Spacetime as a quantum circuit
A. Ramesh Chandra, Jan de Boer, Mario Flory, Michal P. Heller, Sergio Hörtner, Andrew Rolph
PPrepared for submission to JHEP
Spacetime as a quantum circuit
A. Ramesh Chandra, a Jan de Boer, a Mario Flory, b,c
Michal P. Heller, d, SergioHörtner, a and Andrew Rolph a a Institute for Theoretical Physics, University of Amsterdam, PO Box 94485, 1090 GL Amsterdam,The Netherlands b Institute of Physics, Jagiellonian University, 30-348 Kraków, Poland c Instituto de Física Téorica IFT-UAM/CSIC, Universidad Autonoma de Madrid, 28049, Madrid,Spain d Max Planck Institute for Gravitational Physics (Albert Einstein Institute), 14476 Potsdam-Golm,Germany
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We propose that finite cutoff regions of holographic spacetimes represent quan-tum circuits that map between boundary states at different times and Wilsonian cutoffs,and that the complexity of those quantum circuits is given by the gravitational action. Theoptimal circuit minimizes the gravitational action. This is a generalization of both the “com-plexity equals volume” conjecture to unoptimized circuits, and path integral optimizationto finite cutoffs. Using tools from holographic T ¯ T , we find that surfaces of constant scalarcurvature play a special role in optimizing quantum circuits. We also find an interestingconnection of our proposal to kinematic space, and discuss possible circuit representationsand gate counting interpretations of the gravitational action. On leave of absence from:
National Centre for Nuclear Research, 02-093 Warsaw, Poland a r X i v : . [ h e p - t h ] J a n ontents T ¯ T Quantum information theoretic concepts such as entanglement entropy have proven to beof fundamental importance for our understanding of quantum gravity, most notably in thecontext of the AdS/CFT correspondence [1, 2]. However, it has also been claimed that “en-tanglement is not enough” [3], such as in the inability of holographic entanglement entropyto probe the late time linear growth of the Einstein-Rosen bridge in eternal black holes,and that other concepts, in particular state complexity, are needed for a more completeunderstanding.The notion of state complexity in quantum mechanics refers to a setup where one isgiven an initial state, a final state, a margin of error and a list of allowed unitary operations.The smallest number of unitaries needed to obtain the final state from the initial state upto the margin of error is an indication of how difficult it is to obtain the final state fromthe initial state. If the initial state is a fixed and very simple state, i.e. an unentangledproduct state, one can simply refer to the complexity of the final state as state complexity.The idea to build interesting quantum states using a limited set of operations hasbeen successful in condensed matter physics leading to e.g. tensor network representationsof states [4]. Moreover, a specific relation between tensor networks and gravity has beenproposed [5, 6] in which one interprets constant time slices of AdS spacetime as a so-calledmultiscale entanglement renormalization (MERA) tensor network [7]. As a first piece ofevidence for this relation one notices that the tensor network in question indeed closely– 1 –esembles a lattice discretization of an equal time slice of AdS. If one imagines that thisMERA tensor network is the optimal network to obtain the ground state of a (discretized)CFT, then the number of tensors needed equals the volume of the equal time slice. Thissubsequently led to a much more general “complexity equals volume” proposal [8] whereone proposes that the complexity of any viable state in holographic quantum field theoriescan be obtained from the minimal volume of a slice of the geometry which is anchored atthe relevant fixed boundary time slice. Other complexity proposals include those wherecomplexity is computed from the action [9, 10] or spacetime volumes [11] evaluated in bulkWheeler-deWitt (WdW) patches. All these holographic complexity proposals share certainqualitative features that any notion of state complexity should possess, while still lackinga precise microscopic definition in the dual CFT.A continuum version of MERA [12] was an important factor in the realization of [13, 14],that a natural way to count gates and define complexity in QFT is by assigning a metricto a suitable group underlying the state preparation of interest [15]. While this is arguablythe most promising way to define and prove holographic complexity proposals, or to findother gravitational manifestations of complexity, in this paper we will not directly attemptto find a precise microscopic definition of complexity in holographic CFTs. Instead, we willpropose a significant refinement of the relation between geometry and complexity as follows:we suggest that any spacetime region can be interpreted as a quantum circuit, with thegravitational action providing a notion of complexity for this particular quantum circuit.
Figure 1 . We consider a subregion M of Euclidean Poincaré AdS . We introduce two time-slices t = t i and t = t f corresponding to the field theory ground states | (cid:105) z i and | (cid:105) z f , which are preparedfor different values of the radial cutoff. The radial boundary is at finite cutoff, z = ρ ( t ) . Ourproposal is that the complexity of the circuit that maps between these ground states with differentfinite Wilsonian cutoffs is given by the gravitational action on M . Let us make our proposal more precise. Take an Euclidean asymptotically AdS geom-etry with radial coordinate z , the asymptotic boundary being at z = 0 , and a spacetime– 2 –egion M given by t i ≤ t ≤ t f and z ≥ ρ ( t ) for some function ρ ( t ) , see figure 1. The AdSgeometry describes the time evolution of a given state, and the region z ≥ ρ ( t ) knows aboutthe state, but only up to a UV cutoff set by ρ ( t ) . Here, we use the well-known relationbetween the radial distance in AdS and the UV cutoff in the theory [16] and its recentrefinement [17]. The latter development links gravity in AdS spacetimes with a finite radialcutoff to finite irrelevant deformations of dual CFTs by a T ¯ T operator, and this will bea useful way of thinking about the UV cutoff in the remainder of this paper. The bulkgeometry of interest can be thought of as describing a sequence of states, which are relatedto each other by both Euclidean time evolution and a change of UV cutoff. One can askwhat the complexity of this particular process is, i.e. how many operations would be re-quired to recover a (discretized version of) this sequence of states. We propose that thenumber of operations, the complexity, is given by the gravitational action itself, evaluatedwith suitable boundary condition on the boundary of the spacetime region.Our proposal, after specializing to two boundary dimensions, and keeping only the firsttwo orders in a Taylor expansion of the Wilsonian cutoff, coincides with the path integraloptimization proposal [18–22], and so for holographic CFTs can be seen as generalization ofthat proposal to any dimension and finite cutoff. In path integral optimization one considersdifferent preparations of a fixed state using CFT path integrals on background geometriesdiffering by a Weyl factor. The proposal is to regard the change in the unnormalized pathintegral measure – which for AdS is given by the Liouville action [23] – as a cost function.However, as recognized in [22], minimization of such a cost function is not consistent withkeeping only the terms which remain finite as the UV cutoff is removed. Our proposal avoidsthis problem altogether by considering the full gravitational action, with