Proposed Complexity Bounds and Their Counterexamples

A complexity-growth bound is meaningful only after the computational quantity, time, energy above a specified ground state, charges, normalization, and subtraction scheme are fixed. The familiar rate $2E/\pi$ is therefore not a theorem about every notion of complexity. It is a proposed comparison scale whose status must be tested separately for boundary circuits, complexity equals action (CA), complexity equals volume (CV), and quantum-corrected geometries.

Required background. Complexity Equals Action Proposals supplies the complete Wheeler–DeWitt action and its null-boundary conventions; Complexity Equals Volume Proposals supplies the independent maximal-volume prescription and its normalization ambiguity.

Helpful background. Complexity, Chaos, and Computational Claims distinguishes circuit statements from chaos diagnostics; Finite-c Corrections and Nonperturbative Questions identifies the quantum-gravity regime of the BTZ examples; Operational Preparation Cost and Energy-Constrained Bounds owns general energy-constrained task definitions.

Lloyd-type rates are hypotheses, not invariant theorems

The Margolus–Levitin orthogonalization time implies a rate scale $2E/\pi$ for a system with average energy $E$ above its ground state. Lloyd used related quantum-speed-limit reasoning to estimate a maximum rate of elementary logical operations Lloyd 2000, pp. 1047–1049. Neither statement, by itself, bounds the derivative of a Nielsen geodesic distance or of a regulated QFT state complexity. A circuit complexity may depend on the gate set $\mathcal G$, penalty metric $F$, reference state $|\psi_{\mathrm R}\rangle$, tolerance $\epsilon$, and whether ancillas or time-dependent gates are allowed.

The CA conjecture suggested the late-time comparison

$$\frac{d\mathcal C_{\mathrm A}}{dt} \stackrel{?}{\leq} \frac{2}{\pi}\bigl(E-E_{\mathrm{gs}}(Q_i)\bigr), \qquad \mathcal C_{\mathrm A}=\frac{I_{\mathrm{WDW}}}{\pi},$$

where the ground state must lie in the same charge sector $Q_i$ and the boundary time normalization must be held fixed Brown et al. 2016, §§ VI–VII. This is a proposal-specific conjecture. For rotating or charged holes, several inequivalent replacements—using $M-\Omega J-\mu Q$, subtracting an extremal state, or taking a difference between horizon generators—appear in the literature. They must not be silently interchanged.

CV has an additional obstruction. Since

$$\mathcal C_{\mathrm V}=\frac{V(\Sigma_{\max})}{G_N\ell_{\mathrm V}},$$

rescaling the freely chosen length $\ell_{\mathrm V}$ rescales the rate. A numerical energy bound on $d\mathcal C_{\mathrm V}/dt$ is consequently non-invariant until $\ell_{\mathrm V}$ is fixed by an independent dictionary. Agreement after fitting $\ell_{\mathrm V}$ is not a test of such a bound.

First application: classical and quantum-corrected BTZ

Consider a nonrotating classical BTZ black hole with boundary time normalized by the asymptotic metric,

$$f(r)=\frac{r^2-r_+^2}{L^2}, \qquad M=\frac{r_+^2}{8G_3L^2}.$$

Using the standard Einstein–Hilbert CA normalization, the complete late-time Wheeler–DeWitt action gives

$$\lim_{t\to\infty}\frac{dI_{\mathrm{WDW}}}{dt}=2M, \qquad \lim_{t\to\infty}\frac{d\mathcal C_{\mathrm A}}{dt}=\frac{2M}{\pi}.$$

Thus classical neutral BTZ saturates the candidate CA rate when the zero-mass BTZ state is used as the energy origin. This is one successful model calculation, not a derivation of a boundary circuit theorem.

