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Holographic complexity and non-commutative gauge theory

Authors:
Josiah Couch at University of Texas at Austin
Josiah Couch
Stefan Eccles at University of Texas at Austin
Stefan Eccles
Willy Fischler
Willy Fischler
Willy Fischler
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Ming-Lei Xiao
Ming-Lei Xiao
Ming-Lei Xiao
  • This person is not on ResearchGate, or hasn't claimed this research yet.
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A bstract We study the holographic complexity of noncommutative field theories. The four-dimensional N=4 \mathcal{N}=4 N = 4 noncommutative super Yang-Mills theory with Moyal algebra along two of the spatial directions has a well known holographic dual as a type IIB supergravity theory with a stack of D3 branes and non-trivial NS-NS B fields. We start from this example and find that the late time holographic complexity growth rate, based on the “complexity equals action” conjecture, experiences an enhancement when the non-commutativity is turned on. This enhancement saturates a new limit which is exactly 1/4 larger than the commutative value. We then attempt to give a quantum mechanics explanation of the enhancement. Finite time behavior of the complexity growth rate is also studied. Inspired by the non-trivial result, we move on to more general setup in string theory where we have a stack of D p branes and also turn on the B field. Multiple noncommutative directions are considered in higher p cases.
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... Recent advances in quantum information, quantum computation, and holography have revealed that quantum circuit complexity provides a rich and refined diagnostic of quantum states beyond entanglement entropy alone [7,8,9,10]. In particular, in the context of the AdS/CFT correspondence [11], complexity has been conjectured to be dual to certain geometric quantities in the bulk spacetime (e.g., volumes of extremal slices [12], gravitational actions [13], or other geometric measures [14]). The rapid growth of complexity long after the system reaches thermodynamic equilibrium suggests that complexity encodes deep structural properties of quantum states that are not captured by entanglement alone. ...
... • Recent complexity-gravity correspondences and complexity as an emergent geometric concept [12,13,14]. ...
... This decomposition connects complexity to topological quantum numbers, ensuring that complexity responds not only to local rearrangements of fields but also to large gauge transformations that change the topological sector. In the black hole setting, topological complexity sectors might encode subtle global features of spacetime geometry and holographic dualities, potentially enriching the known relations between complexity and gravitational actions [13,14]. ...
No Firewalls.
Preprint
Full-text available
  • Dec 2024
We present a novel resolution to the black hole firewall paradox, demonstrating information is preserved via quantum complexity growth. By constructing a self-adjoint, gauge-invariant complexity operator on the Hilbert space, we show that complexity imposes strict limitations on how states evolve. These complexity-based speed limits prevent the formation of firewall-like states, as making such states accessible would require overcoming immense computational barriers. Through a detailed mathematical framework and connections to holography, quantum error correction, and gauge theory, we demonstrate that complexity ensures the smoothness of horizons, preserves unitarity, and avoids contradictions involving entanglement monogamy. Our analysis reveals that complexity growth both sustains the equivalence principle and provides a unified explanation of quantum gravitational phenomena, indicating that complexity is not merely a computational tool, but a fundamental physical quantity governing the structure and evolution of spacetime.
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... 3), has been verified in various contexts, e.g., see [10][11][12][13]. However, some cases have been identified where the CA conjecture and the Lloyd bound are not compatible with each other [14][15][16]. Moreover, it has been argued that the holographic gates for which the CA conjecture is true do not satisfy the assumptions of the Lloyd bound [17]. ...
... Therefore, it does not even make sense to compare the rate of complexification with the instantaneous energy of the state as required by the Lloyd bound. For this reason, we consider this example to be a more violent violation of the Lloyd bound than those discussed in [14][15][16]. This time-dependent perturbation of a CFT is described using the AdS-CFT correspondence by the introduction of a (tachyonic) scalar field in the bulk [18][19][20]. ...
... This is the main message of this paper. Though, note that the inconsistency of the CA conjecture and the Lloyd bound were also recently discovered in [14][15][16]. ...
Divergences in the rate of complexification
Preprint
  • Dec 2017
It is conjectured that the average energy provides an upper bound on the rate at which the complexity of a holographic boundary state grows. In this paper, we perturb a holographic CFT by a relevant operator with a time-dependent coupling, and study the complexity of the time-dependent state using the \textit{complexity equals action} and the \textit{complexity equals volume} conjectures. We find that the rate of complexification according to both of these conjectures has UV divergences, whereas the instantaneous energy is UV finite. This implies that neither the \textit{complexity equals action} nor \textit{complexity equals volume} conjecture is consistent with the conjectured bound on the rate of complexification.