the Liouville actionmerely capturing the leading two terms in the limit where one takes the cutoff to infinity.Coming back to our tensor network motivation, it is interesting to note that in thecourse of the past several years substantial progress has been achieved in obtaining MERAfrom a systematic coarse-graining procedure rather than using it merely as an efficientvariational ground state ansatz for critical spin chains. The key idea is to employ anentanglement-based coarse-graining of the discretized Euclidean path-integral [24]. Thisprocedure is closely related to one where one puts the corresponding conformal field theories,which arise in the continuum limit of the critical spin chains, on curved geometries [25, 26].We find the apparent connection between these ideas and our setup quite intriguing. Inparticular, in the language of [25] one would be tempted to call part of our circuit associatedwith moving in t as composed from “euclideons”, whereas the motion in the z direction wouldhave to do with “isometries” and possibly “disentanglers”.It would be certainly very interesting to make this association more quantitative, per-haps using the results of [27], as there are currently several distinct proposals for associatinggeometries to MERA. It has been suggested to connect MERA to a hyperbolic geometry ofan equal time slice of AdS as mentioned above, to a light-cone [26] and to an auxiliary dSgeometry [28] as in [29]. The latter was motivated by work trying to probe the bulk geome-try using non-local CFT observables such as entanglement entropy of spherical subregions,which gave rise to the kinematic space program [30–32]. This confusing state of affairs wasone of our motivations to try to sharpen the relation between gravity and quantum circuits.– 3 –t is important to emphasize that we are not considering arbitrary circuits: all circuitsare essentially composed of time evolution and changes in the local cutoff starting with agiven initial state. One could certainly imagine more general circuits, but these would notbe captured by a single semi-classical geometry and would require off-shell gravitationalconfigurations. The latter are typically exponentially suppressed and we will not considerthem in this paper. One can still try to find the optimal circuit within a given semi-classicalgeometry, by varying over t i and over ρ ( t ) . In particular, taking ρ ( t i ) = ∞ corresponds to aCFT state where the CFT has a momentum cutoff brought down to zero, so this is akin tomaking the state at t = t f from “nothing” [33]. We find that optimizing complexity over thisrestricted set of circuits gives results quite similar to other holographic complexity proposals.Perhaps in these holographic situations, there is nothing to be gained (from a complexitypoint of view) by considering circuits that involve different semiclassical geometries.Finally, let us emphasize that there was earlier work, such as [21, 34–39], advocatingfor the relation between quantum circuits and holographic geometry. In the present workwe are building on these earlier developments to bring in three important new elements intothis discussion: being explicit about an initial state, realizing the need of keeping UV cut-offfinite and interpreting it in terms of a T ¯ T deformation and, last but not least, making aconnection with the kinematic space program.The outline of this paper is as follows. In section 2, we will first describe the gen-eral setup and then give an explicit example in vacuum AdS , where we will find that ournotion of complexity agrees with the complexity equals volume proposal and, in the limit ρ ( t ) → , also with path-integral optimization. Subsequently, we discuss various finer pointsassociated with our proposal in sections 2.2 and 2.3. This brings us to section 3, where wedescribe the relation between a change in the spacetime region and T ¯ T -deformations usingbulk flow equations. Considerations in this section will also allow us to argue that com-plexity is optimized if the boundary of the spacetime region has constant scalar curvature.Finally, we will discuss some ideas to more directly connect the gravitational action to agate counting procedure in section 4, and end with some conclusions and suggestions forfuture work in section 5. The main idea of our construction is as follows: We can produce states in a CFT usingpath integrals over Euclidean manifolds with a boundary and operator insertions. Similarly,path integrals over manifolds with two boundaries can be interpreted as objects mappingstates to states. We would like to think of these path integrals as describing a circuit whichprepares states or maps states to states, and associate a notion of cost function to themwhich measures the effort it takes to perform these CFT operations in a given way. Todefine such a cost function it seems inevitable to introduce some sort of cutoff in the fieldtheory. This cutoff defines a local lattice spacing and provides the natural length scale atwhich to define tensors which make up an approximate tensor network description of theCFT operation. The cutoff could in principle be space and time dependent.– 4 –o determine the complexity, we are going to propose to use the unnormalized CFTpath integral. An important issue is how to incorporate the space and time dependentcutoff in this computation. In the field theory, one could try to implement this by includinga space and time dependent T ¯ T -deformation in the CFT which is known to implement aparticular type of finite cutoff [40, 41]. It seems difficult to compute the required pathintegral directly in the CFT, but luckily, for CFTs with a holographic dual we can usethe AdS/CFT correspondence to do the computation. Following [17], the relevant AdSsetup is one where we move the boundary of AdS a finite distance inward, with the time(and possibly space) dependent radial position corresponding to the cutoff or coefficient ofthe T ¯ T deformation. The partition function of the CFT with cutoff is then computed, toleading order, by the on-shell value of the gravitational action with a finite instead of anasymptotic boundary.There are some aspects of this proposal that require clarification. One is the choice ofboundary for the gravitational path integral away from the surface where the cutoff CFTlives. For example, if the cutoff CFT lives on a hemisphere, we need to fill in the boundaryof the hemisphere in AdS. There is in general no canonical slice in the bulk where the state“lives”. In the example that we consider below, there are always natural time-symmetricsurfaces in the bulk which are the natural surfaces where to bound the bulk path integral.Another issue with the construction is whether or not to include the standard countertermsfor AdS/CFT for co-dimension one boundaries when evaluating the bulk action. Due to theexistence of a finite cutoff, there is no strict need to do so, and not including them appearsto be the most natural thing to do as we discuss below. A closely related issue comes fromthe fact that the full bulk region has corners, and one may need to include corner termswhen evaluating the bulk action. We will also address this issue below. For simplicity and concreteness, we are going to consider the preparation of the groundstate of a 2d CFT on a line using the Euclidean path integral. To this end, we take thestandard Euclidean AdS solution, with the curvature scale l AdS = 1 , ds = dz + dt + dx z , (2.1)and the partition function of the CFT equals the exponent of minus the on-shell bulk action I = 1 κ (cid:90) M d x √ G ( R + 2) + 2 κ (cid:90) ∂M d x √ gK + I c . (2.2) M is the bulk region bounded by ρ ( t ) ≤ z ≤ ∞ and t i ≤ t ≤ t f , as shown in figure 1. Thea priori finite function ρ ( z ) interpolates between the values z = z i at t = t i and z = z f at t = t f , with t i ≤ t f and z f < z i . For simplicity we also take the setup to be independentof the transverse direction x . Furthermore, we write κ = 16 πG N , G for the 3d metric on M , g for the induced 2d metric on ∂M , and K is the trace of the extrinsic curvature. ∂M is only piecewise smooth and has a kink or joint at t = t f and t = t i as shown in figure 1.