Complexity in Continuum QFT

The question is: Does an interacting continuum QFT admit a non-arbitrary complexity notion with stated operational or invariant content? A complexity is meaningful only after specifying the task, allowed elementary operations, reference resource, tolerance, and cost. Removing a regulator while leaving those inputs implicit does not produce an invariant; it hides the choices in a divergent answer.

Evidence cutoff. 11 August 2026.

Required background. What Task Does Complexity Answer? distinguishes state preparation, unitary synthesis, and distinguishability tasks. Regulator Dependence and Continuum Complexity tracks cutoff scaling and counterterms.

Helpful background. Cost Geometry, Gate Sets, and Reference States supplies Nielsen-type constructions; Operational Preparation Cost and Energy-Constrained Bounds turns a formal distance into a resource task; and Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets exposes the inputs required before comparing to bulk proposals.

Complexity as a family of resource questions

For a regulated Hilbert space, a circuit complexity may be written schematically as

$$\mathcal C_{G,F,\epsilon}(|\psi\rangle,|R\rangle) =\inf_{U:R\to\psi}\int_0^1 ds\,F\!\left(Y(s)\right),$$

where $G$ is the allowed generator or gate set, $F$ its cost, $|R\rangle$ the reference, and $\epsilon$ the preparation tolerance. Every subscript can change the answer or even the ordering of states. In a continuum QFT, local gates at arbitrarily short distance commonly make $\mathcal C$ diverge with the cutoff.

Candidate notion	What is well defined	What remains arbitrary or conjectural
Lattice circuit complexity	Minimum gate count or cost at fixed lattice, gates, tolerance, and reference	Continuum scaling and equivalence between gate models
Nielsen geometry	A geodesic problem on a chosen unitary group with chosen penalties	Metric, penalties, operator basis, and treatment of unbounded generators
Gaussian-state QFT complexity	Explicit solutions and cutoff divergences for free theories	Interaction, non-Gaussian gates, and operational implementation
Path-integral optimization	An optimized Euclidean preparation geometry within a chosen cost functional	Uniqueness of cost, state normalization, and relation to physical circuits
Holographic volume/action/spacetime-volume proposals	Geometric quantities with characteristic growth in semiclassical bulk states	The exact boundary task and dictionary; counterterm and normalization choices
Energy- or control-constrained preparation	A laboratory-style minimum time, work, or control resource	Apparatus and control algebra dependence; continuum feasibility

Direct assessment

Established in bounded settings: regulated free and Gaussian fields admit calculable circuit costs. Jefferson and Myers exhibited the continuum divergence and gate/reference dependence explicitly for free scalar ground states Jefferson and Myers 2017. Interacting lattice and truncated systems can likewise be assigned a complexity once the resource theory is fixed.

Open for interacting continuum QFT: no included construction supplies a regulator-independent scalar complexity that is simultaneously non-arbitrary, operational, and universal across interacting theories.

Constrained negative result: basis independence cannot be demanded while retaining the characteristic behavior sought from holographic complexity. A no-go argument shows that a basis-independent distance fails to reproduce those features Couch et al. 2018. This does not make complexity meaningless; it makes the basis or gate algebra part of its definition.

Holographic status: “complexity = volume” and “complexity = action” are productive proposals, not established equalities to a unique boundary complexity. The action proposal is explicitly conjectural Brown et al. 2016, and bulk counterterms can remove divergences while leaving finite scheme choices Akhavan and Omidi 2019.

Facts, assumptions, and interpretations

Kind	Statement
Established fact	Complexity is not a state functional until the target relation, reference, operations, cost, and tolerance are fixed.
Established fact	Local continuum degrees of freedom generate ultraviolet-divergent costs in standard circuit geometries.
Assumption	A preferred local operator algebra and penalty schedule is physically selected by the QFT or experimental controls.
Assumption	Removing divergent local terms leaves a finite part stable under allowed changes of regulator and counterterm scheme. This must be proved per proposal.
Interpretation	Similar late-time growth, switchback behavior, or divergence structure identifies the bulk and boundary quantities. Shared qualitative behavior is evidence for a dictionary candidate, not a uniqueness proof.