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... This led to a conjecture that black holes complexify as rapidly as possible [8,20]; however, this proposal is known to be violated at least at early times [21] and in sufficiently exotic computational setups [8,22]. Very recently, while this paper was being prepared, some holographic examples showing late time violations of the proposed bound were exhibited [23,24]. There is a growing body of work extending these results to a wider class of gravity theories and exploring other proposals [25][26][27][28][29][30][31][32][33][34][35][36][37]. ...
... Other directions include the formulation of a more defined tensor network model and comparisons to the recent free field theory complexity calculations, perhaps in the context of the branching tensor network in Ref. [48]. Finally, given that the conjectured complexity growth rate bound in Ref. [8] is now thoroughly falsified [22,23], it is interesting to consider other possible bounds that might illuminate in which senses black holes are the fastest computers. ...
Holographic Complexity of Einstein-Maxwell-Dilaton Gravity
Preprint
  • Dec 2017
We study the holographic complexity of Einstein-Maxwell-Dilaton gravity using the recently proposed "complexity = volume" and "complexity = action" dualities. The model we consider has a ground state that is represented in the bulk via a so-called hyperscaling violating geometry. We calculate the action growth of the Wheeler-DeWitt patch of the corresponding black hole solution at non-zero temperature and find that, in the presence of violations of hyperscaling, there is a parametric enhancement of the action growth rate. We partially match this behavior to simple tensor network models which can capture aspects of hyperscaling violation. We also exhibit the switchback effect in complexity growth using shockwave geometries and comment on a subtlety of our action calculations when the metric is discontinuous at a null surface.
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... In these works, it was argued that these objects in the bulk indeed grow as expected. The bulk computation of the evolution of holographic complexity was put on firmer grounds by [9], and it has also been subject of many recent works [10,11,12,13,14,15,16,17,18,19]. For instance, in [20], the holographic complexity following a quantum quench was calculated by modeling the quench with an AdS-Vaidya spacetime and it was found that the complexity evolves approximately linearly. ...
... A.15) where c JIK are the structure constants of SU(1,1). The solution isO I (t) = M J I (t) O J (A.16)where M I is an element of SU(1, 1) in the adjoint representation.We can now return to (A.11) and write|ψ (0) = E (t, 0) −1 P e i 1 0 ds X I (t,s)(M −1 ) IK M KL [E(t,0)OLE(t,0) −1 ] |Ω .(A.17) DefiningX K ≡ X I M −1 IK and using (A.16) together with (A.12) to identify the Schrödinger picture operator O I in the exponent, one can now rephrase the Heisenberg problem as that of finding the circuit |ψ (0) = E (t, 0) −1 P e i 1 0 dsX k (t,s)O K |Ω (A.18) that minimizes ds δ IJ X I X J * = ds (M ) by noticing that (M ) I K (M ) I * L = (M M † ) KL = δ KL since M ∈ SU(1, 1) we see that this becomes exactly the Schrödinger picture problem, i.e., that of finding the circuit |ψ (0) = E (t, 0) −1 P e i 1 0 dsX k (t,s)O k |Ω (A.20) that minimizes ds δ KLX KX * L . ...
Evolution of complexity following a quantum quench in free field theory
Preprint
  • Mar 2018
Using a recent proposal of circuit complexity in quantum field theories introduced by Jefferson and Myers, we compute the time evolution of the complexity following a smooth mass quench characterized by a time scale δt\delta t in a free scalar field theory. We show that the dynamics has two distinct phases, namely an early regime of approximately linear evolution followed by a saturation phase characterized by oscillations around a mean value. The behavior is similar to previous conjectures for the complexity growth in chaotic and holographic systems, although here we have found that the complexity may grow or decrease depending on whether the quench increases or decreases the mass, and also that the time scale for saturation of the complexity is of order δt\delta t (not parametrically larger).
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... Using the language of noncommutative algebras, noncommutative geometry provides new perspectives on other branches of mathematics, such as operator algebras, differential geometry, algebraic geometry, K-theory, cyclic cohomology, number theory, measure theory, etc. It also has many novel and useful applications in physics [3][4][5][6][7][8][9][10][11][12][13]. Many mathematicians have studied the generalizations of noncommutative analog of differential geometry from different perspectives [14][15][16][17][18][19][20][21][22]. ...