– 5 –ach joint contributes a term I c = 2 κ (cid:90) dx (cid:112) j α (2.3)to the gravitational action. Herein, √ j is the length element along the joint and α is simplythe angle between the two normal vectors of the two surfaces coming together at the joint(which may have either sign). Joint-terms of this type were studied by Hayward in [42, 43],but in the Euclidean setting, which is of interest here, this was already done earlier in [44],see also the discussion in [45].As discussed above, we are going to interpret the on-shell value of the bulk effectiveaction of the region M as the complexity of the circuit defined by the surface z = ρ ( t ) whichmaps the vacuum state | (cid:105) z i with cutoff z i to the vacuum state | (cid:105) z f with cutoff z f . If weuse the relation between a finite radial cutoff and the coefficient µ of the T ¯ T deformationvia [17], µ ( t ) = κ ρ ( t ) , (2.4)we can reinterpret the states | (cid:105) ρ ( t ) as ground states of the T ¯ T deformed CFT with atime-dependent coefficient µ ( t ) .Concretely, the induced line element on the boundary surface is ds = (1 + ˙ ρ ) dt + dx ρ , (2.5)its Ricci scalar reads R ( d − = 2( ρ ¨ ρ − ˙ ρ (1 + ˙ ρ ))(1 + ˙ ρ ) , (2.6)the trace of the extrinsic curvature reads K = ρ ¨ ρ + 2(1 + ˙ ρ )(1 + ˙ ρ ) / , (2.7)and from (2.2) we obtain I = − κ (cid:90) M d x (cid:90) ∞ z = ρ dzz + 2 κ (cid:90) ∂M d x ρ ¨ ρ + 2(1 + ˙ ρ ) ρ (1 + ˙ ρ ) + I c = 2 V x κ (cid:90) t f t i dt ρ ¨ ρ + (1 + ˙ ρ ) ρ (1 + ˙ ρ ) + I c (2.8)for the on-shell bulk action, where we have introduced V x = (cid:82) dx . For the corner term, wealso find I c = 2 V x κ (cid:18) π/ − arctan ˙ ρ ( t f ) z f + π/ ρ ( t i ) z i (cid:19) . (2.9)Integrating by parts, this action can be written only using first derivatives of ρ , yielding I = 2 V x κ (cid:90) t f t i dt (cid:18) ρ + ˙ ρ arctan ˙ ρρ (cid:19) + πV x κ (cid:18) z f + 1 z i (cid:19) . (2.10)– 6 –he terms which are independent of ρ do not affect the equations of motion, and can alwaysbe removed by a suitable counter term, which we will assume to be done from now on. Webelieve this is justified, as it is known [43, 45] that the joint term can spoil the additivityof the action under combining bulk regions, which besides the formulation of a well definedvariational principle is usually the second main reason for adding boundary terms to theaction (2.2). The equations of motion obtained by extremizing (2.10) read ρ ¨ ρ + (1 + ˙ ρ ) ρ (1 + ˙ ρ ) = 0 . (2.11)The most immediately visible solution to this equation is the one where we formally takethe limit ˙ ρ → ∞ . This corresponds to the boundary surface turning into an equal-timeslice, which is in fact where, based on the intuition surrounding holographic complexityand tensor networks, we expect the most optimised circuit preparing the state | (cid:105) z f to live,see e.g. [38]. The generic solution to (2.11) reads ρ ( t ) = (cid:112) R − ( t − t ) (2.12)and describes circular arcs of radius R centered on the boundary point at t = t . Theformal solution ˙ ρ → ∞ corresponds to the limit of infinite radius.Our proposal is that the Euclidean action (2.10) (excluding the ρ -independent remnantsof the joint terms) is a measure of the complexity of preparing the state | (cid:105) z f from the state | (cid:105) z i using the circuit described by ρ ( t ) . The optimal circuit, with fixed Euclidean timedistance ∆ t = | t f − t i | , is then of the form (2.12), and the complexity of this circuit is givenby evaluating the Euclidean action on this solution. With the explicit boundary conditionsbeing ρ ( t f ) = z f and ρ ( t i ) = z i , the value of the Euclidean action in the first term of (2.10)is I = 2 V x κ (cid:32) z f arctan z i − z f + ∆ t z f ∆ t − z i arctan z i − z f − ∆ t z i ∆ t (cid:33) . (2.13)Note that this result comes entirely from the corner terms, as the first term in (2.8) exactlyvanishes on-shell. Interpreting it as a function of the variable t i ≤ t f while keeping z i (cid:54) = z f fixed, we can verify that the above expression is minimized by t i = t f . This corresponds tothe limit R → ∞ or ˙ ρ → ∞ and hence the equal time slice that is intuitively expected toplay a special role in describing the complexity of the state | (cid:105) z f . Using /κ = c/ [46],the minimum value is given by I min = cπV x (cid:18) z f − z i (cid:19) , (2.14)which is proportional to the spatial volume of the strip z f ≤ z ≤ z i on the equal time sliceat t = t f . Of course, if we send z f → (cid:15) (cid:28) and z i → ∞ , this reproduces the standard Note that in our Euclidean setting, where spacelike surfaces have spacelike normal vectors, the jointsunder consideration are more similar to the timelike joints discussed in [43, 45] than spacelike ones in aLorentzian setting. – 7 –esult of the volume proposal for the complexity of the CFT ground state. Clearly, thisresult also vanishes if z i = z f , which we take as a non-trivial consistency check and furtherjustification for excluding the remnants of the joint terms in (2.10). To close this section, let us compare our results to the ones that can be obtained fromthe Liouville action. For ˙ ρ (cid:28) , equation (2.10) can be approximated as I = 2 V x κ (cid:90) dt (cid:18) ρ + ˙ ρ ρ (cid:19) , (2.15)which, assuming no x -dependence, is equivalent to the Liouville Lagrangian S L = c π (cid:90) dt (cid:90) dx (cid:16) η e ω + ( ∂ t ω ) + ( ∂ x ω ) (cid:17) . (2.16)after a change of variables ρ ( t ) → (1 / √ η ) e − ω ( t ) . Note that the physically interestingsolution ˙ ρ → ∞ falls outside of the range of applicability of the approximation necessary toobtain the Liouville action from (2.10). The equations of motion derived from (2.15) takethe form ρ ¨ ρ + (1 − ˙ ρ ) ρ = 0 . (2.17)As we will see below, these field equations also arise if we introduce a new time coordinatein order to bring the induced metric on the boundary into conformal gauge. There is a subtle but crucial difference between our setup discussed in the previous subsec-tion and the calculations of [38], which we will discuss in this subsection in order to avoidconfusion.In order to do so, we note that [38] investigates a setup similar to the one depicted infigure 1, and up to notation (2.8) also appears in the appendix of that paper. Following[38], we can now introduce a conformal time u , with du = (cid:112) ρ ( t ) dt, (2.18)such that the line element (2.5) is transformed into the conformal gauge form ds = du + dx (cid:37) ( u ) . (2.19)Here, we have introduced a new variable such that (cid:37) ( u ( t )) = ρ ( t ) . Under (2.18), the action(2.10) changes to [38] I = 2 V x κ (cid:90) u f [ (cid:37) ] u i [ (cid:37) ] du (cid:32) (cid:112) − (cid:37) (cid:48) + (cid:37) (cid:48) arcsin (cid:37) (cid:48) (cid:37) (cid:33) . (2.20) As an illustrative example, imagine a Euclidean axisymmetric spacetime, with a spacetime region inthe shape of a regular prism that breaks rotational symmetry around the axis to a discrete subgroup. Inthe limit where the radius of the prism goes to zero, the action on that region may not go to zero, as whilebulk and surface terms vanish in this limit due to the vanishing of bulk volume and surface area, the