Complexity Growth Rate (CGR) Overview

• Complexity Growth Rate (CGR) is defined as the derivative of a complexity functional with respect to an evolution parameter, capturing time-dependent behavior in various holographic proposals.
• It quantifies the rate of change in complexity through frameworks like Complexity = Action and Complexity = Volume, linking it to thermodynamic quantities such as energy, pressure, and volume.
• CGR provides practical insights into nonequilibrium dynamics, phase transitions, and the effects of modified gravity theories, while its interpretation remains proposal-dependent.

Complexity Growth Rate (CGR) usually denotes the derivative of a complexity functional with respect to an evolution parameter. In holographic work this is most often the boundary-time derivative of holographic complexity, written in the Complexity $=$ Action proposal as

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$

while in Complexity $=$ Volume 2.0 it is taken, at late times, to satisfy

$\dot{\mathcal C}\sim \frac{PV}{\hbar}.$

In cMERA it is the rate at which complexity changes along the renormalization-group flow rather than physical time (Tanhayi et al., 2018, Sun et al., 2019, 1803.02356). CGR is therefore not a single universal observable but a proposal-dependent rate quantity whose precise interpretation varies with the underlying notion of complexity.

1. Definitions and scope

In the CA framework, the complexity of a boundary state at time $t$ is identified with the on-shell action on the Wheeler–DeWitt patch,

$$\mathcal C(t)=\frac{\mathcal I_{\mathrm{WDW}(t)}}{\pi},$$

so that CGR is

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d\mathcal I_{\mathrm{WDW}(t)}}{dt}.$$

The action includes the bulk Einstein–Hilbert term, Gibbons–Hawking terms, joint terms, and, when needed, null boundary counterterms (Tanhayi et al., 2018).

In the CV framework, complexity is proportional to the maximal volume of a codimension-one bulk slice,

$\mathcal C_V\sim \frac{V_{\max}}{G_N L},$

and the corresponding CGR is the time derivative of that maximal volume. In CV 2.0, the late-time relation is reformulated as

$\dot{\mathcal C}\sim \frac{PV}{\hbar},$

with $P$ the thermodynamic pressure and $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$0 the thermodynamic volume (Sun et al., 2019).

The “Complexity $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$1 Anything” program broadens this further. A generalized observable is built from a diffeomorphism-invariant scalar integrated over an extremal hypersurface, and different choices can still satisfy the AdS criteria of late-time linear growth and the switchback effect while exhibiting very different behavior in de Sitter static patches (Aguilar-Gutierrez et al., 2023).

In cMERA, complexity is defined operationally from the path integral associated to the cMERA tensor network. The corresponding growth rate is not a boundary-time derivative but a derivative along the continuous MERA scale variable $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$2; for the action-based cMERA functional,

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$3

with $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$4 the cMERA entanglement Hamiltonian (1803.02356).

A distinct literature studies growth rates of aggregate complex systems rather than quantum complexity. There the central object is a system growth rate

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$5

whose fluctuations obey generalized central-limit behavior. This suggests that the acronym overlap does not imply conceptual identity: in that setting the subject is stochastic growth of many subunits, not holographic or circuit complexity (Takayasu et al., 2013).

2. Late-time behavior, bounds, and thermodynamic reformulations

A central reference point for holographic CGR is Lloyd’s bound,

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$6

or, in black-hole language, $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$7. In the multiple-shock Vaidya geometries studied with CA, the bound is respected during the entire thermalization process, and at late time the complexity growth saturates to the value proportional to the energy of the final state (Tanhayi et al., 2018).

In several stationary black-hole settings, late-time CGR reduces to horizon thermodynamics. In Lovelock gravity, the late-time CA result becomes a difference of internal energies associated with the outer and inner horizons,

$$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$8

and in the large-mass relative to charge limit the Einstein result $$\dot{\mathcal C}(t)=\frac{1}{\pi}\frac{d I_{\mathrm{WDW}}}{dt},$$9 is recovered, with corrections proportional to the highest Lovelock coupling in even boundary dimensions (Cano et al., 2018). In Horndeski theory, neutral AdS black holes saturate the Lloyd bound, while charged solutions have only one horizon and the growth rate takes the form $=$0, remaining below the bound (Feng et al., 2018).

The flat-space CA computation yields the same expression as the flat-space limit of the asymptotically AdS result. For bulk dimensions greater than three, the late-time rate approaches Lloyd’s bound from above; for three-dimensional bulks it is a constant differing from the bound by a logarithmic term (Fareghbal et al., 2018). This is one of several cases showing that “saturation” and “respecting the bound” are not interchangeable statements.