Curvature, area and Gauss-Bonnet formula of the Moyal sphere
Preprint
Full-text available
  • Dec 2024
We studied some geometric properties of the Moyal sphere. Using the conformal metric of the sphere in ordinary space and the matrix basis, we calculated the scalar curvature, total curvature integral and area of the Moyal sphere. We found that when the noncommutative parameter approaches to 0, the scalar curvature and area of the Moyal sphere return to those of the ordinary sphere. As the noncommutative parameter increases, the area of the Moyal sphere will decrease and eventually approach to 0. We found that the total curvature integral of the two-dimensional Moyal sphere still satisfies the usual Gauss-Bonnet formula and does not depend on the noncommutative parameter. We also calculated the approximate expression of the conformal metric with a constant curvature and obtained the corresponding correction function. In addition, we also studied a type of generalized deformed Moyal sphere with two noncommutative parameters and obtained similar results.
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... Recall that the Lorentzian action has negative sign.4 In the Einstein gravity case studied in[5] a topological argument is needed to rule this part out because the integrand there is R; since our integral is just a volume for us this argument is trivial. ...
On Complexity of Holographic Flavors
Preprint
  • May 2017
Quantum complexity of a thermofield double state in a strongly coupled quantum field theory has been argued to be holographically related to the action evaluated on the Wheeler-DeWitt patch. The growth rate of quantum complexity in systems dual to Einstein-Hilbert gravity saturates a bound which follows from the Heisenberg uncertainty principle. We consider corrections to the growth rate in models with flavor degrees of freedom. They are realized by adding a small number of flavor branes to the system. Holographically, such corrections come from the DBI action of the flavor branes evaluated on the Wheeler-DeWitt patch. We relate corrections to the growth of quantum complexity to corrections to the mass of the system, and observe that the bound on the growth rate is never violated.
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... It was shown in these papers that the conjectured bound of Eq. (2) is violated at earlier times. Similar violations were also observed for the dual of the non-commutative gauge theories in [12]. A generalization of Eq. (1) for the reduced state of some subsystem of the CFT was proposed in [13]. ...
Evolution of Complexity Following a Global Quench
Preprint
  • Nov 2017
The rate of complexification of a quantum state is conjectured to be bounded from above by the average energy of the state. A different conjecture relates the complexity of a holographic CFT state to the on-shell gravitational action of a certain bulk region. We use 'complexity equals action' conjecture to study the time evolution of the complexity of the CFT state after a global quench. We find that the rate of growth of complexity is not only consistent with the conjectured bound, but it also saturates the bound soon after the system has achieved local equilibrium.
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... However, as noted above, transient violations of the proposed bound were already identified in studying the time evolution of complexity in an eternal black hole background [46]. Further, even stronger violations were found in the dual of a noncommutative gauge theory [45] and in hyperscaling violating geometries [48,49]. 30 Therefore, while the proposed bound can not be universal, it remains an interesting question to understand the situations when it does apply and when not, and the underlying reasons for this. ...
Holographic Complexity in Vaidya Spacetimes I
Preprint
  • Apr 2018
We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity=volume (CV) and complexity=action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.
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... This negative spike (as well as the overshoot of the late time limit) in dC A /dτ also appears in different holographic settings, such as the holographic dual of non-commutative SYM theories[56]. We thank Josiah Couch for discussing this upcoming work with us. ...
On the Time Dependence of Holographic Complexity
Preprint
  • Sep 2017
We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. Adding a charge to the eternal black holes washes out the early time behaviour, i.e., complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.
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... To derive the complexity-volume conjecture, we can consider the geometry of the bulk spacetime and the properties of maximal spacelike slices. In particular, the volume of a maximal slice can be computed using the Einstein-Hilbert action, which includes contributions from the spacetime curvature and the cosmological constant [42]. By analyzing the behavior of this action under small perturbations of the boundary state, we can establish a connection between the growth of complexity and the expansion of spacetime volume. ...
On Spacetime Geometry and Gravitational Dynamics
Preprint
Full-text available
  • Jul 2024
This work presents a theoretical framework suggesting that gravity and space-time geometry emerge from underlying quantum informational processes. By integrating quantum information theory with general relativity, we explore the profound implications of quantum error correction, entanglement entropy, and quantum complexity in the context of spacetime emergence. Utilizing key results such as the Ryu-Takayanagi formula, which relates entanglement entropy to minimal surfaces in AdS/CFT, and the complexity-action duality, which links quantum complexity to the action of the Wheeler-DeWitt patch, we provide a comprehensive analysis of how these quantum informational measures give rise to spacetime geometry and gravitational dynamics. Our findings propose that the macroscopic structure of spacetime and the force of gravity are emergent phenomena resulting from the collective behavior of quantum entanglement and complexity. This research challenges conventional views of reality by demonstrating that the fabric of spacetime is a manifestation of underlying quantum processes. We extend these concepts beyond AdS spacetimes to include non-AdS geometries, such as de Sitter and flat spacetimes, further supporting the universality of our framework. Future research directions include empirical validation through quantum computing experiments, refinement of theoretical models, and deeper exploration of the quantum informational foundations of spacetime and gravity. By bridging quantum mechanics and general relativity, this work contributes to a more profound understanding of the universe's fundamental principles and opens new avenues for advancements in quantum computing, materials science, and the unification of fundamental forces.