jointterms will lead to a contribution proportional to an integral along the axis of symmetry. This remnant termis the analogue of the last bracket in (2.10). – 8 –f we were to just identify the integrand in (2.20) as a Lagrangian and compute naively theEuler equations, we arrive at (cid:37)(cid:37) (cid:48)(cid:48) + 2(1 − (cid:37) (cid:48) ) (cid:37) (1 − (cid:37) (cid:48) ) = 0 , (2.21)which, up to notation and the addition of a nonzero tension term, are the equations whichwhere studied in [38].The subtlety announced at the beginning of the subsection is that (2.18) is a reparametriza-tion of time which is dependent on the variable with respect to which we want to vary theaction, hence formally in going from (2.10) to (2.20) the integration bounds u i and u f become themselves functionals of (cid:37) , and will lead to a nontrivial contribution according toLeibniz’s rule when varying the action. In fact it can be checked that introducing (2.18)and (cid:37) ( u ) in the equation of motion (2.11) gives a result (cid:37)(cid:37) (cid:48)(cid:48) + (1 − (cid:37) (cid:48) ) (cid:37) = 0 (2.22)that is inequivalent to (2.21). Interestingly, (2.22) has the form of the Liouville equation(2.17), just for (cid:37) ( u ) instead of ρ ( t ) .The most commonly known example where a field-dependent reparametrization can beuseful is the Lagrangian for geodesic motion, which becomes a constant when introducingaffine parametrisation. Of course, this does not mean that the equations of motion degen-erate, as the full information about the value of the action – i.e. the length of the curve– is now entirely encoded in the integration domain. Unfortunately, the expression (2.20)rather inelegantly falls into a middle ground between the two possible extremes, as both theintegrand and the integration bounds are functionals of the variable (cid:37) , and for this reasonwe found it intractable to work with.This does not mean that either our work or [38] are wrong , just that we are studying adifferent variational problem. We work with the action (2.10) where explicitly we assumeDirichlet boundary conditions for ρ ( t ) at the fixed values t = t f and t = t i , while [38] workswith the action (2.20) with the implicit assumption of Dirichlet boundary conditions forthe field (cid:37) at fixed values of u i , u f , which is an inequivalent mathematical exercise. We can investigate this issue a bit further. So far, we have essentially considered whatamounts to minisuperspace models, by plugging in an ansatz into the action and derivingequations of motion for the function parametrizing that ansatz, instead of first derivinggeneral equations of motion and then simplifying them with a given ansatz. How can wewrite our equations of motion in a form that is more suggestive for their general meaningand potential origin? We will do this in the next section, but as an aside, we will nowdemonstrate that the semicircle solutions that we found can also be obtained if we interpretthe boundary of the bulk domain as an “end of the world brane” with an energy-momentumtensor describing matter with a very specific equation of state. The covariant equations ofmotion of this end of the world brane will imply the general equation that we will derive– 9 –n the next section. The derivation in the next section does not rely on an end of theworld brane interpretation, and it remains to be seen whether this agreement is more thana technical coincidence.We should also point out that the work of [38] was strongly influenced by the type ofAdS/boundary CFT (BCFT) models introduced in [47, 48]. In such models the boundaryof the space on which the BCFT lives is also extended into the bulk spacetime in the formof an end of the world brane, on which Neumann boundary conditions are imposed. Besidesthe bulk Einstein equations, this leads to an equation of motion of the form K µν − Kg µν = κ T µν (2.23)which determines the embedding of the end of the world brane into the ambient space.These models allow for considerable bottom-up toy-model building freedom, and T µν is theenergy-momentum tensor of any matter that lives in the brane worldvolume. In practice,it is often set to be a constant tension term T µν = λg µν (2.24)with tension λ . As reported in [38], their equation of motion is consistent with (2.23). Aswe ignore tension terms, we would set the right hand side of (2.23) to zero, and apart fromthe equal time slice obtained by ˙ ρ → ∞ , our semicircular embeddings do not satisfy thisequation.Interestingly, in a Lorentzian AdS/BCFT context, semicircular embeddings into PoincaréAdS were derived in [49] for a simple model of T µν given by a perfect fluid with equationof state p = aσ ( p = pressure, σ =energy density) in the limit a → ∞ . So we see thatsemicircular embeddings into a Poincaré AdS do satisfy an equation of the form (2.23), justwith a specific non-trivial right hand side. Due to the peculiar limit in the parameter a , T µν satisfies the condition det[ T µν ] = 0 (2.25)or equivalently T µν T µν − T = 0 , (2.26)and hence det [ K µν − Kg µν ] = 0 , (2.27)respectively ( K µν − Kg µν )( K µν − Kg µν ) − Tr[ K µν − Kg µν ] = K µν K µν − K = 0 (2.28)for our semicircular embeddings (2.12), even though they were not derived from an AdS/BCFTansatz in this paper. We will give a direct derivation of equation (2.28) as a flow equationfor our complexity proposal in the following section.– 10 – Bulk action and T ¯ T We have considered the on-shell action of a cutout region of Poincaré AdS , and interpretedit as a complexity functional of states in T ¯ T -deformed holographic CFTs. The relation (2.4)between the coefficient of the T ¯ T deformation and the radial location has been derivedfor constant radial cutoff [17, 50, 51], but not for time-dependent ρ ( t ) . In this sectionwe consider the flow equations which describe movement of the cutoff surface in a fixedbackground. By integrating these flow equations we should be able to derive a more preciserelation between the coefficient of the T ¯ T deformation and the location of the bulk surface.In addition, these flow equations will tell us how complexity changes as we change thesurface locations, and for which surfaces complexity is optimized while keeping the initialand final state fixed. The relevant flow equation can most easily be derived using the ADM formalism [52]. Wewill keep the number of space-time dimensions free in what follows, and write the metric as ds = N dr + g µν ( x, r )( dx µ + N µ dr )( dx ν + N ν dr ) . (3.1)This contains the usual lapse and shift functions, for which one can locally choose a conve-nient gauge N = 1 and N µ = 0 . Following ADM and choosing units so that κ = 1 , we nowwrite the Lagrangian in terms of canonical variables L = √ g ( π µν ∂ r g µν − N H − N µ H µ ) , (3.2)where the lapse and shift functions appear as Lagrange multipliers enforcing the Hamilto-nian and momentum constraints H = H µ = 0 . (3.3)The canonical momenta are given by [53] π µν = 1 √ g ∂S∂g µν = − ( K µν − Kg µν )= −
12 ( ∂ r g µν − g µν g ρσ ∂ r g ρσ ) , (3.4)where in the second step we used the fact that metric variations are given by the Brown-York tensor, and in the last step we used the explicit form of the extrinsic curvature for themetric (3.1) in the gauge N = 1 , N µ = 0 . Of course, the same result can also be obtainedby explicitly rewriting the action as in (3.2). Using (3.4), we find for the radial derivative ∂ r g µν = − π µν + 2 d − g µν π ρρ (3.5)where d is the total number of space-time dimensions (in this paper we are predominantlyinterested in d = 3 ). The Hamiltonian constraint can be computed from (3.2) and, for unit– 11 –dS radius, one finds H = R ( d − − − ( K − K µν K µν )= R ( d − + ( d − d −
