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Reflections on Virasoro circuit complexity and Berry phase

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In the large central charge limit, the paper claims that Virasoro circuit complexity is minus the integrated Berry connection, and derives from this a log-overlap formula for complexity in two-dimensional CFTs.

desk verdict The central equality (47) has the wrong sign, and the paper's main claims fall with it. read the letter →

arxiv 1908.08514 v1 pith:MOFSBFEC submitted 2019-08-22 hep-th quant-ph

Ibrahim Akal This is my paper · ORCID
classification hep-thquant-ph
keywords circuitcomplexityBerryphaseVirasorogrouptwo-dimensionalconformalfieldtheorylargecentralchargecoadjointorbitsgeometricactionLiouville
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

This paper aims to establish that, in two-dimensional conformal field theories, the computational cost of a circuit built from Virasoro symmetry gates is the negative of the integrated Berry connection of the Virasoro group's unitary representation, at least in the large central charge limit. From that identification it derives a relation between the Virasoro Berry phase, the circuit complexity, and the state overlap: the Berry phase equals minus the complexity minus $|\log\langle\psi_R|\psi_T\rangle|$, so that the complexity is proportional to $-\log|\langle\psi_R|\psi_T\rangle|^2$. A sympathetic reader would care because this turns a logarithmic, state-dependent complexity formula that had previously been proposed on heuristic grounds into a corollary of a concrete circuit construction, and it connects complexity with geometric actions and the Liouville action that appear in holography.

What carries the argument

The carrying object is the Virasoro group manifold, i.e. the centrally extended group of orientation-preserving circle diffeomorphisms, used as the space of circuits: a protocol is a path $g(s,\sigma)$ in the group, and the complexity is the integrated cost functional (24), $C_{h,c}[g](\tau) = \frac{c}{24\pi}\int_0^\tau ds \int_0^{2\pi} d\sigma\, \frac{\dot{g}}{g'}\left(\frac12 - \frac{12h}{c} + \mathrm{Sch}[g,\sigma]\right)$, which coincides with the Kirillov geometric action on coadjoint orbits. The Berry connection is $A_{h,c} = i\langle h | u[\hat{\Theta}_g] | h\rangle$, the expectation value of the centrally extended Maurer-Cartan form in a highest-weight representation built on a primary state of weight $h$ and central charge $c$; it splits into a centreless term (45) and a Bott-cocycle term, i.e. the two-cocycle that centrally extends the circle-diffeomorphism group (46), and the load-bearing step is the claim that their sum equals the negative of the cost integrand. The generalized Berry phase for open paths adds a boundary contribution from the stabilizer of the highest-weight state, which is what converts the endpoint term into $|\log\langle\psi_R|\psi_T\rangle|$.

What would settle it

Evaluate both sides of Eq. (47) for the rotation protocol $g(s,\sigma)=\sigma+s\theta$ with the SL(2,$\mathbb{R}$)-invariant vacuum as reference state: Eq. (45) contributes $+c\theta/24$, while $-C_{0,c}$ from Eq. (24) contributes $-c\theta/24$. This direct mismatch for the simplest allowed protocol is a concrete falsifier of the claimed identity.

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Extended reading notes

Core claim

The paper's central claim is Eq. (47): in the large central charge limit, $\int_g A_{h,c} = - C_{h,c}[g](\tau)$, where the left side is the integral of the Virasoro Berry connection along the protocol path in the group manifold and the right side is the Virasoro circuit complexity for the same path. Using the generalized Berry phase for open paths whose endpoints differ by a rotation, the author rewrites the full Berry phase as $B_{h,c}[g](\tau) = -C_{h,c}[g](\tau) - |\log\langle\psi_R|\psi_T\rangle|$, Eq. (50). Combining left and right chiral sectors gives the proportionality $C_{\mathrm{CFT}} \propto -\log|\langle\psi_R|\psi_T\rangle|^2$, Eq. (53). The paper frames this as showing that the Berry phase of a primary state transported by Virasoro transformations is, up to an endpoint term fixed by the reference-target overlap, just the negative of the symmetry-circuit complexity.

