Complexity, Interiors, and Late-Time Dynamics

Complexity-based interior proposals relate the continued growth of a black hole’s interior to the difficulty of preparing its quantum state. The proposal becomes physical only when both sides are defined well enough to yield a prediction that survives arbitrary choices of gates, reference state, counterterms, foliation, and normalization. Semiclassical volume and action calculations do exhibit robust-looking growth and switchback patterns in controlled geometries. Their identification with one microscopic circuit complexity remains conjectural. This page asks what invariant comparison could separate such a proposal from a productive analogy.

Required background. Quantum Information supplies circuit complexity, state preparation, and error tolerance; Gauge Theory and String-Theory Preparation supplies the holographic states and boundary description used by the proposals.

Helpful background. Semiclassical Gravity supplies stationary black-hole scales and the approximation behind bulk actions; Quantum Fields in Curved Spacetime supplies fixed-background correlators; and Black-Hole Information supplies the recovery and interior questions that complexity is sometimes invoked to address.

What an interior-complexity claim must predict

For a regulated boundary system with reference state $|\psi_0\rangle$, gate set $\mathcal G$, and tolerance $\epsilon$, circuit complexity can be defined as

$$\mathcal C_{\mathcal G,\epsilon}(|\psi\rangle) =\min\left\{N: \min_{\theta\in\mathbb R} \left\lVert U_N\cdots U_1|\psi_0\rangle-e^{i\theta}|\psi\rangle\right\rVert \le\epsilon, \ U_i\in\mathcal G \right\}.$$

The value changes with $\mathcal G$, $|\psi_0\rangle$, the norm, and $\epsilon$. On the gravity side, the complexity-equals-volume proposal uses an extremal spatial volume, schematically

$$\mathcal C_V=\frac{V_{\max}}{G\ell_*},$$

where the length $\ell_*$ is not fixed by the semiclassical geometry alone. The complexity-equals-action proposal instead uses the on-shell action of a Wheeler–DeWitt region,

$$\mathcal C_A=\frac{I_{\rm WDW}}{\pi}.$$

Null-boundary terms, joints, counterterms, and normalization choices must be fixed. Neither equation is a definition of boundary circuit complexity unless a dictionary establishes it.

A discriminating prediction should therefore be relational: for example, a dimensionless ratio of growth rates, a switchback delay under a specified precursor, or a late-time response that is unchanged by admissible regulator and gate choices. Matching only monotonic growth is too weak, because many coarse observables grow for a long time.

Controlled patterns and their interpretation

In eternal AdS black holes, maximal volumes and Wheeler–DeWitt actions often grow approximately linearly at late classical times. Shock-wave geometries add a more structured signal. A precursor $W(t_w)=e^{iHt_w}We^{-iHt_w}$ contains a forward and backward evolution that largely cancels until the perturbation has scrambled; the resulting switchback subtracts a delay of order $t_*$. The geometry shows an analogous delayed response of the extremal surface or action region. Stanford and Susskind calculate this multi-shock structure in Stanford and Susskind 2014, §§ 2–4.

The action proposal and its stationary black-hole growth are developed in Brown et al. 2016, §§ 2–4. Those are semiclassical calculations of $I_{\rm WDW}$ under specified boundary prescriptions. The additional statement that the answer equals a boundary gate count is an interpretation. Agreement of volume and action on qualitative features strengthens the analogy but does not eliminate their different normalizations or divergences.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. The strongest competing positions are a permissive view, on which repeated agreement of growth and switchbacks reveals a genuine interior-complexity duality, and a conservative view, on which the bulk quantities are useful geometric diagnostics until a boundary complexity and error tolerance derive them.