Strongest positions and contrary evidence

Position	Strongest version	Contrary or qualifying evidence
Complexity is inherently relative but physical.	Once an operational control model is fixed, minimum resources can bound preparation time, work, or error and need no universal basis independence.	Different apparatuses define different resources; this does not produce an intrinsic state invariant.
Renormalized complexity has universal data.	Divergent coefficients and selected differences may encode locality, dimension, or state changes, much as entropy differences can.	Finite counterterms and gate penalties can alter finite parts and even rankings unless a protection theorem exists.
Holography selects the correct complexity.	Bulk geometry may define a preferred strong-coupling observable and constrain boundary inputs.	Multiple geometric proposals share qualitative signatures. “Does Complexity Equal Anything?” constructs broad gravitational observables with similar late-time behavior, weakening uniqueness. Belin et al. 2022
No continuum complexity can be meaningful.	Raw gate counts diverge and choices are unavoidable.	Relative, energy-constrained, or task-specific costs can still be finite, monotone, and experimentally meaningful; non-uniqueness is not nonexistence.

Obstructions and uncertainty

• Unbounded operations: local field generators are unbounded; formal geodesics need domains and implementability, not only Lie-algebra notation.
• Continuum tolerance: exact preparation of a state with infinitely many modes is generally an infinite task. An energy norm, local-observable norm, or channel distance must replace unspecified fidelity.
• Gauge constraints: a gate set must preserve Gauss law and specify edge/center data. An extended Hilbert-space circuit may solve a different problem.
• Interaction: Gaussian normal-mode methods do not control non-Gaussian operator growth. Perturbative corrections can be scheme dependent and need remainder estimates.
• Holographic normalization: bulk volumes require a length scale; action prescriptions require null-boundary and counterterm choices. Matching divergences does not fix these uniquely.

What would count as a resolution?

A strong positive result would define a task for a class of interacting continuum QFTs, prove existence and regulator convergence in a declared topology, show stability under a physically motivated equivalence class of gates and schemes, and connect the value to a measurable control or information bound. It should distinguish states that simpler observables do not and survive held-out regulator and interaction tests.

A strong impossibility theorem would identify a natural axiom set—locality, continuity, additivity, monotonicity, covariance, and finite energy—and show that every nontrivial complexity violates at least one. A holographic resolution additionally needs a boundary definition computed independently of the bulk and agreement beyond leading large-$N$, semiclassical order.

Connected methods and tests

• Quantum Information and Entanglement in QFT supplies resource theories; Holography and Quantum Gravity supplies geometric proposals.
• Replica, Modular, and Operator-Algebra Methods offers invariant comparison quantities but does not define circuit complexity; Holographic Reconstruction and Gravitational Path Integrals tracks bulk assumptions.
• A useful continuum benchmark holds the target state fixed while varying the gate set, reference state, penalty schedule, regulator, and renormalization prescription. Comparisons of bulk volume, action, or other geometric proxies should likewise expose their normalization and scheme dependence; agreement in a restricted family is evidence of correlation, not proof of a complexity dictionary.

Evidence boundary

The finite source set was selected through targeted arXiv, INSPIRE, APS, and journal searches for circuit definitions, basis dependence, holographic proposals, and counterterm ambiguities available through 11 August 2026. Papers computing a quantity without stating its reference, cost, or regulator were not used to support invariance; the selection is not exhaustive.

References

• Akhavan, A., and Omidi, F. (2019). “On the Role of Counterterms in Holographic Complexity.” Journal of High Energy Physics 2019, 054. DOI; arXiv:1906.09561.
• Belin, A., Myers, R. C., Ruan, S.-M., Sárosi, G., and Speranza, A. J. (2022). “Does Complexity Equal Anything?” Physical Review Letters 128, 081602. DOI.
• Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., and Zhao, Y. (2016). “Complexity Equals Action.” Physical Review Letters 116, 191301. DOI; arXiv:1509.07876.
• Couch, J., Eccles, S., Jacobson, T., and Nguyen, P. (2018). “Holographic Complexity and Its Basis Dependence.” Physical Review D 98, 046002. DOI.
• Jefferson, R., and Myers, R. C. (2017). “Circuit Complexity in Quantum Field Theory.” Journal of High Energy Physics 2017, 107. DOI; arXiv:1707.08570.