Load-bearing premise

The derivation stands on the sign-matching of Eqs. (45) plus (46) to the negative of the expression for the circuit cost, and this matching already fails for the simplest allowed protocol: rotating the vacuum state gives $+c\theta/24$ on the Berry side and $-c\theta/24$ on the complexity side.

Editorial extensions

If this is right

  • In large-$c$ 2D CFTs, circuit complexity built from Virasoro symmetry gates becomes computable as a group-theoretic holonomy, since it is (minus) the integrated Berry connection along the protocol path.
  • The Virasoro Berry phase is fixed by Eq. (50) to be the negative complexity plus a path-independent endpoint term, so measuring the Berry phase is equivalent to measuring the complexity up to a constant.
  • The logarithmic complexity formula $C_{\mathrm{CFT}}\propto-\log|\langle\psi_R|\psi_T\rangle|^2$ follows from a concrete circuit construction with symmetry gates, rather than being a standalone state-distance proposal.
  • The same relations suggest a proportionality between the Virasoro Berry phase and the classical Liouville action, connecting the Berry-phase formalism to the path-integral optimization measure of complexity.
  • Because the derivation is for continuous CFT systems, the paper predicts the logarithmic formula is not a valid complexity measure for discrete qubit systems; applying it there yields infinite complexity for a single-qubit flip.

Reading between the lines

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

  • Beyond the paper, if Eq. (50) survives, a state-overlap complexity measure must secretly know the gate set: it should be trusted only when the underlying circuit is made of Virasoro symmetry gates, not as a universal state distance.
  • Beyond the paper, the identification of complexity with $-\int A_{h,c}$ suggests complexity inherits gauge freedom from the Berry connection; rephasing the reference or target states should shift the endpoint term, a concrete prediction that can be checked within the same group-manifold formalism.
  • Beyond the paper, a natural testable extension is to take non-rotation paths, such as special conformal transformations, and check whether the endpoint term in Eq. (50) remains path-independent; if it varies, the formula holds only for rotation protocols.
  • Beyond the paper, one could compare the time evolution of $-\log|\langle\psi_R|\psi_T\rangle|^2$ after a global quench with known holographic complexity growth; agreement would suggest that holographic complexity probes Virasoro Berry phases.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper investigates a relation between circuit complexity defined on Virasoro group manifolds and Berry phases in unitary representations of the Virasoro group. After reviewing the geometric complexity framework of Caputa and Magan [61] and the group-theoretic Berry phase construction of Oblak [75], the paper proposes that in the large central charge limit the Virasoro computational cost coincides with the negative of the Virasoro Berry connection (Eq. (47)). From this it derives that the generalized Berry phase equals minus the complexity plus an endpoint term (Eq. (48)), and then, for rotation-type protocols, a proportionality between full CFT complexity and -log|overlap|^2 (Eqs. (50), (52), (53)). The final discussion suggests a connection between the Berry phase and the Liouville action in the path-integral optimization proposal.

Significance. If valid, the identification would link three independent notions—circuit complexity, Berry phase, and state overlap—in two-dimensional CFTs, and would provide a symmetry-circuit origin for logarithmic complexity formulas that mimic holographic and path-integral complexity predictions. The paper has positive features: it uses no free parameters, it builds on explicit formulas from Ref. [61] and Ref. [75], and the final proportionalities are concrete and falsifiable. The central identification, however, fails a direct check using the paper's own equations, and the subsequent derivations do not survive that failure.

