CFT Complexity and Penalty Factors

Nicolas Chagnet; Osher Shoval; Shira Chapman; Stefano Baiguera

REVIEW 3 major objections 5 minor 1 cited by

CFT Complexity and Penalty Factors

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives a penalty-aware metric for CFT state complexity by projecting weighted generator costs onto the space of coherent states, and extracts exact and perturbative complexity values for one- and two-dimensional CFTs.

desk verdict The submersion framework and the exact results are solid and citable; the perturbative expansion in §4.3.2 has a factor-of-2 error that contradicts the paper's own J2=0 exact result and needs correction before the numerics can be trusted. read the letter →

arxiv 2507.22118 v1 pith:I3N3H5NF submitted 2025-07-29 hep-th quant-ph

Stefano Baiguera , Nicolas Chagnet , Shira Chapman , Osher Shoval This is my paper

classification hep-thquant-ph

keywords quantumcomplexityNielsenpenaltyfactorsconformalfieldtheorypseudo-RiemanniansubmersioncoherentstatesSL(2R)holography

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum complexity is normally defined as if every gate costs the same. This paper removes that assumption for conformal field theories: it assigns penalty factors to the generators of the global conformal group and defines state complexity as the shortest path in the metric this weighted cost induces on the space of coherent states. The core construction is a projection from the group manifold to the coset space $G/H$ by setting $f^a=0$, which the authors show is a pseudo-Riemannian submersion for compact and non-compact groups, making the induced metric unique. Applying this to one- and two-dimensional CFTs gives exact formulas in special cases, a perturbative expansion in the weights, numerical geodesics, and a condition on the penalties ($J_1 > \max(J_3 \pm 2J_2)$) under which the metric is positive definite and hence interpretable as complexity. The interest is that weighted gates, not just isotropic ones, are the realistic setting for holography and for diagnosing chaotic dynamics.

What carries the argument

The load-bearing object is the pseudo-Riemannian submersion from the unitary-group manifold to the coset space $G/H$ of quantum states. The paper decomposes any right-invariant norm with penalties as $F_I^2 = (F_I^{\mathrm{state}})^2 + \tilde I_{ab} f^a f^b$, where the $f^a$ encode the stabilizer degrees of freedom that move the unitary but not the state; the projection $f^a=0$ kills those directions. Because an isometric group action on a pseudo-Riemannian manifold defines a unique quotient metric, this condition fixes the induced state metric unambiguously, and for abelian stabilizers it is equivalent to setting the conserved charge $K_a = \tilde I_{ab} f^b$ to zero. This same machinery carries the penalty weights from the group down to the coherent states, where geodesic length becomes Nielsen complexity.

What would settle it

Take the $J_2=0$ case and compute the geodesic length numerically for a fixed target radius $r_T$ with $J_1=100$ and $J_3=1$; the analytic claim $C=\sqrt{J_3}\,C_{\mathrm{FS}}$ predicts a length independent of $J_1$, so any measurable dependence of the geodesic length on $J_1$ at this point would falsify the central analytic result.

Watch

Extended reading notes

Core claim

The paper's central claim is that a right-invariant cost function with penalty factors on a Lie group $G$ projects uniquely onto the space of states via the condition $f^a=0$, giving the metric $(F_{\mathrm{state}}^I)^2 = (\tilde I_{ij} - \tilde I_{ic}(\tilde I^{-1})^{ca}\tilde I_{aj}) u^i u^j$ on the coset $G/H$, and that this projection is a pseudo-Riemannian submersion for both compact and non-compact groups. Specializing to $G=\mathrm{SO}(1,2)\simeq\mathrm{SL}(2,\mathbb{R})$ acting on a primary state $|\Delta\rangle$, the state space is the unit disk of generalized coherent states, and geodesics of the projected metric are interpreted as optimal circuits. The paper finds the analytic complexity $C=\sqrt{J_3}\,C_{\mathrm{FS}}$ when the anisotropy penalty $J_2$ vanishes, $C=2\,\mathrm{arctanh}(r_T)\sqrt{\min(I_-,I_+)}$ when dilatations are free, and a positivity window $J_1>\max(J_3\pm 2J_2)$ inside which the complexity interpretation is viable. For two-dimensional CFTs, coupling the two copies by a dilatation-dilatation penalty $J_0$ leaves the complexity unchanged at least through second order in perturbation theory, with a small higher-order effect visible numerically that can either raise or lower the complexity depending on the penalty regime. The construction also connects to stabilizer-direction minimization and to coadjoint-orbit methods used in earlier penalty-free treatments.

Load-bearing premise

All complexity values are computed for generalized coherent states obtained by acting with the global conformal group on a single primary state, not for the full CFT Hilbert space with its infinite tower of higher conformal-tower states, and the Virasoro extension is deferred; the framework itself does not require the full Hilbert space, but any claim about full CFT state complexity would depend on that truncation.