2) + π µν π µν − d − π ρρ ) . (3.6)It is fairly straightforward to include matter fields in the discussion; the Hamiltonian con-straint will then also contain the Hamiltonian of the matter sector, but we will for simplicityrestrict to the purely gravitational case. To describe the flow we imagine starting with asurface at constant r and moving the cutoff slightly so that r → r + (cid:15) ( x ) . For any surface,we can always locally find coordinates such that the surface is located at fixed value of r and the metric is in the ADM gauge, so there is no loss of generality in this assumption.Then δ (cid:15) S = (cid:90) (cid:15) ( x ) ∂ r g µν ∂S∂g µν = (cid:90) √ g(cid:15) ( x ) ∂ r g µν π µν = 2 (cid:90) √ g(cid:15) ( x ) (cid:18) π µν π µν − d − π ρρ ) (cid:19) , (3.7)where we used equation (3.5) for the radial dependence of the metric in terms of momenta.Interestingly, this is precisely of T ¯ T form, but with T and ¯ T defined with respect to themetric variations of the finite surface, not the boundary at infinity. A more coordinate in-dependent way of stating the result is that as we move a surface in a given AdS background,we turn on a local T ¯ T -deformation with a coefficient given by the orthogonal distance be-tween the original and deformed surface. If we could relate the local T and ¯ T on a givensurface to the T and ¯ T as defined at infinity, we could integrate these flow equations andwrite the final result in terms of a finite T ¯ T deformation of the theory at infinity. We leavea further exploration of this interesting question to future work but thinking of finite T ¯ T deformations in terms of a change in the boundary conditions for the metric we expect itto involve the linearized Einstein equations around the background [54].Clearly, using (3.4) for d = 3 the variation of the action vanishes if equation (2.28) issatisfied. As is clear from the Hamiltonian constraint, this condition can also be phrasedas R ( d − + ( d − d −
2) = 0 , i.e. the boundary surface has constant scalar curvature.Therefore, to optimize the complexity of the process we should use constant scalar curvaturesurfaces; the metric on a Euclidean AdS d − manifold precisely has the required scalarcurvature. This is consistent with the observation that complexity is minimized if we take t i = t f and consider a purely radial surface at the t = t f constant timeslice in section 2. So far the discussion has used the standard bulk AdS action without the inclusion of addi-tional counterterms, which would render the on-shell value of the action finite as one takesthe surface to the asymptotic boundary. As alluded to in the beginning, in the originalappearance of Liouville theory as defining path integral complexity, the absence of the vol-ume counterterm was important. Here we briefly discuss what happens if we add a volume– 12 –erm for the boundary surface with an arbitrary coefficient. In our discussion of the on-shellvalue of the action, it would add an extra term S c.t = − λ (cid:90) d x √ g = − λ (cid:90) dtdx (cid:112) ρ ρ . (3.8)Adding the counterterm modifies the field equations to ρ (1 + ˙ ρ ) (cid:18) ( ρ ¨ ρ + 1 + ρ ) − λ (cid:112) ρ (cid:18) ρ ¨ ρ + 1 + ˙ ρ (cid:19)(cid:19) = 0 (3.9)We can also reconsider the flow equations in the presence of the volume counterterm.Denoting the volume counterterm as S vol = − λ ( d − (cid:90) ∂M √ g (3.10)so that λ = 1 is precisely the counterterm which would cancel the volume divergence nearthe AdS boundary, we now introduce ˜ π µν = π µν − λ ( d − g µν so that these are precisely thecanonical momenta in the presence of the extra boundary volume term. The Hamiltonianconstraint can be rewritten as H = R ( d − + (1 − λ )( d − d −
2) + ˜ π µν ˜ π µν − d − π ρρ ) − λ ˜ π ρρ = 0 (3.11)We can now consider two types of flows. We can consider the variation of the action as wechange the radial surface in a given background, but we can also consider the variation ofthe action as we perform a conformal rescaling of the metric on the radial surface. In d = 3 ,the latter does not require an adjustment of the bulk geometry, but in higher dimensionsthis is no longer true. It is therefore not clear whether conformal rescalings of the inducedmetric on the boundary surface are in general compatible with keeping the initial and finalstates fixed in d > . Regardless, the change of the action under the first type of flow nowreads δ (cid:15) S = (cid:90) (cid:15) ( x ) ∂ r g µν ∂ ˜ S∂g µν = (cid:90) √ g(cid:15) ( x ) ∂ r g µν ˜ π µν = 2 (cid:90) √ g(cid:15) ( x ) (cid:18) ˜ π µν ˜ π µν − d − π ρρ ) − λ ˜ π ρρ (cid:19) (3.12)and for the second type of flow with δg µν = (cid:15) ( x ) g µν δ (cid:15) ˜ S = (cid:90) (cid:15) ( x ) g µν ∂ ˜ S∂g µν = (cid:90) √ g(cid:15) ( x )˜ π ρρ = 12 λ (cid:90) √ g(cid:15) ( x ) (cid:18) R ( d − + (1 − λ )( d − d −
2) + ˜ π µν ˜ π µν − d − π ρρ ) (cid:19) . (3.13)– 13 –e see that both flows take the form of T ¯ T deformations, with various extra terms suchas the scalar curvature and the trace of the stress tensor. Just as in the case withoutcounterterm ( λ = 0 ) it would be interesting to integrate these flows to finite flows startingat the AdS boundary.The first flow is extremized when the surface obeys R ( d − + ( d − d − − λ ( d − K = 0 (3.14)which still holds for an AdS d − equal time slice in AdS d . As expected, for our setup (3.14) isequivalent to (3.9). The second flow, on the other hand, is extremized when K = λ ( d − .This does not have an extremum for an AdS d − equal time slice in AdS d unless λ = 0 .Moreover, as we indicated above, it is not clear whether the initial state and final state arekept fixed along the flow, and therefore the precise interpretation of this flow is somewhatunclear. In any case, it would be interesting to explore whether surfaces obeying (3.14) or K = λ ( d − have the potential to define a new notion of complexity.Finally, we notice that it is also possible to add higher order counterterms, but forthose the connection to T ¯ T deformations becomes more complicated. The bulk computation from section 2 and illustrated in figure 1 can be viewed in the lightof the results from the previous section as the following non-unitary circuit acting on theinitial state | (cid:105) z f = P exp (cid:20) i (cid:90) t f t i dt ( H ρ ( t ) + ˙ ρ [ T ¯ T ] ρ ( t ) ) (cid:21) | (cid:105) z i . (4.1)In the above expression, H ρ ( t ) represents Euclidean time evolution in a CFT with cutoffspecified by ρ ( t ) , which in the bulk would correspond to moving in the t -direction whilekeeping ρ ( t ) fixed. The other term represents the operator that implements a change inthe scale of the theory, which we have schematically denoted by [ T ¯ T ] ρ ( t ) . In the bulk thiswould correspond to changing ρ while keeping t fixed. What is important is that each layerof (4.1) in general uses operators from a different theory.Following gate counting ideas of [13–15, 22], due to spatial homogeneity of our setupone might be tempted to regard H ρ ( t ) and [ T ¯ T ] ρ ( t ) as two classes of elementary operationswith ρ ( t ) playing two independent roles. The first role played by ρ ( t ) lies in labelling theelementary operations we are using (as already mentioned, both H ρ ( t ) and [ T ¯ T ] ρ ( t ) aredifferent operators for each value of ρ ( t ) ). The second role stems from dt ˙ ρ being relatedto the number of times the operator [ T ¯ T ] ρ ( t ) is applied in a given layer of the circuit.Correspondingly, H ρ ( t ) is applied simply dt number of times.As a result, a naïve way of counting insertions of H ( ρ t ) would be (cid:82) t f t i dt and (cid:82) t f t i dt | ˙ ρ | when it comes to [ T ¯ T ] ρ ( t ) with the total number of insertions being the sum of the twocontributions. Note that since ρ ( t ) is just a label, in principle the contribution from everylayer can be weighted by some non-negative function of ρ ( t ) – a penalty factor that weights– 14 –ardness of applications of particular transformations. The above logic was based on an L norm of the vector { , ˙ ρ } , but in principle any norm would do.However, our proposal is to view the action (2.8) or (2.10) as a cost function for thecircuit (4.1). It seems quite straightforward to associate a suitable weight to H ρ ( t ) . Ifwe imagine a CFT with a fixed cutoff or lattice spacing ∼ ρ , and we count the number oflattice points in a given Euclidean volume (which we interpret as suitable tensor operations),then we immediately obtain an answer proportional to (cid:82) dtdxρ − = (cid:82) dtV x ρ − , which isindeed proportional to the potential term in the action (2.10). Within the logic outlined inthe previous paragraph, this corresponds to using L norm with the penalty factor equalto V x ρ − .The second term in the action (2.10) is tricky to interpret within the framework of [13–15]. Following the above logic, one would be naturally inclined to associate this term withthe presence of [ T ¯ T ] ρ ( t ) insertions in the circuit (4.1), however, this is difficult. Writing therelevant contribution as V x | arctan ˙ ρ | ρ | ˙ ρ | one does not recognize a standard penalty factor infront of | ˙ ρ | within an L norm. To this end, the penalty factor is not supposed to knowabout what circuits do at other layers, and its dependence on ˙ ρ via arctan ˙ ρ induces sucha dependence. This is very much reminiscent of the discussion in [22] about viewing theLiouville action as a bona fide cost function.Following this thread, our interpretation of the “potential” and “kinetic” terms in thegravity action (2.10) is quite similar to an earlier discussion about the aforementionedqualitative interpretation of the Liouville action as a complexity of a tensor network [19, 20].In our study, the “potential” term studies euclideons, whereas the “kinetic” term is associatedwith changes in cut-off and might be related to isometries or full layers of MERA.Note also that the circuit (4.1) is very similar to the expression usually written downfor cMERA [12], which takes the form of a path-ordered exponent of infinitesimal unitariesand dilatation operators. An immediate issue with this expression is the precise meaningof the operator [ T ¯ T ] ρ ( t ) in the CFT, as we have seen that a careful construction of thisoperator requires one to integrate the flow equations from the AdS boundary to the bulksurface. If we knew the precise definition of this operator in the CFT, we could try toassign a number to this path-ordered exponent, for example by computing the length of thetrajectory in a suitable space of operators. It is not inconceivable that such a computationis possible, as we know the commutation relations between the Hamiltonian and the T ¯ T operator in the CFT, and it would be interesting to explore this further.Furthermore, it is intriguing to note that arctan ˙ ρ is the angle that the surface makes inthe z, t -plane, so it looks like this term is measuring the amount of effort it takes to rotatethe surface in the z, t -plane. It would be very interesting to understand this observationbetter.Since our interpretation of the gravity action in terms of gate counting is more on thequalitative side, in the following we want to propose another way of arriving at (2.8). In the above, we have often tacitly assumed that the information about the bulk surface z = ρ ( t ) is encoded locally in the boundary theory. However, as our discussion of flows– 15 –hows, it is highly questionable whether this is a reasonable assumption. A better way toencode the information of the surface z = ρ ( t ) in the boundary theory is through pairs ofpoints ( t ( t ) , t ( t )) (with x =0) on the boundary, such that the geodesic that starts at t ( t ) and ends at t ( t ) is tangent to the bulk surface at the point ( z = ρ ( t ) , t, , see figure 2. Figure 2 . We can parametrize a generic bulk curve ρ ( t ) by the pairs of boundary points ( t ( t ) , t ( t )) ,such that a bulk geodesic connecting these two points is tangent to the bulk curve at z = ρ ( t ) . Thisway, the profile ρ ( t ) is encoded as a path in kinematic space, the space of bulk geodesics. This construction has the benefit of being covariant, and viewing Euclidean time asanother spatial coordinate, these geodesics encode precisely the entanglement wedges whichtouch the surface but do not cross it. In other words, they precisely encode the informationabout those regions of spacetime we try to omit in our bulk path integral construction.One can ask whether there is a natural geometry associated to the pairs of points of thistype, and the answer is yes. Conformal invariance produces a natural metric on the spaceof pairs of points, also known as kinematic space [30]. For the case at hand it is given upto an undetermined constant prefactor by the 2d de Sitter metric ds ks = − dt dt ( t − t ) . (4.2)In the spirit of defining complexity by assigning a metric to a group of transforma-tions [13–15], we can now ask what the length of the path in this geometry associated with ρ ( t ) is. To compute it explicitly, we need the explicit form of t ( t ) and t ( t ) . These aregiven by t , ( t ) = t + ρ ˙ ρ ± ρ (cid:112) ˙ ρ + 1 . (4.3)Consider now the action S = (cid:90) dxρ ds ks ( t ) , (4.4)– 16 –here we included the coordinate x in units of the cutoff ρ , and the distance ds obtainedfrom (4.2) upon inserting (4.3). This results in S = (cid:90) dtdx (cid:12)(cid:12)(cid:12)(cid:12) ρ ¨ ρ + (1 + ˙ ρ ) ρ (1 + ˙ ρ ) (cid:12)(cid:12)(cid:12)(cid:12) , (4.5)which agrees precisely with the bulk action in the form (2.8) as long as ¨ ρ ≥ − ρ − (1 +˙ ρ ) . This is related to the fact that the kinematic space is a Lorentzian manifold and thecondition in question is the one that one moves there along a timelike path.This strongly suggests that the relevant circuit geometry for these types of finite bulksurface computations is a version of kinematic space. Note that on-shell, (4.5) vanishesexactly for the semi-circular arcs that solve (2.11), as they are also geodesics in AdS-space.In other words, for these solutions the path traversed in kinematic space shrinks to a point.We will come back to this point in section 5. Let us mention that generalizing this kinematicspace consideration to more complicated geometries is not obvious as minimal geodesics donot necessarily penetrate the whole spacetime. In the case of geodesics computing theentanglement entropy, these are entanglement shadows [55] and they appear, for example,in the case of double-sided black holes.Finally, let us mention that the relation between kinematic space and complexity wasexplored earlier in two different instances in [56] and [57], however, these proposals aredistinct from ours and use a standard entanglement-based kinematic space. In this paper we have discussed the idea that finite spacetime regions correspond to quantumcircuits with a complexity given by the on-shell value of the gravitational action. We foundseveral intriguing results, but much more work remains to be done to put our results ona firmer footing. Perhaps the most pressing of these is to find a more precise circuitinterpretation along the lines we discussed in the previous section. Some other obviousaspects to explore are the impact of counterterms, higher derivative terms and matterfields on the computations. We list some further open issues and ideas for future workbelow.