Thermodynamic reformulations make CGR proposal-dependent but structurally compact. In CV 2.0,

$=$1

and with $=$2 and $=$3, this gives

$=$4

or $=$5 in natural units (Sun et al., 2019). In a related thermodynamic program, the butterfly velocity satisfies

$=$6

so for classes of AdS black holes one can write

$=$7

or, more generally,

$=$8

with the precise form fixed by the Smarr relation (Mansoori et al., 2017).

These formulations establish that CGR is often expressible in terms of $=$9, $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$0, $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$1, free energy, grand potential, or horizon potentials, but the equivalence is conditional on the complexity prescription and background.

3. Nonequilibrium evolution: shocks, quenches, and cosmological backgrounds

The CA treatment of Vaidya geometries provides an explicit nonequilibrium notion of CGR. For double black hole–Vaidya and AdS–Vaidya with one, two, or $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$2 shocks, the WDW patch intersects the shock surfaces at joints $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$3, and the time dependence of these joints controls the full rate (Tanhayi et al., 2018). In the thermal initial state, the explicit formula contains negative corrections depending on $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$4, $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$5, and the shock-joint position $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$6, ensuring $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$7. At early times the growth behaves as $\dot{\mathcal C}\sim \frac{PV}{\hbar}.$8, while at late times

$\dot{\mathcal C}\sim \frac{PV}{\hbar}.$9

The saturation value is independent of the initial temperature, whereas the transient rate is not: the rate of complexity evolution after a shock in the case of thermal initial state is always less than the similar rate for the vacuum initial state and by multi shocks it gets smaller (Tanhayi et al., 2018).

Cosmological backgrounds introduce a different kind of time dependence. In FLRW realizations from AdS–Schwarzschild, both CV and CA separate into an interaction term and a term proportional to the change of the spatial volume of the universe (An et al., 2019). For the realization on the asymptotic boundary, the leading divergent term of the CGR obeys a volume law, which is natural from the field theory viewpoint. In the brane-universe realization, the behavior depends on the conjecture: for the flat universe both CV and CA give positive, increasing CGR dominated by the volume contribution, whereas for the closed universe CV and CA behave differently, and the CA result can become inconsistent when the brane crosses the black-hole horizon (An et al., 2019).

A related FLRW analysis in Gauss–Bonnet gravity and massive gravity using CV finds three kinds of contributions in the dual FLRW universe. In Gauss–Bonnet gravity these are a finite interaction term from a conserved charge, a term from the spatial volume of the universe, and a curvature-dependent term in which the Gauss–Bonnet effect plays a vital role. In massive gravity, the graviton mass contributes both to the finite term and to additional divergent terms, including one saturating an area law (Pan et al., 2020).

These time-dependent studies show that CGR can encode thermalization, repeated energy injection, expansion or contraction of the ambient universe, and the detailed structure of the gravitational theory.

4. Dependence on gravity theory, topology, and couplings

Modified-gravity computations make explicit that CGR is sensitive to curvature corrections, parity violation, and horizon topology. In Gauss–Bonnet gravity, the “Complexity–Volume” and CV 2.0 analyses show that for $t$0 and $t$1 the Gauss–Bonnet term suppresses the growth rate as expected, while for $t$2 the effect may be opposite to what is expected (An et al., 2018). For planar horizons, the late-time CV rate admits a small-$t$3 expansion whose leading correction is negative, and the full time evolution starts from zero, grows monotonically, and approaches a suppressed plateau (An et al., 2018).

Three-dimensional massive gravities provide a different deformation pattern. In Topologically Massive Gravity, decreasing the parameter $t$4, which increases the effect of the Chern–Simons term and increases chirality, increases the rate of growth of complexity (Ghodrati, 2017). By contrast, New Massive Gravity is parity-preserving, so the dependence of CGR on angular variables remains symmetric. The same work reports a stronger correlation between complexity growth and temperature rather than complexity growth and entropy (Ghodrati, 2017).

Lovelock and Horndeski examples further illustrate how horizon structure enters. In Lovelock gravity the late-time CA rate reduces to $t$5, and in some cases there is a minimum mass below which complexity remains effectively constant, even if the black hole contains a non-degenerate horizon (Cano et al., 2018). In Horndeski theory, the existence of only one horizon in the charged case prevents the usual difference-of-horizon-potentials formula familiar from Reissner–Nordström–AdS, yet the bound is still not violated (Feng et al., 2018).

A plausible implication is that late-time CGR is often less universal than the existence of late-time linear growth itself: the latter recurs across many models, while the coefficient and even the sign of corrections depend sharply on the higher-curvature sector, the parity structure, and the topology $t$6.

