowed thermodynamics (CWT) extends classical statistical mechanics by admitting a finite quantum circuit complexity budget $\Xi$. Here, only those microstates accessible with $C^* \leq \Xi$ contribute to the windowed entropy $S_\Xi(E, \Xi)$, guaranteeing smoothness in energy, monotonicity under increasing budget, and ultimately, removing classical singularities (e.g., at phase transitions). CWT modifies the first law to include a complexity generation potential, $\Pi_\Xi$, a non-negative work term conjugate to $\Xi$. Two universal bounds emerge: the action required to generate complexity must scale at least as $S_\text{proc} \gtrsim k_S C^* \hbar$, and the minimal time for complexity growth is $$t_\text{proc} \geq \frac{C^* \hbar}{2\pi k_B T_\Xi}$$, extending the logic of the chaos bound to computational resource constraints. CWT predicts that the universe’s maximum generatable complexity should be of the same order as its holographic entropy, establishing a deep connection between quantum computational, thermodynamic, and gravitational principles (Liu et al., 8 Jun 2025).

7. Physical and Conceptual Implications

Complexity growth at finite temperature is governed by system-specific algebraic, statistical, and dynamical constraints reflecting underlying symmetries, topological structures, physical resource limitations, and algorithmic implementation choices. The emergence of complexity maxima, nontrivial scaling regimes, chaos bounds, and resource-based smoothing of otherwise singular critical points are universal themes across models from spin chains and quantum fields to holographic duals and quantum simulations. This body of work provides not only a hierarchy of diagnostics and bounds but also a bridge from information processing and quantum computation to critical phenomena and cosmological physics, where complexity serves as both a diagnostic and a limiting principle for physical dynamics and simulation.