Editorial extensions

If this is right

In one-dimensional CFTs with equal costs for $P$ and $K$, the state complexity is exactly $\sqrt{J_3}$ times the Fubini-Study complexity, independent of the dilatation penalty $J_1$.
If the dilatation generator is free ($I_0=0$), every target state's complexity is $2\,\mathrm{arctanh}(r_T)\sqrt{\min(I_-,I_+)}$, achieved by moving along the least-penalized axis and then rotating freely.
Outside the positivity window $J_1>\max(J_3\pm 2J_2)$ the state-space metric is indefinite, so no global complexity interpretation exists for all coherent states, defining a kind of landscape and swampland for penalty choices.
In two-dimensional CFTs with a dilatation coupling between left and right sectors, the complexity is the quadratic sum of the per-sector complexities at least through second order; the coupling first shows up at higher order and can either increase or decrease the complexity.
Because the projection works for any symmetric reductive coset, the same construction provides a template for weighted state complexity of the full conformal group $\mathrm{SO}(d,2)/(\mathrm{SO}(2)\times\mathrm{SO}(d))$ in higher dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the paper leaves implicit is that the near-insensitivity to the dilatation penalty is an artifact of the projection: the stabilizer control function is solved for and substituted, so the cost of $D$ never appears independently, and any complexity prescription that minimizes over the stabilizer will inherit a similarly mild dependence on the Hamiltonian direction.
The constraint $J_1>\max(J_3\pm 2J_2)$ resembles a positivity bound that might ultimately be derivable from CFT data if penalty factors are realized as charges of a deformed algebra; the paper stops at listing the constraint rather than deriving it.
A testable extension is to run the same submersion for $d\ge 3$ CFTs, where $\mathrm{SO}(d,2)/(\mathrm{SO}(2)\times\mathrm{SO}(d))$ is symmetric: formula (3.10) applies immediately, and the positivity conditions should produce accessible regions with a boundary analogous to the examples in appendix C.
The CFT2 numerical result that $J_0$ enters only at third order suggests a perturbative factorization of complexity into left and right sectors; resumming those terms might expose a simple dependence on $J_0$ and the penalty gap $J_1-J_3$.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the Section 3 framework is the real contribution and I think it holds up. The perturbative section has a concrete error. The stress-test note is right: eq. (4.42) does not reduce to the paper's own exact J2 = 0 result, eq. (4.39). Expand sqrt(1+εβ3): the linear term is εβ3/2, but (4.42) gives εβ3, and the '1 + εβ3 ≈ sqrt(1+εβ3)' in eq. (4.43) is just false. The quadratic coefficient −β3²/8 does match, which is probably how the mismatch slipped through. The factor of two then propagates to (4.44) and to the interpretation of figures 3–5, which use J3 − 2J2 cos(2θT) as the leading complexity ratio. It doesn't matter which formula the numerics actually support: (4.42) fails the β2 = 0 check, so at most one of (4.42)/(4.44) is right, and the exact result can't serve as the stated consistency check until (4.42) is corrected.

What is new and good. The pseudo-Riemannian submersion extension to non-compact groups, with the f^a = 0 map recovering the [34] minimization as an extremization, is a genuine generalization of the [37] technology. The algebra in Section 3.1 is consistent and the uniqueness claim via the quotient theorem is sound. The exact results check out: C = sqrt(J3) C_FS for J2 = 0, the free-dilatation formula 2 arctanh(rT) sqrt(min(I±)), and the isotropic CFT2 factorized result (5.18). The positivity landscape (4.29) and the two-case structure in (5.12)–(5.13) are useful, and the authors are explicit that the principal-minor analysis is not exhaustive. Citations to [37] and [81] are fair; the extension is clearly separated from what those papers did.

Soft spots, in proportion. The perturbative error is the main one: it sits in a headline result (table 1 lists §4.3.2) and it directly affects the numerical interpretation. It looks fixable — recompute the O(ε) term and propagate. Second, the abstract's 'generic Lie groups' overstates the derivation, which uses the reductive symmetric-space grading (3.6); the text admits this, the abstract should too. Third, all CFT results live on a single coherent-state orbit (or a product of two), not the full Hilbert space with Virasoro descendants. The paper says this and defers Virasoro to future work, so it is an honest scope limitation, but it narrows the title's promise. Minor repro point: the numerics are described but the code and data are not shipped.

Who it is for: people in Nielsen complexity for CFTs and holographic complexity. The framework and exact formulas will get cited. I would send it to a serious referee as is — the error is concrete and fixable, not fatal. Recommendation: revise, with the perturbative expansion corrected and the abstract tightened.

Referee Report

3 major / 5 minor

Summary. This paper develops a Nielsen-style geometric framework for quantum state complexity with non-isotropic penalty factors, based on projecting a right-invariant (pseudo-)Riemannian cost function from a Lie group G to a coset G/H by setting f^a=0, which is argued to define a (pseudo-)Riemannian submersion. The general construction is applied to SO(1,2)/SO(2) coherent states for one-dimensional CFTs and to two copies of such systems, with a dilatation-dilatation coupling, for two-dimensional CFTs. Exact results are derived in the isotropic J2=0 case, perturbative expansions in small penalty deviations are presented, and numerical solutions are computed for the general case. The paper also analyzes the positivity constraints on penalty factors and offers a holographic interpretation of the submersion condition.