Global AdS and trivial initial state
It is straightforward to repeat our computations in global AdS, as opposed to the Poincarépatch of AdS. There are no major conceptual changes, except that we can now choosea smooth surface without the need to pick an initial state. Stated differently, we havechosen a trivial initial state in the CFT with infinite cutoff, or equivalently, we have a no-boundary type construction of the state at later times. The optimization proceeds exactlyas in Poincaré coordinates, and complexity is optimized if the spacetime region collapsesonto an equal time disc, with complexity proportional to the volume of the disc.
Choice of time slice
States in gravity are not associated to a unique time slice. In some sense, states areassociated to complete causal diamonds in the Lorentzian signature. There is therefore– 17 –o canonical choice of initial and final time slices which bound the spacetime region. Forstationary spacetimes, it seems reasonable to take fixed time slices, but it is not clear whatto do for more general spacetimes. Our proposal is to use slices with vanishing extrinsiccurvature K , as these are covariantly defined, and lead to a vanishing contribution of theGibbons-Hawking boundary term. This choice will give rise to corner contributions, butthose seem unavoidable for any choice, and as we saw in the case where the spacetimeregion collapses to a disc, they are a feature rather than a bug. One could, alternatively,try to extend the spacetime region indefinitely into the past or future, and subtract thecontributions of these semi-infinite pieces later, but this procedure has exactly the sameambiguity in it. It would be interesting to have a better understanding of the various choicesone can make for the future and past boundaries and what the implications of these choicesare. It might for example also be natural to take time slices with constant scalar curvatureas complexity is locally extremized for that choice of time slice. A finite deformation of Liouville
The effective action for a finite bounded region in AdS is of independent interest, as itcomputes the partition function for the CFT with a cutoff and particular curved manifolds.In the limit where the bounded region approaches the boundary of AdS, we recover the CFTpartition function (including divergent terms), which in 2d is given by the Polyakov action,and in conformal gauge becomes the Liouville action. It is interesting to see that (2.20) isapparently a finitely deformed version of Liouville theory for a space-independent Liouvillefield (cid:37) ( u ) = exp( − φ ( u )) . If we insert this and take φ ( u ) → ∞ , we indeed recover Liouvilletheory, see also the discussion in [38]. One might think that (2.20) describes a finite T ¯ T deformation of Liouville theory and it would be interesting to make that connection precise.A possible route to address this matter is to cast the on-shell action in a form involving thescalar curvature of the cutoff surface, which seems feasible in the ADM formalism, comparewith Polyakov’s non-local form of the effective action, and identify the relevant deformation. Other dimensions
In higher dimensions, the computation is more or less the same, and we will not presentthe relevant details here. An exception is AdS , where after a partial integration the actionbecomes proportional to I ∼ (cid:82) dtρ − , which suggests that the coarse graining operationhas no cost associated to it. This is perhaps a consequence of the peculiar nature of theAdS /CFT correspondence, where AdS is merely dual to the ground states of the CFT and is of limited relevance. It would be interesting to repeat the computation for JTgravity [58, 59] and to compare to flows in spaces of Hamiltonians, which are much easierto control than T ¯ T deformations in higher dimensions, and might lead to a more precisegate counting interpretation.In T ¯ T -deformed quantum mechanics, the Hamiltonian is mapped to a function of it-self [60, 61], H (cid:55)→ f ( H ) . (5.1)– 18 –uppose we wish to quantify the complexity of the circuit created by Euclidean time evo-lution, U ( t ) = exp( − H t ) . (5.2)Given U ( t ) of the undeformed theory, we in principle know the operator U f ( t ) of thedeformed theory, U f ( t ) = exp( − tf ( − ∂ t log U ( t ))) , (5.3)but even given this simple relation, it is not clear how to relate the complexities of U ( t ) and U f ( t ) .One puzzle arises when combining complexity and holographic T ¯ T . Increasing the T ¯ T deformation is dual to bringing in the cutoff surface, which reduces the volume of themaximal boundary anchored volume slice, and by the CV conjecture would say impliesthat the complexity of the state similarly reduces. Assuming the volume of the maximalvolume bulk slice monotonically decreases as the boundary is brought in, then this impliesthat the complexity is monotonically decreasing too under the flow. Is there somethingspecial about the holographic T ¯ T deformation such that the complexity of geometric statesmonotonically decreases under its flow, or is the CV proposal incorrect at finite cutoff? Lorentzian geometries
We could repeat our computation in Lorentzian signature, but then several new featuresarise. First, there is the qualitative difference of whether or not z = ρ ( t ) describes a timelikeor spacelike surface. In the timelike case the region is delimited by the lightfronts t = ± z ,and the on-shell action takes the form I = 2 κ (cid:90) ∂M d x ρ ¨ ρ + (1 − ˙ ρ ) ρ (1 − ˙ ρ ) (5.4)Integration by parts yields I = 2 κ (cid:90) ∂M d x (cid:20) ρ −
12 ( log (1 − ˙ ρ ) − log (1 + ˙ ρ )) (cid:21) (5.5)It is easy to see that this expression diverges in the limit ˙ ρ → ± , i.e. when the surface istangent to the lightfronts t = ± z . It is possible to properly define gravitational actions inthe presence of null boundaries [45], and in order for our proposal to make sense we shouldmodify it so that in the null limit it approaches the answer of [45]. With this modificationwe would then be in agreement with the complexity equals action proposal.If we start with a spacelike surface and start optimizing, then there are two possibilities,we either find a constant scalar curvature surface, or we encounter the same null boundariesas in the previous timelike case. Which of the two optimizes the gravitational action dependson whether we choose + S of − S to optimize, and since it is e iS which appears in the pathintegral, it is not a priori clear which one of the two we should take in the absence of aprecise gate counting interpretation. One would be inclined though to pick the sign suchthat the term proportional to /ρ and independent of ˙ ρ has a positive sign so that timeevolution at fixed cutoff has positive complexity. Regardless, we seem to universally findeither constant scalar curvature surfaces or null surfaces as extrema of the extremizationproblem. – 19 – TZ black hole