major comments (3)
  1. [V.B, Eq. (47)] The claimed equality ∫_g A_{h,c} = -C_{h,c}[g](τ) is already false for the simplest admissible protocol. Take the rigid rotation path g(s,σ) = σ + (s/τ)θ with reference state |h>. Then g' = 1, g'' = 0, and Sch[g,σ] = 0. Eq. (45) gives ∫_g A_{h,c} = θ(c/24 - h) (the second piece, Eq. (46), vanishes), while Eq. (24) gives C_{h,c}[g] = θ(c/24 - h). The paper's own formulas therefore give ∫_g A_{h,c} = +C_{h,c}[g], not -C_{h,c}[g]. The relative sign is independent of c, so the large-c limit cannot repair the identification. Since Eqs. (48), (50), (52), and (53) all inherit this sign, the central claim is unsupported.
  2. [V.A/V.B, Eqs. (43) and (50)] The sign error is not merely aesthetic; it contradicts the paper's own vanishing-holonomy statement. For the same rotation path, Eq. (49) gives U_θ|h> = exp(iθ(h - c/24))|h>, so Eq. (43) yields ∫_g A_{h,c} - i log<h|U[...]|h> = θ(c/24 - h) + θ(h - c/24) = 0. With Eq. (47) instead the right-hand side becomes -2θ(c/24 - h) ≠ 0 (for h ≠ c/24), in conflict with the statement in Sec. IV.B that the generalized Berry phase vanishes for any path of the form g(s) = g(0)h(s) with h(s) ∈ G_φ. The rotation protocol is exactly such a path.
  3. [V.C, Eq. (50)] The replacement of -i log<ψ_R|ψ_T> by -|log<ψ_R|ψ_T>| is not an algebraic identity and is false for the rotation protocol. For the state in Eq. (49), |<ψ_R|ψ_T>| = 1, so -|log<ψ_R|ψ_T>| = 0 if the logarithm of the modulus is meant, whereas the boundary term in Eq. (48) equals θ(h - c/24); if the absolute value of the complex logarithm is meant, the sign and magnitude generally do not match. Thus Eqs. (52) and (53) are not derived consequences of Eq. (48).
minor comments (3)
  1. [III.A, Eq. (17)] The Schwarzian is written as g'''/g' - (√3 g'')^2/(√2 g')^2; this is correct but needlessly opaque, and writing (3/2)(g''/g')^2 would improve readability.
  2. [IV.B] The text contains the typo 'Mauer-Cartan'; it should read 'Maurer-Cartan' in both occurrences.
  3. [I, near Eq. (27)] In the Introduction, 'under a diabatic variations' should presumably be 'under adiabatic variations'; the terminology is otherwise standard throughout the paper.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantial circularity: the central claims are algebraic consequences of externally cited formulas, not fitted or self-referential reductions.

full rationale

The paper's central identification (Eq. 47) compares two previously derived expressions: the Virasoro complexity (Eq. 24, taken from Caputa–Magan [61]) and the Berry-connection integrals (Eqs. 45–46, taken from Oblak [75]). No free parameter is fitted, and the final log-overlap relation (Eqs. 50, 52, 53) follows algebraically from that identification together with the Virasoro representation property (Eq. 49). The paper does not use the desired conclusion to fix any constant or to select a cost function; the cost function and Berry connection are independent inputs from earlier work. The only self-citation, [39], is used to note an equivalence between different cost functions and is not load-bearing. Even if Eq. (47) suffers a sign problem on rotation paths, as the reader's consistency check suggests, that would be a physical or mathematical correctness defect, not circularity, because the paper's logic compares rather than fits its inputs. The paper also explicitly labels Eq. (53) as extrapolated from Eq. (52), which further shows the log-overlap formula is treated as a consequence rather than an assumed premise. Therefore no specific circular step can be quoted; the score reflects only the presence of a minor, non-load-bearing self-citation.

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

No free parameters are fitted; c, h, and θ are inputs. The main burden sits on an unproved matching of two cited expressions, which is internally inconsistent. No new physical entities are postulated.

assumptions (5)
  • domain assumption Large central charge limit makes the one-norm and two-norm cost functions equivalent, F2 ≈ F1 ≡ F.
    Invoked in Sec. III.B to fix the cost function (23); valid only when c is large compared to the other scales and relevant for holographic CFTs.
  • domain assumption The reference state is a Virasoro primary |h⟩ with an SL(2,R) invariant vacuum.
    Used throughout Secs. III and V to evaluate cost and Berry connection; the whole derivation is restricted to highest weight representations.
  • ad hoc to paper The protocol path satisfies g^{-1}(0)∘g(τ) = rotation by θ(τ), so the projection to Diff(S^1)/S^1 closes.
    Introduced in Secs. V.A and V.C to evaluate the boundary term; restricts the class of circuits to those whose target differs from the reference by a phase.
  • ad hoc to paper Central terms and Bott cocycle contributions in the Virasoro group product do not contribute to the computational cost.
    Stated in Sec. III.B; needed to assign zero complexity to the identity and to compare with the Berry connection. Not derived from the circuit model.
  • standard math The generalized Berry phase formula (40) applies to open paths with a boundary term canceling the real phase.
    Standard result for group-theoretic Berry phases, reviewed in Sec. IV.B and used to write Eq. (43).