Significance. If correct, the framework would give a systematic method for including generator-dependent costs in CFT complexity computations, extending earlier SU(N) results to non-compact groups. The algebraic core is clear and largely self-contained: the Schur-complement decomposition in Eqs. (3.7)-(3.10), the conserved-charge interpretation in Section 3.2, and the exact J2=0 results are nontrivial and checkable. The penalty factors are inputs rather than fitted parameters, and the results provide concrete predictions for geodesic lengths on the coherent-state manifold. The comparison with the isotropic Fubini-Study case and the discussion of positivity boundaries are useful. The main limitations are the restriction to reductive symmetric cosets, the coherent-state truncation of the Hilbert space, and a concrete error in the perturbative expansion; these do not invalidate the exact results but require correction and clarification.

major comments (3)

[Section 4.3.2, Eqs. (4.42)-(4.43)] The claimed consistency check with the exact J2=0 result is incorrect. Setting beta_2=0 in Eq. (4.42) gives hat C = 1 + epsilon beta_3 - epsilon^2 beta_3^2/8 + O(epsilon^3), whereas the exact result hat C = sqrt(1+epsilon beta_3) expands as 1 + epsilon beta_3/2 - epsilon^2 beta_3^2/8 + O(epsilon^3). The linear coefficient is off by a factor of two, and the statement in Eq. (4.43) that '1 + epsilon beta_3 approximately equals sqrt(1+epsilon beta_3)' is false beyond O(epsilon). Since Eq. (4.44) and the comparisons in Figures 3-5 inherit this expression, the perturbative results and their numerical interpretation need to be recomputed or re-expressed, for example as an expansion of hat C^2 rather than hat C if that is what was intended.
[Section 3.1, Eqs. (3.6)-(3.10)] The derivation leading to Eq. (3.7) relies on the reductive symmetric condition (3.6), in particular the assumption [b,b] is contained in h, which is used to conclude that the stabilizer velocity v has no coset components. The text immediately after Eq. (3.10) states that the result is valid 'for any Lie group, either compact or non-compact', and the abstract promises a framework for generic Lie groups. For a coset that is not reductive symmetric, this separation does not follow and the completion-of-the-square step in Eq. (3.7) may fail. The claim should be restricted to reductive symmetric homogeneous spaces, or a separate proof should be supplied for the general case. The explicit examples in the paper, SO(1,2)/SO(2) and the product used in the two-dimensional CFT, are within the symmetric class, so the applications are not affected.
[Section 4.1 and Section 6] The complexity results are computed only on the homogeneous space of generalized coherent states generated by acting with the global conformal group on a single primary state, as defined in Eqs. (4.4)-(4.5). The full CFT Hilbert space, including the infinite tower of Virasoro descendants, is not explored, and Section 6 explicitly defers the Virasoro extension to future work. Because the abstract and introduction speak of 'state complexity for states in one- and two-dimensional CFTs' without this qualification, the intended scope should be stated explicitly in the abstract. This is not a defect of the geometric construction itself, but it is essential for interpreting the analytic and numerical results as statements about CFT state complexity.

minor comments (5)

[Section 5.1.2] The first sentence of Section 5.1.2 refers to 'plugging the above isotropy conditions on the penalties inside eq. (5.14)', but Eq. (5.14) is the resulting cost function, not the starting expression; the reference should be to Eq. (5.6).
[Eq. (4.20b)] Equation (4.20b) contains a stray 'm' in the expression for I_+, which should read I_+ = J_3 - 2J_2; this is a typographical error but should be corrected.
[Section 5.1.1, Eqs. (5.9)-(5.11)] The positivity analysis for the coupled two-dimensional metric is acknowledged to be non-exhaustive, but the wording 'the following condition should hold' followed by a list of expressions required to 'have the same sign' is too ambiguous to be checked; the authors should specify the exact inequalities used and state clearly which conditions are necessary and which are sufficient for positive definiteness.
[Section 2.2] There is a typo in the last paragraph of Section 2.2: 'Riemanninan' should be 'Riemannian'.
[Table 3] The check and cross symbols in Table 3 are not explained in the caption; adding a sentence such as 'check marks indicate dependence of the quantity on the corresponding coordinates or velocities' would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the complexity metric and all reported formulas are derived from explicitly chosen penalty inputs and standard submersion geometry, not from fitted data or author-imported uniqueness.

full rationale

The derivation chain is self-contained. In Section 3.1, the projected metric (F_state^I)^2 is obtained as the horizontal part of the right-invariant cost function after completing the square in the stabilizer velocities; setting f^a=0 is the extremization over stabilizer directions, and it is not secretly equivalent to the target complexity. The uniqueness invoked in Section 3.1 and Appendix A is Theorem A.3, quoted from standard textbooks [106-108], not from the authors' prior work. The CFT results in Sections 4 and 5 are consequences of geodesic lengths for fixed penalty factors: formulas such as C = sqrt(J3) C_FS for J2=0 and the two-dimensional quadratic-sum results are computed from the stated metrics, and the numerical/perturbative solutions solve the Euler-Lagrange equations for those metrics. No parameter is fitted to a subset of data and then renamed as a prediction. The self-citations to [37] and [81] are contextual: the FS metric is independently re-derived in eq. (4.8), and the coadjoint-orbit connection is presented as a comment/future direction. Even if those citations were removed, the projection algebra and the CFT computations would stand. The restriction to generalized coherent states of the global conformal group is an explicit scope limitation acknowledged in Section 6, not a circularity. One technical caveat: the perturbative consistency check in eqs. (4.42)-(4.43) appears to contain a Taylor-expansion error in the linear coefficient, but this is a correctness issue, not a circular derivation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; the only invented objects are mathematical metrics and penalty parameters. The central claim rests on the right-invariant cost assumption, the reductive symmetric quotient structure, and the choice of f^a=0 as the projection rule.

free parameters (5)