Based on general arguments, there are several key features that measures of complexityshould possess, such as aforementioned asymptotic linear growth in time in black holebackgrounds and the switchback effect [62]. As a first heuristic check, we can investigateconstant scalar curvature slices (with the right value for the scalar curvature) in the BTZblack hole. In Kruskal coordinates, the BTZ black hole looks like ds = − du dv (1 + uv ) + (1 − uv ) (1 + uv ) dφ (5.6)with the asymptotic AdS boundaries located at uv = − . The relevant constant curvatureslices turn out to take the simple form uv + λu + µv − . Consider the special case uv + ( u + v ) / sinh ξ − which intersects the boundary at u = e ξ , v = − e − ξ and onthe other boundary of the eternal black hole at the point obtained by interchanging u, v .Shifting ξ is therefore like shifting time upwards on both asymptotic boundaries, and weare interested in the behavior at late ξ . The midpoint of the slice is at u = v = tanh ξ/ ,which indeed moves towards the singularity at uv = 1 as ξ → ∞ . Therefore, constantscalar curvature slices do correctly probe the growing Einstein-Rosen bridge. The optimalspacetime region in this case is the region between the maximal volume slice with K = 0 (which is where we propose to end the spacetime region, as discussed above) and theconstant curvature slice. We have not computed the gravitational action associated to thisregion but expect it to reproduce the required late time growth. As the maximal volumeslice is also explicitly computable [63], we leave this interesting exercise to future work. Higher curvature corrections
We have proposed that the complexity of the circuit that maps between ground states intwo EFTs with different finite cutoffs is given by the on-shell gravitational action. Consid-ering the effect of higher curvature corrections on the gravitational action, and thereforecomplexity, would be a natural extension to this proposal. Higher curvature corrections tothe holographic complexity=volume proposal were recently studied in [64]. The simplestexample to study is Gauss-Bonnet (GB) gravity in AdS . The GB correction L GB = R − R µν R µν + R µνρσ R µνρσ (5.7)is a constant in the vacuum AdS geometry we have studied in this work, so simply rescalesthe contribution from the volume of M to the on-shell action. In general, however, weexpect non-trivial contributions from the correction to the boundary Lagrangian, and fromthe bulk Lagrangian L GB for perturbed geometries. Seeing whether the resulting on-shellaction could be interpreted as the complexity of a circuit in T ¯ T -deformed CFT with a (cid:54) = c would be interesting.To our knowledge the generalization of holographic T ¯ T to higher curvature gravity hasnot been studied. There is general method to determine the deformation to a holographicCFT needed to put the dual gravity theory at finite cutoff [50, 51]. To start we note thatin a theory with only one dimensionful parameter µ (assuming it to have no spacetime– 20 –ependence) such as a T ¯ T deformed CFT, the effective action changes under infinitesimallength rescaling as ∆ µ µ∂ µ W = (cid:90) d d x √ γ (cid:104) Tr T (cid:105) (5.8)where ∆ µ is the scaling dimension of µ . The flow of the boundary action under changing µ is thus determined by the trace of the boundary stress tensor, which is related to thebulk Brown-York stress tensor through T ij = r d − c ˜ T ij . The essence of the method is to usethe Hamiltonian constraint to eliminate extrinsic curvature terms in the Brown-York stresstensor appearing in (5.8). Following this procedure for pure Einstein gravity gives the flowequation ∂W∂µ = (cid:90) d d x √ γ (cid:18) T ij T ij − d − T ii ) (cid:19) (5.9)This is the field theory deformation flow equation for Einstein gravity at finite cutoff. Whenwe add higher curvature corrections, the method of substituting out extrinsic curvatureterms in the Brown-York stress tensor for powers of the stress tensor using the Hamiltonianconstraint does not change, but the Brown-York stress tensor and the radial Hamiltonianconstraint do [65], and the final result will differ from (5.9). Relation to entanglement wedge reconstruction
Finally, it is interesting to ask whether it is a mathematical coincidence or not that thesolutions (2.12) to (2.11) are semicircular arcs, just like geodesics in the Poincaré AdS-background, even though in our setup they describe the embeddings of co-dimension onesurfaces, not entanglement entropy. In section 4.2, we addressed this question from akinematic space perspective. There, we showed that a generic bulk profile ρ ( t ) can as wellbe described as a path in kinematic space (neglecting a possible x -dependence as throughoutthe paper), but equations (2.11) and (2.12) enforce this path to shrink down to a lengthzero point for fixed boundary conditions t i , t f , z i , z f .Assuming that our results can be generalised to the Lorentzian case, it will be inter-esting to investigate whether these semicircular embeddings are a consequence of entan-glement wedge reconstruction or entanglement wedge nesting [66–68]. For example, onemight imagine that the optimization of the path integral (subject to the boundary condi-tions t i , t f , z i , z f ) drives the bulk surface as deep into the bulk as possible until it leavesthe entanglement wedge and cannot move further. In the fully Lorentzian case, it wouldbe interesting to determine whether this only reproduces the extremal surface at the edgeof the entanglement wedge, or also its null boundaries. At least extremal area surfaces areexpected to play a role of special importance in quantum gravity for quite generic reasons[69]. As said in section 2.2, such semicircular boundaries have also been found in [49] assolutions of an AdS/BCFT toy-model with non-trivial matter content living in the world-volume of the end of the world brane. In that paper, it was shown as a consequence ofphysical energy conditions on the worldvolume matter fields that the corresponding branesgenerally have to be extremal surface barriers in the sense of [70], and the branes withsemicircular embedding profile where precisely the ones staying as close to the boundary aspossible without violating this condition. – 21 –hile all these different observations seem to point to a nontrivial quantum informa-tion theoretic reason for the embeddings (2.12) being the correct ones, we leave furtherinvestigation of this to future work. Besides extending our work to the Lorentzian case,investigating similar setups in higher dimensions or on nontrivial backgrounds such as BTZmay yield further insight. Acknowledgments
We would like to thank Shira Chapman and Ignacio Reyes for being involved in the initialpart of this collaboration and Bartek Czech for useful discussions. The Gravity, QuantumFields and Information (GQFI) group at AEI is supported by the Alexander von Hum-boldt Foundation and the Federal Ministry for Education and Research through the SofjaKovalevskaja Award. AR is supported by the Stichting Nederlandse Wetenschappelijk On-derzoek Instituten (NWO-I). JdB is supported by the European Research Council underthe European Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agree-ment ADG 834878. The work of MF was supported by the Polish National Science Centre(NCN) grant 2017/24/C/ST2/00469 until November 16th 2020 and through the grantsSEV-2016-0597 and PGC2018-095976-B-C21 from MCIU/AEI/FEDER, UE since Decem-ber 1st 2020. RC acknowledges support from the Netherlands Organisation for ScientificResearch (NWO-I). SH holds a fellowship from the Ramon Areces Foundation (Spain).
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