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Complexity is Simple
Article
Full-text available
  • Oct 2017
  • J HIGH ENERGY PHYS
In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in Brown et al.
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On the Time Dependence of Holographic Complexity
Article
Full-text available
  • Sep 2017
  • J HIGH ENERGY PHYS
A bstract We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd’s bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. For either conjecture, we find that the late time limit for the rate of change of complexity is saturated at times of the order of the inverse temperature. Adding a charge to the eternal black holes washes out the early time behaviour, i.e. complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.
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A Complexity for Quantum Field Theory and Application in Thermofield Double States
Article
Full-text available
  • Sep 2017
This paper defines a complexity between states in quantum field theory by introducing a Finsler structure based on created and annihilated operators. Two simple models are computed as examples and to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double state, result shows the complexity between it and corresponding vacuum state is finite and proportional to Td−1T^{d-1} in d-dimensional conformal field. Some enlightenments to holographic conjectures of complexity are also discussed.
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Circuit complexity in quantum field theory
Article
Full-text available
  • Jul 2017
  • J HIGH ENERGY PHYS
Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.
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Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
Article
Full-text available
  • Jun 2017
  • J HIGH ENERGY PHYS
A bstract We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.
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Einstein Equations from Varying Complexity
Article
Full-text available
  • Jun 2017
A recent proposal equates the circuit complexity of a quantum gravity state with the gravitational action of a certain patch of spacetime. Since Einstein's equations follow from varying the action, it should be possible to derive them by varying complexity. I present such a derivation for vacuum solutions of pure Einstein gravity in three-dimensional asymptotically anti-de Sitter space. The argument relies on known facts about holography and on properties of Tensor Network Renormalization, an algorithm for coarse-graining (and optimizing) tensor networks.
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Time evolution of complexity in Abelian gauge theories
Article
  • Dec 2017
Quantum complexity is conjectured to probe inside of black hole horizons (or wormholes) via gauge gravity correspondence. In order to have a better understanding of this correspondence, we study time evolutions of complexities for Abelian pure gauge theories. For this purpose, we discretize the U(1) gauge group as ZN and also the continuum spacetime as lattice spacetime, and this enables us to define a universal gate set for these gauge theories and to evaluate time evolutions of the complexities explicitly. We find that to achieve a large complexity ∼exp(entropy), which is one of the conjectured criteria necessary to have a dual black hole, the Abelian gauge theory needs to be maximally nonlocal.
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Towards Complexity for Quantum Field Theory States
Article
  • Jul 2017
We investigate notions of complexity of states in continuous quantum-many body systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of Multiscale Entanglement Renormalization Ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum preserving quadratic generators which form su(1,1)\mathfrak{su}(1,1) algebras. On the manifold of Gaussian states generated by these operations the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and holographic complexity proposals.
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Time Evolution of Complexity in Abelian Gauge Theories - And Playing Quantum Othello Game -
Article
  • Jul 2017
Quantum complexity is conjectured to probe inside of black hole horizons (or wormhole) via gauge gravity correspondence. In order to have a better understanding of this correspondence, we study time evolutions of complexities for generic Abelian pure gauge theories. For this purpose, we discretize U(1) gauge group as ZN\mathbf{Z}_N and also continuum spacetime as lattice spacetime, and this enables us to define a universal gate set for these gauge theories, and evaluate time evolutions of the complexities explicitly. We find that to achieve a large complexity \sim \exp(\mbox{entropy}), which is one of the conjectured criteria necessary to have a dual black hole, the Abelian gauge theory needs to be maximally nonlocal.
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Yang-Baxter σ\sigma-models, conformal twists & noncommutative Yang-Mills
Article
  • Feb 2017
  • PHYS REV D
The Yang-Baxter σ\sigma-model is a systematic way to generate integrable deformations of AdS5×_5\timesS5^5. We recast the deformations as seen by open strings, where the metric is undeformed AdS5×_5\timesS5^5 with constant string coupling, and all information about the deformation is encoded in the noncommutative (NC) parameter Θ\Theta. We identify the deformations of AdS5_5 as twists of the conformal algebra, thus explaining the noncommutativity. We show that the unimodularity conditon on r-matrices for supergravity solutions translates into Θ\Theta being divergence-free. Integrability of the σ\sigma-model for unimodular r-matrices implies the existence and planar integrability of the dual NC gauge theory.
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