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Cite this review

Pith. "Pith review of Reflections on Virasoro circuit complexity and Berry phase." pith.science (2026). https://pith.science/paper/MOFSBFEC

@misc{pith2026190808514,
  author       = {Pith},
  title        = {Pith review of: Reflections on Virasoro circuit complexity and Berry phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MOFSBFEC}},
  note         = {Machine review of arXiv:1908.08514}
}
read the original abstract

Recently, the notion of circuit complexity defined in symmetry group manifolds has been related to geometric actions which generally arise in the coadjoint orbit method in representation theory and play an important role in geometric quantization. On the other hand, it is known that there exists a precise relation between geometric actions and Berry phases defined in group representations. Motivated by these connections, we elaborate on a relation between circuit complexity and the group theoretic Berry phase. As the simplest setup relevant for holography, we discuss the case of two dimensional conformal field theories. In the large central charge limit, we identify the computational cost function with the Berry connection in the unitary representation of the Virasoro group. We then use the latter identification to express the Berry phase in terms of the Virasoro circuit complexity. The former can be seen as the holonomy of the Berry connection along the path in the group manifold which defines the protocol. In addition, we derive a proportionality relation between Virasoro circuit complexity and the logarithm of the inner product between a particularly chosen reference state and the prepared target state. In this sense, the logarithmic formula turns out to be approximating the complexity up to some additive constant if the building blocks of the circuit are taken to be the underlying symmetry gates. Predictions based on this formula have recently been shown to coincide with the holographic complexity proposals and the path integral optimization procedure. The found connections may therefore help to better understand such coincidences. We also discuss that our findings, put together with earlier observations, may suggest a connection between the Virasoro Berry phase and the complexity measure in the path integral optimization proposal.

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Reference graph

Works this paper leans on

90 extracted references · 16 canonical work pages · cited by 1 Pith paper

  1. [61]

    Apart from th e complexity approaches described above, alternative field theoretic developm ents have been introduced in [ 62, 63]

    which makes explicit use of the symmetries in the CFT. Apart from th e complexity approaches described above, alternative field theoretic developm ents have been introduced in [ 62, 63]. In the present paper, we focus on the circuit complexity approach in the particular case of two dimensional CFTs. Instead of considering geometries in the spa ce of unitar...

  2. [75]

    T. F. Jordan, Berry phases and unitary transformations , Journal of mathematical physics 29 (1988), no. 9 2042–2052

    work page 1988

  3. [1]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergr avity, Int. J. Theor. Phys. 38 (1999) 1113–1133, [ hep-th/9711200]. [Adv. Theor. Math. Phys.2,231(1998)]

    arXiv 1999

  4. [2]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B428 (1998) 105–114, [ hep-th/9802109]

    arXiv 1998

  5. [3]

    gates at earlier times are applied first, see for instance the string of left oriented g ates in ( 2)

    is chosen to be left oriented, i.e. gates at earlier times are applied first, see for instance the string of left oriented g ates in ( 2). The circuit complexity of the unitary, C(U), can be defined by introducing an inner product for the corresponding algebra. Let {iOI} be a basis for the latter, then the inner product may be defined such that it leads to a...

  6. [4]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CF T, Phys. Rev. Lett. 96 (2006) 181602, [ hep-th/0603001]

    arXiv 2006

  7. [5]

    Ryu and T

    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy , JHEP 08 (2006) 045, [ hep-th/0605073]

    arXiv 2006

  8. [6]

    V. E. Hubeny, M. Rangamani, and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062, [ arXiv:0705.0016]

    arXiv 2007

Show all 90 references
  1. [7]

    Van Raamsdonk, Building up spacetime with quantum entanglement , Gen

    M. Van Raamsdonk, Building up spacetime with quantum entanglement , Gen. Rel. Grav. 42 (2010) 2323–2329, [ arXiv:1005.3035]. [Int. J. Mod. Phys.D19,2429(2010)]

    2010 arXiv

  2. [8]

    Lashkari, M

    N. Lashkari, M. B. McDermott, and M. Van Raamsdonk, Gravitational dynamics from entanglement ’thermodynamics’, JHEP 04 (2014) 195, [ arXiv:1308.3716]. 25

    2014 arXiv

  3. [9]

    can be translated into an analogous group equation of the form [ 61] g(τ +dτ ) = gQg·g(τ ), g Qg≡eQg(τ)dτ (10) where for two elements g1,g 2∈ G the product g1·g2 corresponds to the group product. The group theoretic infinitesimal gate gQg in the homologous equation ( 10) depend...