J1 (dilatation penalty in CFT1) = not fitted; examples 1, 1.5
Chosen by hand in Sections 4 and 5; the whole point is to test arbitrary relative weights. No physical principle fixes it.
J2 (anisotropy between L+ and L-) = not fitted; examples 0, 0.05, 0.1
Hand-selected to probe anisotropic costs; controls the difference I- minus I+.
J3 (common coset penalty) = not fitted; examples 0.5, 1, 1.5
Hand-selected; rescales Fubini-Study complexity when J2=0.
J0 (D-\bar D coupling in CFT2) = not fitted; examples 0.1, 4.0
Hand-selected to probe coupling between left and right copies; small and large regimes in Section 5.3.
\bar J1, \bar J2, \bar J3 (right-copy penalties) = not fitted; set equal to left-copy values in symmetric examples
Independent in general, but chosen symmetric in the worked cases; hand-selected inputs.

assumptions (6)

standard math Inner product on the Lie algebra is given by the trace form (Killing form), and the Ad-invariance property (B.11) holds
Used throughout Section 3; standard for semisimple Lie algebras and checked in appendix B for SO(1,2).
domain assumption G/H is a reductive symmetric space: [h,h] subset h, [h,b] subset b, [b,b] subset h
Needed so v^i=0 and the Schur-complement formula (3.10) holds; valid for the SO(1,2)/SO(2) examples but not for every Lie group, despite the abstract's 'generic Lie groups' language.
domain assumption Cost function is right-invariant and constant-penalty in a fixed generator basis
Section 2.2; this excludes state-dependent or left-invariant costs and is a modeling choice, not a consequence of quantum mechanics.
ad hoc to paper State-complexity metric is defined by the horizontal projection f^a=0
Section 3.1 and eq. (3.9). For non-compact groups this is an extremum and saddle choice, not a minimum; it is a definitional postulate of the framework.
domain assumption Circuit space is restricted to the global conformal group and states are coherent states of one primary
Section 4.1, ansatz (4.4) to (4.5). This truncates the CFT Hilbert space and excludes Virasoro descendants; all reported CFT complexities are on this orbit.
domain assumption Two-dimensional CFT cost is a sum of single-copy costs plus a chosen D-\bar D coupling
Section 5.1 eq. (5.5). Other couplings are analyzed only in appendix C and generally give indefinite metrics; the analysis is not exhaustive.

how reviews work

0 comments

Cite this review

Pith. "Pith review of CFT Complexity and Penalty Factors." pith.science (2026). https://pith.science/paper/I3N3H5NF

@misc{pith2026250722118,
  author       = {Pith},
  title        = {Pith review of: CFT Complexity and Penalty Factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3N3H5NF}},
  note         = {Machine review of arXiv:2507.22118}
}

read the original abstract

Quantum complexity of conformal field theory (CFT) states has recently gained significant attention, both as a diagnostic tool in condensed matter systems and in connection with holographic observables probing black hole interiors. Previous studies have primarily focused on cases where all generators of the conformal group contribute equally to the cost of building a circuit. In this work, we present a general framework for studying the complexity of circuits in generic Lie groups, where penalty factors assign relative weights to different generators. Our approach constructs a metric on the coset space of quantum states, induced from a (pseudo-)Riemannian norm on the space of unitary circuits. The geodesics of this metric are interpreted as optimal circuits. The method builds on the formalism of (pseudo-)Riemannian submersions and connects naturally to other prescriptions in the literature, including cost function minimization along stabilizer directions and constructions based on coadjoint orbits. As a concrete application, we compute state complexity for states in one- and two-dimensional CFTs. For specific choices of penalty factors, our prescription yields a positive-definite metric with a viable interpretation as complexity; in other cases, the resulting metric is indefinite. In the viable regime, we derive analytic results when a specific penalty factor is turned off, develop perturbative expansions for small values of the penalty factors, and provide numerical results in the general case. We comment on the relation of our measure of complexity to holography.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Geometry of Quantum Complexity in Open Systems
quant-ph 2026-07 conditional novelty 7.0 of 10

Open-system quantum complexity is governed by a sub-Finslerian geometry whose curvature depends on the cost penalties for unitary and dissipative controls.

Reference graph

Works this paper leans on

112 extracted references · 21 canonical work pages · cited by 1 Pith paper

[34]

A. R. Brown and L. Susskind,Complexity geometry of a single qubit, Phys. Rev. D100 (2019) 046020 [1903.12621]

arXiv 2019
[37]

Auzzi, S

R. Auzzi, S. Baiguera, G. B. De Luca, A. Legramandi, G. Nardelli and N. Zenoni,Geometry of quantum complexity, Phys. Rev. D103 (2021) 106021 [2011.07601]

arXiv 2021
[81]

Chagnet, S

N. Chagnet, S. Chapman, J. de Boer and C. Zukowski,Complexity for Conformal Field Theories in General Dimensions, Phys. Rev. Lett.128 (2022) 051601 [2103.06920]

arXiv 2022
[1]

Susskind,Three Lectures on Complexity and Black Holes, 10, 2018

L. Susskind,Three Lectures on Complexity and Black Holes, 10, 2018. 10.1007/978-3-030-45109-7

doi:10.1007/978-3-030-45109-7 2018
[2]

Chapman and G

S. Chapman and G. Policastro,Quantum computational complexity from quantum information to black holes and back, Eur. Phys. J. C82 (2022) 128 [2110.14672]

arXiv 2022
[3]

Baiguera, V

S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science, 2503.10753

arXiv
[4]

Harlow and P

D. Harlow and P. Hayden,Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [1301.4504]. – 58 –

arXiv 2013
[5]

