Holographic Complexity Proposals and Diagnostics

Holographic complexity proposals ask which bulk quantity could measure the cost of preparing a boundary state. Volume, action, and path-integral constructions give different answers unless their tasks, normalizations, and regulators are matched; this chapter teaches how to compare them.

Enter this chapter

Begin with Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets to define the state-preparation task before choosing a bulk prescription.

Helpful background. What Task Does Complexity Answer? and Regulator Dependence and Continuum Complexity supply the operational task and its continuum limitations.

Topics

1. Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets — Defines the target, reference state, allowed gates, and error tolerance.
2. Complexity Equals Volume Proposals — Relates a maximal bulk slice to a proposed state complexity.
3. Complexity Equals Action Proposals — Evaluates action on a Wheeler–DeWitt patch with its boundary terms.
4. Path-Integral and Tensor-Network Complexity — Compares preparation costs defined by path integrals and tensor networks.
5. Complexity of Formation and Time Growth — Separates reference-state subtraction from time-dependent growth.
6. Divergences, Counterterms, and Scheme Dependence — Tracks which finite answers depend on regulators and counterterms.
7. Shockwaves, Switchbacks, and Scrambling Diagnostics — Tests delayed growth after perturbations against scrambling times.
8. Proposed Complexity Bounds and Their Counterexamples — Checks proposed inequalities against their precise assumptions and counterexamples.
9. Operational Meaning, Nonuniqueness, and Evidence Status — Assesses what operational interpretation each bulk prescription supports.

How the ideas fit together

A shockwave can produce a delay followed by linear growth in both a geometric proposal and a circuit model. Agreement tests the response to scrambling, but identifying the two quantities also requires a common reference state, cost function, and normalization. The switchback calculation makes this distinction concrete Stanford and Susskind 2014, §§ 2–3. Likewise, subtracting a reference geometry can define a complexity of formation without removing every finite scheme choice. A comparison is informative only when those choices are held fixed.

Claim-domain comparison

Compare the assumptions and strongest supported conclusion for each case.

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Representative claim domains and validity boundaries for Holographic Complexity Proposals and Diagnostics

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
complexity equals volume	Declare slice, length scale, and subtraction; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: boundary complexity task → CV, CA, or path proposal → regulator and counterterms → growth and switchback tests → proposal comparison. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “late-growth and formation checks” check is counterevidence to the promoted claim.	With boundary task, reference, normalization, and subtraction fixed, the claimed cost growth or formation disagrees with the volume prediction beyond stated errors.	Identifying the volume proposal with the action proposal as one observable.	one geometric proposal
complexity equals action	Declare WdW patch and null-boundary terms; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: boundary complexity task → CV, CA, or path proposal → regulator and counterterms → growth and switchback tests → proposal comparison. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “joint, counterterm, and switchback check” check is counterevidence to the promoted claim.	The claimed answer changes under a null reparametrization that its stated boundary-term prescription should cancel, or fails its fixed-task switchback prediction.	Treating an action prescription as scheme-independent complexity.	one action-based proposal
complexity bound	Declare gate set or bulk prescription; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: boundary complexity task → CV, CA, or path proposal → regulator and counterterms → growth and switchback tests → proposal comparison. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “counterexample and normalization test” check is counterevidence to the promoted claim.	An admissible state violates the proposed inequality with the same fixed gate set or bulk prescription, normalization, and energy reference.	Promoting a model-dependent bound to a universal operational theorem.	a bound in the stated model

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Where to go next

Continue to AdS3/CFT2 and Three-Dimensional Gravity for a setting where symmetry, spectra, and gravitational saddles can be compared more explicitly.

References

• Stanford, D., and Susskind, L. (2014). “Complexity and Shock Wave Geometries.” Physical Review D 90, 126007. DOI.