  4. [10]

    Witten, Anti-de Sitter space and holography , Adv

    E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2 (1998) 253–291, [hep-th/9802150]

    1998 arXiv

  5. [11]

    Casini, M

    H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036, [ arXiv:1102.0440]

    2011 arXiv

  6. [12]

    X. Dong, A. Lewkowycz, and M. Rangamani, Deriving covariant holographic entanglement , JHEP 11 (2016) 028, [ arXiv:1607.07506]

    2016 arXiv

  7. [13]

    Hartman and J

    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014, [ arXiv:1303.1080]

    2013 arXiv

  8. [14]

    Susskind, Entanglement is not enough , Fortsch

    L. Susskind, Entanglement is not enough , Fortsch. Phys. 64 (2016) 49–71, [arXiv:1411.0690]

    2016 arXiv

  9. [15]

    Faulkner, M

    T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van R aamsdonk, Gravitation from Entanglement in Holographic CFTs , JHEP 03 (2014) 051, [ arXiv:1312.7856]

    2014 arXiv

  10. [16]

    generalizes to Ad(g,α )(X,β ) = ( Adg(X),β− 1 24π ∫ 2π 0 dσ Sch[g,σ ]X(σ) ) . (17) As expected, the first entry in ( 17) corresponds to the centerless case in ( 16), whereas the second one is the additional entry due to the presence of the Bott cocycle C and depends on the usua...

  11. [17]

    Rangamani and T

    M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy , Lect. Notes Phys. 931 (2017) pp.1–246, [ arXiv:1609.01287]

    2017 arXiv

  12. [18]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y . Zhao, Complexity, action, and black holes , Phys. Rev. D93 (2016), no. 8 086006, [ arXiv:1512.04993]

    2016 arXiv

  13. [19]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y . Zhao, Holographic Complexity Equals Bulk Action? , Phys. Rev. Lett. 116 (2016), no. 19 191301, [ arXiv:1509.07876]

    2016 arXiv

  14. [20]

    Couch, W

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

    2017 arXiv

  15. [21]

    Susskind, Computational Complexity and Black Hole Horizons , Fortsch

    L. Susskind, Computational Complexity and Black Hole Horizons , Fortsch. Phys. 64 (2016) 44–48, [ arXiv:1403.5695]. [Fortsch. Phys.64,24(2016)]

    2016 arXiv

  16. [22]

    Stanford and L

    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries , Phys. Rev. D90 (2014), no. 12 126007, [ arXiv:1406.2678]

    2014 arXiv

  17. [23]

    D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, JHEP 03 (2015) 051, [arXiv:1409.8180]

    2015 arXiv

  18. [24]

    M. A. Nielsen, A geometric approach to quantum circuit lower bounds , arXiv:quant-ph/0502070 (2005)

    2005 arXiv

  19. [25]

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311 (2006), no. 5764 1133–1135

    2006

  20. [26]

    M. R. Dowling and M. A. Nielsen, The geometry of quantum computation , Quantum Information & Computation 8 (2008), no. 10 861–899

    2008

  21. [27]

    Almheiri and J

    A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014, [ arXiv:1402.6334]

    2015 arXiv

  22. [28]

    Papadodimas and S

    K. Papadodimas and S. Raju, Black Hole Interior in the Holographic Correspondence and the Information Paradox , Phys. Rev. Lett. 112 (2014), no. 5 051301, [ arXiv:1310.6334]

    2014 arXiv

  23. [29]

    Papadodimas and S

    K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D89 (2014), no. 8 086010, [ arXiv:1310.6335]. 26

    2014 arXiv

  24. [30]