I. H. Kim, E. Tang and J. Preskill,The ghost in the radiation: robust encodings of the black hole interior (invited paper), JHEP 06 (2020) 031 [2003.05451]

arXiv 2020

Show all 112 references

[6]

Hayden and J

P. Hayden and J. Preskill,Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [0708.4025]

2007 arXiv
[7]

Yoshida and A

B. Yoshida and A. Kitaev,Efficient decoding for the Hayden-Preskill protocol, 1710.03363

arXiv
[8]

F. Liu, S. Whitsitt, J. B. Curtis, R. Lundgren, P. Titum, Z.-C. Yang et al.,Circuit complexity across a topological phase transition, Phys. Rev. Res.2 (2020) 013323 [1902.10720]

2020 arXiv
[9]

Yang and A

J. Yang and A. R. Frey,Complexity, scaling, and a phase transition, JHEP 09 (2023) 029 [2307.08229]

2023 arXiv
[10]

Susskind,Computational Complexity and Black Hole Horizons, Fortsch

L. Susskind,Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [1403.5695]

2016 arXiv
[11]

Stanford and L

D. Stanford and L. Susskind,Complexity and Shock Wave Geometries, Phys. Rev. D90 (2014) 126007 [1406.2678]

2014 arXiv
[12]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett.116 (2016) 191301 [1509.07876]

2016 arXiv
[13]

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Complexity, action, and black holes, Phys. Rev. D93 (2016) 086006 [1512.04993]

2016 arXiv
[14]

Lehner, R

L. Lehner, R. C. Myers, E. Poisson and R. D. Sorkin,Gravitational action with null boundaries, Phys. Rev. D94 (2016) 084046 [1609.00207]

2016 arXiv
[15]

Carmi, R

D. Carmi, R. C. Myers and P. Rath,Comments on Holographic Complexity, JHEP 03 (2017) 118 [1612.00433]

2017 arXiv
[16]

Chapman, H

S. Chapman, H. Marrochio and R. C. Myers,Complexity of Formation in Holography, JHEP 01 (2017) 062 [1610.08063]

2017 arXiv
[17]

Couch, W

J. Couch, W. Fischler and P. H. Nguyen,Noether charge, black hole volume, and complexity, JHEP 03 (2017) 119 [1610.02038]

2017 arXiv
[18]

Carmi, S

D. Carmi, S. Chapman, H. Marrochio, R. C. Myers and S. Sugishita,On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [1709.10184]

2017 arXiv
[19]

Belin, R

A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi and A. J. Speranza,Does Complexity Equal Anything?, Phys. Rev. Lett.128 (2022) 081602 [2111.02429]

2022 arXiv
[20]

Belin, R

A. Belin, R. C. Myers, S.-M. Ruan, G. Sárosi and A. J. Speranza,Complexity equals anything II, JHEP 01 (2023) 154 [2210.09647]

2023 arXiv
[21]

D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman,A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [1812.08657]

2019 arXiv
[22]

Nandy, A

P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rept. 1125-1128 (2025) 1 [2405.09628]

2025 arXiv
[23]

Rabinovici, A

E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner,Krylov Complexity, 2507.06286

arXiv
[24]

Susskind and Y

L. Susskind and Y. Zhao,Switchbacks and the Bridge to Nowhere, 1408.2823. – 59 –

arXiv
[25]

A. R. Brown, L. Susskind and Y. Zhao,Quantum Complexity and Negative Curvature, Phys. Rev. D 95 (2017) 045010 [1608.02612]

2017 arXiv
[26]

M. A. Nielsen,A geometric approach to quantum circuit lower bounds, quant-ph/0502070

arXiv
[27]

M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty,Quantum computation as geometry, Science 311 (2006) 1133

2006
[28]

M. R. Dowling and M. A. Nielsen,The geometry of quantum computation, quant-ph/0701004

arXiv
[29]

A. R. Brown, M. H. Freedman, H. W. Lin and L. Susskind,Universality in long-distance geometry and quantum complexity, Nature 622 (2023) 58 [2111.12700]

2023 arXiv
[30]

A. R. Brown,Polynomial Equivalence of Complexity Geometries, Quantum 8 (2024) 1391 [2205.04485]

2024 arXiv
[31]

Lloyd,Universal quantum simulators, Science (1996)

S. Lloyd,Universal quantum simulators, Science (1996)

1996
[32]

A. R. Brown and L. Susskind,Second law of quantum complexity, Phys. Rev. D97 (2018) 086015 [1701.01107]

2018 arXiv
[33]

Balasubramanian, M

V. Balasubramanian, M. DeCross, A. Kar and O. Parrikar,Binding Complexity and Multiparty Entanglement, JHEP 02 (2019) 069 [1811.04085]

2019 arXiv
[35]

Bernamonti, F

A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan and J. Simón,First Law of Holographic Complexity, Phys. Rev. Lett.123 (2019) 081601 [1903.04511]

2019 arXiv
[36]

Balasubramanian, M

V. Balasubramanian, M. Decross, A. Kar and O. Parrikar,Quantum Complexity of Time Evolution with Chaotic Hamiltonians, JHEP 01 (2020) 134 [1905.05765]

2020 arXiv
[38]

R. J. Caginalp and S. Leutheusser,Complexity in One- and Two-Qubit Systems, 2010.15099

2010 arXiv
[39]