Now keep the same asymptotic time and energy convention in the braneworld quantum-BTZ family, where a three-dimensional black hole is coupled to a large-$c$ CFT and its backreaction is captured by a four-dimensional classical bulk. The expansion parameter measures quantum backreaction on the brane; it is not an $\alpha'$ expansion of a specified string compactification. Emparan, Frassino, and Sasieta found that generalized CA receives a contribution controlled by the quantum-corrected singular region and fails to recover the expected classical CA behavior smoothly as the backreaction is removed, whereas generalized CV has a controlled perturbative classical limit Emparan, Frassino, and Sasieta 2022, §§ 4–6.

The correct inference is sharp: with the energy origin, boundary clock, action prescription, and limiting family fixed, this is a counterexample to the proposed universal CA behavior. One may respond by restricting CA to geometries whose ultraviolet completion controls the singular region, or by proposing additional quantum terms, but either move changes a hypothesis and requires independent justification. Renormalizing the target rate after seeing the answer would make the test circular. Rotating quantum BTZ calculations further show that inner-horizon structure and the order of late-time and weak-backreaction limits matter Chen, Liu, and Yu 2024, §§ 3–5.

What survives a counterexample

A useful proposed bound should be written as a tuple

$$(\mathcal C,\ t,\ H,\ Q_i,\ E_{\mathrm{gs}},\ \text{regulator},\ \text{counterterms},\ \text{state class}),$$

not merely as an inequality. It should also specify whether it is expected at all times, only at late time, or only after averaging. The following distinctions then remain available:

• Classical Einstein solutions can satisfy or saturate a CA rate without making the rate universal.
• A quantum correction can violate a proposed inequality while leaving qualitative linear growth intact.
• A finite local counterterm may shift a finite complexity or formation cost; it cannot be tuned separately for each member of a family and still count as a fixed scheme.
• Higher-derivative corrections require the corresponding gravitational action and boundary terms. Taking $\alpha'/L^2\to0$ or $G_N/L^{d-1}\to0$ only tests a classical limit when all other ultraviolet-sensitive terms are controlled.
• Model-dependent generalized bounds may hold in other semiclassical quantum black-hole families. A 2026 dyonic example illustrates that charge sector and ground-state subtraction can materially change the comparison Parihar and Punia 2026, §§ 4–6; it does not erase the quantum-BTZ counterexample.

Adversarial control: freeze the target before the calculation

Pre-register the mass definition, charge sector, ground state, boundary time, $\mathcal C=I/\pi$ normalization, null-normal counterterm, cutoff removal, and order of limits. Evaluate classical BTZ and quantum BTZ without changing that tuple. Then repeat with an allowed single finite counterterm shared by the whole family. If the classical calibration is preserved but the weak-backreaction CA rate remains discontinuous, the claimed universal bound or smooth classical limit fails. If rescuing it requires a state-dependent finite term, a new energy zero, or a different time normalization, the rescue is a different conjecture.

The evidence ceiling is therefore a proposal- and family-specific growth statement. Existing calculations establish useful successes and explicit failures of CA/CV diagnostics; they do not establish a universal energy bound on operational boundary complexity. General quantum speed limits belong to the boundary task, while live quantum-black-hole comparisons hand off to dated research review.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., and Zhao, Y. (2016). “Complexity, Action, and Black Holes.” Physical Review D 93, 086006. DOI.
• Chen, B., Liu, Y., and Yu, B. (2024). “Holographic Complexity of Rotating Quantum Black Holes.” Journal of High Energy Physics 2024(1), 055. DOI.
• Emparan, R., Frassino, A. M., and Sasieta, B. (2022). “Holographic Complexity in Quantum Black Holes.” Journal of High Energy Physics 2022(2), 204. DOI.
• Lloyd, S. (2000). “Ultimate Physical Limits to Computation.” Nature 406, 1047–1054. DOI.
• Parihar, M. K., and Punia, A. (2026). “Holographic Complexity of Dyonic Quantum Black Holes.” Journal of High Energy Physics 2026(2), 248. DOI.