    Bhattacharyya, A

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

    2018 arXiv

  25. [31]

    D. W. F. Alves and G. Camilo, Evolution of complexity following a quantum quench in free field theory , JHEP 06 (2018) 029, [ arXiv:1804.00107]

    2018 arXiv

  26. [32]

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

    2018 arXiv

  27. [33]

    Jefferson and R

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

    2017 arXiv

  28. [34]

    R. Khan, C. Krishnan, and S. Sharma, Circuit Complexity in Fermionic Field Theory , Phys. Rev. D98 (2018), no. 12 126001, [ arXiv:1801.07620]

    2018 arXiv

  29. [35]

    Hackl and R

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

    2018 arXiv

  30. [36]

    Jiang and X

    J. Jiang and X. Liu, Circuit Complexity for Fermionic Thermofield Double states , Phys. Rev. D99 (2019), no. 2 026011, [ arXiv:1812.00193]

    2019 arXiv

  31. [37]

    Sinamuli and R

    M. Sinamuli and R. B. Mann, Holographic Complexity and Charged Scalar Fields , arXiv:1902.01912

    1902 arXiv

  32. [38]

    F. Liu, R. Lundgren, P. Titum, J. R. Garrison, and A. V. Go rshkov, Circuit Complexity across a Topological Phase Transition , arXiv:1902.10720. 27

    1902 arXiv

  33. [39]

    Chapman, J

    S. Chapman, J. Eisert, L. Hackl, M. P. Heller, R. Jefferson , H. Marrochio, and R. C. Myers, Complexity and entanglement for thermofield double states , arXiv:1810.05151

    arXiv

  34. [40]

    T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim, a nd N. Moynihan, Time Evolution of Complexity: A Critique of Three Methods , arXiv:1810.02734

    arXiv

  35. [41]

    T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim, a nd N. Moynihan, Post-Quench Evolution of Distance and Uncertainty in a Topological Syst em: Complexity, Entanglement and Revivals , arXiv:1811.05985

    arXiv

  36. [42]

    Nozaki, S

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

    2012 arXiv

  37. [43]

    Mollabashi, M

    A. Mollabashi, M. Nozaki, S. Ryu, and T. Takayanagi, Holographic Geometry of cMERA for Quantum Quenches and Finite Temperature , JHEP 03 (2014) 098, [ arXiv:1311.6095]

    2014 arXiv

  38. [44]

    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), no. 12 121602, [arXiv:1707.08582]

    2018 arXiv

  39. [45]

    Akal, Weighting gates in circuit complexity and holography , arXiv:1903.06156

    I. Akal, Weighting gates in circuit complexity and holography , arXiv:1903.06156

    1903 arXiv

  40. [46]

    Bhattacharyya, P

    A. Bhattacharyya, P. Nandy, and A. Sinha, Renormalized Circuit Complexity , arXiv:1907.08223

    1907 arXiv

  41. [47]

    Haegeman, T

    J. Haegeman, T. J. Osborne, H. Verschelde, and F. Verstr aete, Entanglement Renormalization for Quantum Fields in Real Space , Phys. Rev. Lett. 110 (2013), no. 10 100402, [ arXiv:1102.5524]

    2013 arXiv

  42. [48]

    Indeed, as we will see, this gives rise to a simple formula which explicitly depends on the overlap between reference state and the target state

    even further by working out a more explicit form for the logarithmic boundary term. Indeed, as we will see, this gives rise to a simple formula which explicitly depends on the overlap between reference state and the target state. In order to arrive at the latter stage, recall ...

    2019

  43. [49]

    T. Ali, A. Bhattacharyya, S. S. Haque, E. H. Kim, N. Moyni han, and J. Murugan, Chaos and Complexity in Quantum Mechanics , arXiv:1905.13534

    1905 arXiv

  44. [50]

    A. R. Brown and L. Susskind, Second law of quantum complexity , Phys. Rev. D97 (2018), no. 8 086015, [ arXiv:1701.01107]

    2018 arXiv

  45. [51]

    H. A. Camargo, P. Caputa, D. Das, M. P. Heller, and R. Jeffer son, Complexity as a novel probe of quantum quenches: universal scalings and purificat ions, Phys. Rev. Lett. 122 (2019), no. 8 081601, [ arXiv:1807.07075]