Basteiro, J

P. Basteiro, J. Erdmenger, P. Fries, F. Goth, I. Matthaiakakis and R. Meyer,Quantum complexity as hydrodynamics, Phys. Rev. D106 (2022) 065016 [2109.01152]

2022 arXiv
[40]

Balasubramanian, M

V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li and O. Parrikar,Complexity growth in integrable and chaotic models, JHEP 07 (2021) 011 [2101.02209]

2021 arXiv
[41]

Baiguera, S

S. Baiguera, S. Chapman, G. Policastro and T. Schwartzman,The Complexity of Being Entangled, Quantum 8 (2024) 1472 [2311.04277]

2024 arXiv
[42]

Milnor,Curvatures of left invariant metrics on lie groups, Advances in Mathematics21 (1976) 293

J. Milnor,Curvatures of left invariant metrics on lie groups, Advances in Mathematics21 (1976) 293

1976
[43]

O’Neill,The fundamental equations of a submersion., Michigan Mathematical Journal13 (1966) 459

B. O’Neill,The fundamental equations of a submersion., Michigan Mathematical Journal13 (1966) 459

1966
[44]

O’Neill,Submersions and geodesics, Duke Mathematical Journal34 (1967) 363

B. O’Neill,Submersions and geodesics, Duke Mathematical Journal34 (1967) 363

1967
[45]

Vidal,Entanglement Renormalization, Phys

G. Vidal,Entanglement Renormalization, Phys. Rev. Lett.99 (2007) 220405 [cond-mat/0512165]. – 60 –

2007 arXiv
[46]

Vidal,Class of Quantum Many-Body States That Can Be Efficiently Simulated, Phys

G. Vidal,Class of Quantum Many-Body States That Can Be Efficiently Simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099]

2008 arXiv
[47]

Haegeman, T

J. Haegeman, T. J. Osborne, H. Verschelde and F. Verstraete,Entanglement Renormalization for Quantum Fields in Real Space, Phys. Rev. Lett.110 (2013) 100402 [1102.5524]

2013 arXiv
[48]

Nozaki, S

M. Nozaki, S. Ryu and T. Takayanagi,Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [1208.3469]

2012 arXiv
[49]

J. I. Cirac, D. Perez-Garcia, N. Schuch and F. Verstraete,Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys.93 (2021) 045003 [2011.12127]

2021 arXiv
[50]

Couch, S

J. Couch, S. Eccles, T. Jacobson and P. Nguyen,Holographic Complexity and Volume, JHEP 11 (2018) 044 [1807.02186]

2018 arXiv
[51]

Jefferson and R

R. Jefferson and R. C. Myers,Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [1707.08570]

2017 arXiv
[52]

Chapman, M

S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski,Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett.120 (2018) 121602 [1707.08582]

2018 arXiv
[53]

Hashimoto, N

K. Hashimoto, N. Iizuka and S. Sugishita,Time evolution of complexity in Abelian gauge theories, Phys. Rev. D96 (2017) 126001 [1707.03840]

2017 arXiv
[54]

R. Khan, C. Krishnan and S. Sharma,Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [1801.07620]

2018 arXiv
[55]

Hackl and R

L. Hackl and R. C. Myers,Circuit complexity for free fermions, JHEP 07 (2018) 139 [1803.10638]

2018 arXiv
[56]

Chapman, J

S. Chapman, J. Eisert, L. Hackl, M. P. Heller, R. Jefferson, H. Marrochio et al.,Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [1810.05151]

2019 arXiv
[57]

H. A. Camargo, P. Caputa, D. Das, M. P. Heller and R. Jefferson,Complexity as a novel probe of quantum quenches: universal scalings and purifications, Phys. Rev. Lett.122 (2019) 081601 [1807.07075]

2019 arXiv
[58]

M. Guo, J. Hernandez, R. C. Myers and S.-M. Ruan,Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [1807.07677]

2018 arXiv
[59]

Bhattacharyya, A

A. Bhattacharyya, A. Shekar and A. Sinha,Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [1808.03105]

2018 arXiv
[60]

J. M. Magán,Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [1805.05839]

2018 arXiv
[61]

Bueno, J

P. Bueno, J. M. Magan and C. S. Shahbazi,Complexity measures in QFT and constrained geometric actions, JHEP 09 (2021) 200 [1908.03577]

2021 arXiv
[62]

Caceres, S

E. Caceres, S. Chapman, J. D. Couch, J. P. Hernandez, R. C. Myers and S.-M. Ruan, Complexity of Mixed States in QFT and Holography, JHEP 03 (2020) 012 [1909.10557]

2020 arXiv
[63]

Chapman and H

S. Chapman and H. Z. Chen,Charged Complexity and the Thermofield Double State, JHEP 02 (2021) 187 [1910.07508]

2021 arXiv
[64]

Ge and G

D. Ge and G. Policastro,Circuit Complexity and 2D Bosonisation, JHEP 10 (2019) 276 [1904.03003]. – 61 –

2019 arXiv
[65]

Bernamonti, F

A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan and J. Simón,Aspects of The First Law of Complexity, J. Phys. A53 (2020) 29 [2002.05779]

2020 arXiv
[66]

Bernamonti, F

A. Bernamonti, F. Bigazzi, D. Billo, L. Faggi and F. Galli,Holographic and QFT complexity with angular momentum, JHEP 11 (2021) 037 [2108.09281]

2021 arXiv
[67]