Proposal or adverse evidence	Result	Scope or limitation
Maximal-volume conjecture in Susskind 2016, §§ 4–5	Long wormhole volume tracks a long boundary evolution and motivates $\mathcal C_V$.	The scale $\ell_*$ and precise boundary complexity are not derived; absolute values are prescription dependent.
Wheeler–DeWitt action in Brown et al. 2016, §§ 2–4	Specified stationary geometries show computable action growth with thermodynamic-scale relations.	Null terms, counterterms, matter content, charge, and late-time limits affect the result.
Shock and precursor calculation in Stanford and Susskind 2014, §§ 2–4	Switchback delays reproduce a scrambling-time subtraction for controlled multi-shock geometries.	The boundary circuit interpretation and gate robustness are assumed, and the setting is not generic evaporation.
Finite-dimensional complexity analysis in Harlow 2016, Appendix E	Typical state preparation can require exponentially many gates and eventually saturates in a regulated system.	The estimate depends on gate set and tolerance; it does not uniquely select volume, action, or an interior observable.

The matrix supports nontrivial structural agreement, especially the switchback effect. It does not settle whether either geometric functional is the unique dual of a microscopic complexity, whether the dual extends to one-sided evaporation, or whether long-time ensemble averages describe one Hamiltonian.

Adverse checks and invariant content

Change the allowed gate set from geometrically local two-qubit gates to a set containing long products as elementary operations. The absolute circuit complexity and its growth rate can change parametrically while the classical interior geometry does not. This is a direct counterexample to an unqualified equality. A defensible claim must either derive the preferred gate geometry from the boundary theory or identify a feature insensitive to a controlled class of such changes.

Another independent check compares volume and action for the same perturbed state. A feature is stronger evidence when both prescriptions give the same dimensionless switchback delay after their ultraviolet terms and normalizations are fixed. Disagreement is informative rather than an instruction to tune a scale: it identifies which claimed interior property is prescription dependent.

Late-time extrapolation is especially delicate. A finite quantum system has a maximum complexity and recurrences on enormous timescales, whereas a single classical wormhole can appear to grow indefinitely. Quantum saddles, nonperturbative effects, or a changed dictionary must enter before that limit. It remains unresolved which mechanism supplies the correct single-theory completion.

This page does not teach generic field-theory complexity, stable black-hole thermodynamics, or explicit string constructions. It also does not infer a smooth interior from a large gate count. Its bounded conclusion is that a complexity-based interior proposal needs a defined boundary quantity and at least one regulator-robust, state-specific prediction beyond qualitative growth.

The conclusions could change after a microscopic dictionary derives a unique bulk functional for a specified boundary cost, or a controlled one-sided evaporating model reproduces or falsifies the switchback relation, or a nonperturbative single-theory calculation resolves the late-time saturation mismatch without changing the declared state and error criterion.

Exercises

Suppose two admissible gate sets give complexities related by $\mathcal C_2=a\mathcal C_1+b$ with positive constants $a$ and $b$. Which of an absolute complexity, a time delay between two matched switchback curves, and a raw growth rate can be compared directly with one geometric prediction?

Solution

The absolute value changes under both $a$ and $b$, and the raw growth rate changes under $a$, so neither is invariant without a normalization dictionary. A time delay extracted by aligning two curves is unchanged by a common affine vertical transformation and is therefore the strongest of the three comparisons. It still requires the two gate sets to represent the same state, precursor, tolerance, and time variable; otherwise even the delay need not be comparable.

References

• Brown, Adam R., Daniel A. Roberts, Leonard Susskind, Brian Swingle, and Ying Zhao. 2016. “Complexity, Action, and Black Holes.” Physical Review D 93, 086006. arXiv:1509.07876.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88, 015002. arXiv:1409.1231.
• Stanford, Douglas, and Leonard Susskind. 2014. “Complexity and Shock Wave Geometries.” Physical Review D 90, 126007. arXiv:1406.2678.
• Susskind, Leonard. 2016. “Computational Complexity and Black Hole Horizons.” Fortschritte der Physik 64, 24–43. DOI. arXiv:1402.5674.