5. Phase transitions, criticality, and scale dependence

Several works treat CGR as a diagnostic of phase structure. In the thermodynamic CV 2.0 framework, Schwarzschild–AdS and Reissner–Nordström–AdS black holes display Hawking–Page and Van der Waals-like signatures directly in $t$7: for RN–AdS, $t$8 exhibits the characteristic swallow-tail structure below the critical pressure, and this disappears at the critical point (Sun et al., 2019).

A broader CA-based survey finds a strong connection between phase transitions and the discontinuities in the complexity growth rates. For dyonic black holes dual to van der Waals fluids, the behavior of CGR displays a similar first-order phase transition. In the Gubser models of QCD, the complexity growth rate tracks crossover, first-order, and second-order transitions, and its temperature dependence parallels the behavior of entropy and speed of sound; in the first-order case the slope can become negative in the spinodal region (Ghodrati, 2018). The same study connects CGR qualitatively to the Schwinger effect and to quasinormal-mode behavior in confining and non-confining backgrounds (Ghodrati, 2018).

Recent “Complexity $t$9 Anything” work pushes this farther by analyzing jumps in the CGR. In that framework, bulk fields are in charge of the height and the location of these jumps, with boundary counterparts given by divergence of energy momentum tensor, shear viscosity, and the Weyl anomaly. Near the jumps, CGR shows critical behavior and critical exponents, and satisfies a Callan–Symanzik-like equation (Shahbazi et al., 18 Aug 2025). The resulting scale equation suggests that the speed of information processing could be changed by the scale of energy. This suggests a renormalization-group interpretation of CGR rather than a purely kinematic one.

These results argue against treating complexity growth as a featureless late-time linear slope. In several models it behaves more like an order-parameter-like quantity, or at least like a sensitive nonequilibrium diagnostic, with discontinuities, critical scaling, and branch changes.

6. Alternative dynamical interpretations and conceptual limits

In cMERA, CGR has an exact operator interpretation. For the action-based cMERA complexity,

$$\mathcal C(t)=\frac{\mathcal I_{\mathrm{WDW}(t)}}{\pi},$$0

while for the $$\mathcal C(t)=\frac{\mathcal I_{\mathrm{WDW}(t)}}{\pi},$$1 circuit length,

$$\mathcal C(t)=\frac{\mathcal I_{\mathrm{WDW}(t)}}{\pi},$$2

These equalities were presented as complexity growth limits analogous to Margolus–Levitin and Mandelstam–Tamm quantum speed limits, but saturated by the optimized cMERA circuit (1803.02356). The same framework identifies the cMERA complexity action with a Liouville action, so the growth of complexity along the renormalization flow is simultaneously an entanglement-generation rate and a two-dimensional gravity action (1803.02356).

De Sitter holography has sharpened the issue of proposal dependence. Standard CV, CA, and CV 2.0 in de Sitter static patches exhibit “hyperfast growth,” namely finite-time divergence of complexity or its derivative. By contrast, a family of CMC-slice observables within “Complexity $$\mathcal C(t)=\frac{\mathcal I_{\mathrm{WDW}(t)}}{\pi},$$3 Anything” yields late-time linear or exponential growth that persists classically forever (Aguilar-Gutierrez et al., 2023). Hyperfast growth is therefore not a necessity. This is a direct challenge to any attempt to define a background-independent universal CGR.

Several persistent open issues follow from this literature. One is the status of Lloyd-type bounds away from the stationary AdS setting: they are respected in multiple-shock Vaidya geometries and in several modified gravities, but they can be approached from above, altered by logarithmic terms, or become problematic in cosmological brane setups (Tanhayi et al., 2018, Fareghbal et al., 2018, An et al., 2019). Another is the ambiguity among CA, CV, CV 2.0, generalized volume functionals, and CAny observables: these proposals can agree on broad trends while disagreeing on detailed time dependence or even on whether growth is finite or hyperfast (An et al., 2018, Aguilar-Gutierrez et al., 2023). A further limitation is that most calculations are classical-gravity computations using thin shells, planar or constant-curvature horizons, and highly symmetric states.

Taken together, the literature defines CGR as a family of rate observables that track how rapidly a chosen notion of complexity changes. In stationary AdS black holes it often reduces to a compact thermodynamic quantity; in quenches it becomes a detailed measure of thermalization history; in cosmology it splits into interaction and volume pieces; in modified gravity it probes higher-curvature and parity-violating couplings; in cMERA it is exactly tied to the entanglement Hamiltonian; and in de Sitter it exposes the extent to which holographic complexity is proposal-dependent rather than unique (Cano et al., 2018, Sun et al., 2019, 1803.02356, Aguilar-Gutierrez et al., 2023).