Chowdhury, M

S. Chowdhury, M. Bojowald and J. Mielczarek,Geometric quantum complexity of bosonic oscillator systems, JHEP 10 (2024) 048 [2307.13736]

2024 arXiv
[68]

Caputa, N

P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe,Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett.119 (2017) 071602 [1703.00456]

2017 arXiv
[69]

Caputa, N

P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe,Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [1706.07056]

2017 arXiv
[70]

Bhattacharyya, P

A. Bhattacharyya, P. Caputa, S. R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-Integral Complexity for Perturbed CFTs, JHEP 07 (2018) 086 [1804.01999]

2018 arXiv
[71]

Takayanagi,Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [1808.09072]

T. Takayanagi,Holographic Spacetimes as Quantum Circuits of Path-Integrations, JHEP 12 (2018) 048 [1808.09072]

2018 arXiv
[72]

Cotler, M

J. Cotler, M. R. Mohammadi Mozaffar, A. Mollabashi and A. Naseh,Renormalization Group Circuits for Weakly Interacting Continuum Field Theories, Fortsch. Phys. 67 (2019) 1900038 [1806.02831]

2019 arXiv
[73]

H. A. Camargo, M. P. Heller, R. Jefferson and J. Knaute,Path integral optimization as circuit complexity, Phys. Rev. Lett.123 (2019) 011601 [1904.02713]

2019 arXiv
[74]

Boruch, P

J. Boruch, P. Caputa and T. Takayanagi,Path-Integral Optimization from Hartle-Hawking Wave Function, Phys. Rev. D103 (2021) 046017 [2011.08188]

2021 arXiv
[75]

Boruch, P

J. Boruch, P. Caputa, D. Ge and T. Takayanagi,Holographic path-integral optimization, JHEP 07 (2021) 016 [2104.00010]

2021 arXiv
[76]

Caputa and J

P. Caputa and J. M. Magan,Quantum Computation as Gravity, Phys. Rev. Lett.122 (2019) 231302 [1807.04422]

2019 arXiv
[77]

Flory and M

M. Flory and M. P. Heller,Geometry of Complexity in Conformal Field Theory, Phys. Rev. Res. 2 (2020) 043438 [2005.02415]

2020 arXiv
[78]

Flory and M

M. Flory and M. P. Heller,Conformal field theory complexity from Euler-Arnold equations, JHEP 12 (2020) 091 [2007.11555]

2020 arXiv
[79]

Erdmenger, M

J. Erdmenger, M. Gerbershagen and A.-L. Weigel,Complexity measures from geometric actions on Virasoro and Kac-Moody orbits, JHEP 11 (2020) 003 [2004.03619]

2020 arXiv
[80]

Erdmenger, J

J. Erdmenger, J. Kastikainen and T. Schuhmann,Towards complexity of primary-deformed Virasoro circuits, JHEP 03 (2025) 127 [2409.08319]

2025 arXiv
[82]

C. Lv, R. Zhang and Q. Zhou,Building Krylov complexity from circuit complexity, Phys. Rev. Res. 6 (2024) L042001 [2303.07343]. – 62 –

2024 arXiv
[83]

S. E. Aguilar-Gutierrez and A. Rolph,Krylov complexity is not a measure of distance between states or operators, Phys. Rev. D109 (2024) L081701 [2311.04093]

2024 arXiv
[84]

Erdmenger, M

J. Erdmenger, M. Flory, M. Gerbershagen, M. P. Heller and A.-L. Weigel,Exact Gravity Duals for Simple Quantum Circuits, SciPost Phys. 13 (2022) 061 [2112.12158]

2022 arXiv
[85]

Erdmenger, A.-L

J. Erdmenger, A.-L. Weigel, M. Gerbershagen and M. P. Heller,From complexity geometry to holographic spacetime, Phys. Rev. D108 (2023) 106020 [2212.00043]

2023 arXiv
[86]

de Boer, V

J. de Boer, V. Godet, J. Kastikainen and E. Keski-Vakkuri,Quantum information geometry of driven CFTs, JHEP 09 (2023) 087 [2306.00099]

2023 arXiv
[87]

Craps, M

B. Craps, M. D. Clerck, O. Evnin and P. Hacker,Integrability and complexity in quantum spin chains, SciPost Phys. 16 (2024) 041

2024
[88]

Craps, M

B. Craps, M. De Clerck, O. Evnin, P. Hacker and M. Pavlov,Bounds on quantum evolution complexity via lattice cryptography, SciPost Phys. 13 (2022) 090 [2202.13924]

2022 arXiv
[89]

Castellani, R

L. Castellani, R. D’Auria and P. Fre,Supergravity and superstrings: A Geometric perspective. Vol. 1: Mathematical foundations, 1991

1991
[90]

Gilmore,Lie Groups, Lie Algebras, and Some of Their Applications, Dover Books on Mathematics

R. Gilmore,Lie Groups, Lie Algebras, and Some of Their Applications, Dover Books on Mathematics. Dover Publications, 2012

2012
[91]

Alekseev and S

A. Alekseev and S. Shatashvili,Path integral quantization of the coadjoint orbits of the virasoro group and 2-d gravity, Nuclear Physics B323 (1989) 719

1989
[92]

Alekseev and S

A. Alekseev and S. L. Shatashvili,Coadjoint Orbits, Cocycles and Gravitational Wess–Zumino, 1801.07963

arXiv
[93]

de Alfaro, S

V. de Alfaro, S. Fubini and G. Furlan,Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569

1976
[94]