    2019 arXiv

  46. [52]

    Hashimoto, N

    K. Hashimoto, N. Iizuka, and S. Sugishita, Time evolution of complexity in Abelian gauge theories, Phys. Rev. D96 (2017), no. 12 126001, [ arXiv:1707.03840]

    2017 arXiv

  47. [53]

    Cotler, M

    J. Cotler, M. R. Mohammadi Mozaffar, A. Mollabashi, and A. Naseh, Renormalization Group Circuits for Weakly Interacting Continuum Field Theories , arXiv:1806.02831

    arXiv

  48. [54]

    Balasubramanian, M

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

    2019 arXiv

  49. [55]

    Caputa, N

    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, and K. Wat anabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theorie s, Phys. Rev. Lett. 119 (2017), no. 7 071602, [ arXiv:1703.00456]

    2017 arXiv

  50. [56]

    Caputa, N

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

    2017 arXiv

  51. [57]

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

    2018 arXiv

  52. [58]

    Yang and K.-Y

    R.-Q. Yang and K.-Y. Kim, Time evolution of the complexity in chaotic systems: concret e examples, arXiv:1906.02052. 28

    1906 arXiv

  53. [59]

    Vidal, Class of quantum many-body states that can be efficiently simul ated, Physical review letters 101 (2008), no

    G. Vidal, Class of quantum many-body states that can be efficiently simul ated, Physical review letters 101 (2008), no. 11 110501

    2008

  54. [60]

    Miyaji, T

    M. Miyaji, T. Takayanagi, and K. Watanabe, From path integrals to tensor networks for the AdS/CFT correspondence, Phys. Rev. D95 (2017), no. 6 066004, [ arXiv:1609.04645]

    2017 arXiv

  55. [62]

    Yang, Complexity for quantum field theory states and applications t o thermofield double states , Phys

    R.-Q. Yang, Complexity for quantum field theory states and applications t o thermofield double states , Phys. Rev. D97 (2018), no. 6 066004, [ arXiv:1709.00921]

    2018 arXiv

  56. [63]

    Czech, Einstein Equations from Varying Complexity , Phys

    B. Czech, Einstein Equations from Varying Complexity , Phys. Rev. Lett. 120 (2018), no. 3 031601, [ arXiv:1706.00965]

    2018 arXiv

  57. [64]

    H. A. Camargo, M. P. Heller, R. Jefferson, and J. Knaute, Path integral optimization as circuit complexity, Phys. Rev. Lett. 123 (2019), no. 1 011601, [ arXiv:1904.02713]

    2019 arXiv

  58. [65]

    Milsted and G

    A. Milsted and G. Vidal, Tensor networks as path integral geometry , arXiv:1807.02501

    arXiv

  59. [66]

    These relations ou tlined above therefore suggest a connection between optimal quantum computation in two dimensional CFTs and gravity [61]

    on the coadjoint orbits of the Virasoro group. These relations ou tlined above therefore suggest a connection between optimal quantum computation in two dimensional CFTs and gravity [61]. IV. BERR Y PHASE IN GROUP REPRESENT A TIONS A. General aspects In this section, we introd...

  60. [67]

    Milsted and G

    A. Milsted and G. Vidal, Geometric interpretation of the multi-scale entanglement renormalization ansatz, arXiv:1812.00529

    arXiv

  61. [68]

    Caputa and J

    P. Caputa and J. M. Magan, Quantum Computation as Gravity , Phys. Rev. Lett. 122 (2019), no. 23 231302, [ arXiv:1807.04422]

    2019 arXiv

  62. [69]

    Belin, A

    A. Belin, A. Lewkowycz, and G. Srosi, Complexity and the bulk volume, a new York time story, JHEP 03 (2019) 044, [ arXiv:1811.03097]

    2019 arXiv

  63. [70]

    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 B 323 (1989), no. 3 719–733

    1989

  64. [71]

    Alekseev and S

    A. Alekseev and S. Shatashvili, From geometric quantization to conformal field theory , Communications in mathematical physics 128 (1990), no. 1 197–212

    1990

  65. [72]