J. M. Maldacena, J. Michelson and A. Strominger,Anti-de Sitter fragmentation, JHEP 02 (1999) 011 [hep-th/9812073]

1999 arXiv
[95]

Jensen, S

K. Jensen, S. Kachru, A. Karch, J. Polchinski and E. Silverstein,Towards a holographic marginal Fermi liquid, Phys. Rev. D84 (2011) 126002 [1105.1772]

2011 arXiv
[96]

Jensen,Chaos in AdS2 Holography, Phys

K. Jensen,Chaos in AdS2 Holography, Phys. Rev. Lett.117 (2016) 111601 [1605.06098]

2016 arXiv
[97]

Minwalla,Restrictions imposed by superconformal invariance on quantum field theories, Adv

S. Minwalla,Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys.2 (1998) 783 [hep-th/9712074]

1998 arXiv
[98]

A. M. Perelomov,Generalized coherent states and some of their applications, Soviet Physics Uspekhi 20 (1977) 703

1977
[99]

A. R. Chandra, J. de Boer, M. Flory, M. P. Heller, S. Hörtner and A. Rolph,Spacetime as a quantum circuit, JHEP 21 (2021) 207 [2101.01185]

2021 arXiv
[100]

A. R. Chandra, J. de Boer, M. Flory, M. P. Heller, S. Hörtner and A. Rolph,Cost of holographic path integrals, SciPost Phys. 14 (2023) 061 [2203.08842]

2023 arXiv
[101]

R. V. Meter, W. J. Munro, K. Nemoto and K. M. Itoh,Arithmetic on a distributed-memory quantum multicomputer, J. Emerg. Technol. Comput. Syst.3 (2008) . – 63 –

2008
[102]

Beals, S

R. Beals, S. Brierley, O. Gray, A. W. Harrow, S. Kutin, N. Linden et al.,Efficient distributed quantum computing, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469 (2013) 20120686

2013
[103]

Caleffi, M

M. Caleffi, M. Amoretti, D. Ferrari, D. Cuomo, J. Illiano, A. Manzalini et al.,Distributed quantum computing: A survey, Comput. Net. 254 (2024) 110672 [2212.10609]

2024 arXiv
[104]

Parzanchevski and P

O. Parzanchevski and P. Sarnak,Super-golden-gates for pu(2), Advances in Mathematics327 (2018) 869

2018
[105]

Letter to Aaronson and Pollington on the Solvay-Kitaev theorem and golden gates, 2015

P. Sarnak, “Letter to Aaronson and Pollington on the Solvay-Kitaev theorem and golden gates, 2015.”

2015
[106]

Petersen,Riemannian Geometry, Graduate Texts in Mathematics

P. Petersen,Riemannian Geometry, Graduate Texts in Mathematics. Springer New York, 2006

2006
[107]

A. L. Besse,Einstein Manifolds. Springer-Verlag, Berlin, Heidelberg, New York, 1987

1987
[108]

O’Neill,Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics)

B. O’Neill,Semi-Riemannian Geometry With Applications to Relativity, 103, Volume 103 (Pure and Applied Mathematics). Academic Press, 1983

1983
[109]

Luscher and G

M. Luscher and G. Mack,Global Conformal Invariance in Quantum Field Theory, Commun. Math. Phys. 41 (1975) 203

1975
[110]

Group theory

A. Tomasiello, “Group theory.”
[111]

M. E. Peskin and D. V. Schroeder,An Introduction to quantum field theory. Addison-Wesley, Reading, USA, 1995, 10.1201/9780429503559

1995 doi
[112]

Dorn and G

H. Dorn and G. Jorjadze,On particle dynamics in AdS(N+1) space-time, Fortsch. Phys. 53 (2005) 486 [hep-th/0502081]. – 64 –

2005 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [34]

A. R. Brown and L. Susskind,Complexity geometry of a single qubit, Phys. Rev. D100 (2019) 046020 [1903.12621]

arXiv 2019

[2] [37]

Auzzi, S

R. Auzzi, S. Baiguera, G. B. De Luca, A. Legramandi, G. Nardelli and N. Zenoni,Geometry of quantum complexity, Phys. Rev. D103 (2021) 106021 [2011.07601]

arXiv 2021

[3] [81]

Chagnet, S

N. Chagnet, S. Chapman, J. de Boer and C. Zukowski,Complexity for Conformal Field Theories in General Dimensions, Phys. Rev. Lett.128 (2022) 051601 [2103.06920]

arXiv 2022

[4] [1]

Susskind,Three Lectures on Complexity and Black Holes, 10, 2018

L. Susskind,Three Lectures on Complexity and Black Holes, 10, 2018. 10.1007/978-3-030-45109-7

doi:10.1007/978-3-030-45109-7 2018

[5] [2]

Chapman and G

S. Chapman and G. Policastro,Quantum computational complexity from quantum information to black holes and back, Eur. Phys. J. C82 (2022) 128 [2110.14672]

arXiv 2022

[6] [3]

Baiguera, V

S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller et al., Quantum complexity in gravity, quantum field theory, and quantum information science, 2503.10753

arXiv

[7] [4]

Harlow and P

D. Harlow and P. Hayden,Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [1301.4504]. – 58 –

arXiv 2013

[8] [5]

I. H. Kim, E. Tang and J. Preskill,The ghost in the radiation: robust encodings of the black hole interior (invited paper), JHEP 06 (2020) 031 [2003.05451]

arXiv 2020