    A. A. Kirillov, Lectures on the orbit method , vol. 64. American Mathematical Soc., 2004

    2004

  66. [73]

    Kostant, Quantization and unitary representations , in Lectures in modern analysis and applications III , pp

    B. Kostant, Quantization and unitary representations , in Lectures in modern analysis and applications III , pp. 87–208. Springer, 1970. 29

    1970

  67. [74]

    Witten, Coadjoint orbits of the virasoro group , Comm

    E. Witten, Coadjoint orbits of the virasoro group , Comm. Math. Phys. 114 (1988), no. 1 1–53

    1988

  68. [76]

    Vinet, Invariant berry connections , Physical Review D 37 (1988), no

    L. Vinet, Invariant berry connections , Physical Review D 37 (1988), no. 8 2369

    1988

  69. [77]

    Strahov, Berrys phase for compact lie groups , Journal of Mathematical Physics 42 (2001), no

    E. Strahov, Berrys phase for compact lie groups , Journal of Mathematical Physics 42 (2001), no. 5 2008–2022

    2001

  70. [78]

    M. V. Berry, Quantal phase factors accompanying adiabatic changes , Proceedings of the Royal Society of London. A. Mathematical and Physical Scien ces 392 (1984), no. 1802 45–57

    1984

  71. [79]

    Baggio, V

    M. Baggio, V. Niarchos, and K. Papadodimas, Aspects of Berry phase in QFT , JHEP 04 (2017) 062, [ arXiv:1701.05587]

    2017 arXiv

  72. [80]

    Czech, L

    B. Czech, L. Lamprou, S. Mccandlish, and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT , Phys. Rev. Lett. 120 (2018), no. 9 091601, [ arXiv:1712.07123]

    2018 arXiv

  73. [81]

    Oblak, Berry Phases on Virasoro Orbits , JHEP 10 (2017) 114, [ arXiv:1703.06142]

    B. Oblak, Berry Phases on Virasoro Orbits , JHEP 10 (2017) 114, [ arXiv:1703.06142]

    2017 arXiv

  74. [82]

    Yang, Y.-S

    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang, and K.-Y. Kim, To be unitary-invariant or not?: a simple but non-trivial proposal for the complexity betwee n states in quantum mechanics/field theory , arXiv:1906.02063

    1906 arXiv

  75. [83]

    A. R. Brown and L. Susskind, The Complexity Geometry of a Single Qubit , arXiv:1903.12621

    1903 arXiv

  76. [84]

    Bueno, J

    P. Bueno, J. M. Magan, and C. S. Shahbazi, Complexity measures in QFT and constrained geometric actions , arXiv:1908.03577

    1908 arXiv

  77. [85]

    Neeb, Central extensions of infinite-dimensional lie groups , in Annales de l’institut Fourier, vol

    K.-H. Neeb, Central extensions of infinite-dimensional lie groups , in Annales de l’institut Fourier, vol. 52, pp. 1365–1442, 2002

    2002

  78. [86]

    Hollands, Relative entropy for coherent states in chiral CFT , arXiv:1903.07508

    S. Hollands, Relative entropy for coherent states in chiral CFT , arXiv:1903.07508

    1903 arXiv

  79. [87]

    Born and V

    M. Born and V. Fock, Beweis des adiabatensatzes , Zeitschrift f¨ ur Physik51 (1928), no. 3-4 165–180

    1928

  80. [88]

    Mickelsson, String Quantization on Group Manifolds and the Holomorphic Geometry of Diff S1 / S1 , Commun

    J. Mickelsson, String Quantization on Group Manifolds and the Holomorphic Geometry of Diff S1 / S1 , Commun. Math. Phys. 112 (1987) 653

    1987

  81. [89]

    L. J. Boya, A. M. Perelomov, and M. Santander, Berry phase in homogeneous k¨ ahler manifolds with linear hamiltonians , Journal of Mathematical Physics 42 (2001), no. 11 5130–5142. 30

    2001

  82. [90]

    Toffoli, Quo vadimus? — much hard work is still needed , Physica D: Nonlinear Phenomena 120 (1998), no

    T. Toffoli, Quo vadimus? — much hard work is still needed , Physica D: Nonlinear Phenomena 120 (1998), no. 1-2 1–11. 31

    1998

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