Contents

Complexity change under conformal transformations in

[.5ex] Mario Flory111mflory@th.if.uj.edu.pl, Nina Miekley222nina.miekley@physik.uni-wuerzburg.de

Institute of Physics, Jagiellonian University,

Łojasiewicza 11, 30-348 Kraków, Poland

Lehrstuhl für Theoretische Physik III, Institut für Theoretische Physik und Astrophysik,

Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

Using the volume proposal, we compute the change of complexity of holographic states caused by a small conformal transformation in . This computation is done perturbatively to second order. We give a general result and discuss some of its properties. As operators generating such conformal transformations can be explicitly constructed in CFT terms, these results might allow for a comparison between holographic methods of defining and computing computational complexity and purely field-theoretic proposals in the future.

1 Introduction

Suppose that a scientist in possession of a quantum computer is given a specific task like, for example, applying a certain operator to an initial reference state in order to obtain the resulting state

(1)

In [1, 2] it was proposed to define the quantum computational complexity (short complexity from here on) of the operator by geometric methods, defining a distance measure on the space of unitary operators and equating the complexity of , , as the (minimal) distance between and the identity operator according to this distance function. Equivalently, the complexity of with respect to , , could be defined to be the minimal complexity of any operator such that .333 might be equal to , or it may be a more efficient operator in terms of complexity. Hence in general. The idea behind this is that in order to implement the operation, the programmer of the quantum computer would have to subdivide into a product of allowed universal gates that implement the operation step by step, until the end-state agrees with the desired outcome at least within a certain error tolerance. The notion of complexity is meant to count the minimal number of gates that the programmer would have to utilise even when using an optimal program. This definition of the complexity would hence neccessarily depend on the following explicit or implicit choices (see also [3, 4]):

  • A choice of the reference state , which often is assumed to be a simple product state, without spacial entanglement. This should not be confused with the groundstate of a given physical system.

  • A choice of the set of allowed gates , such that the operation can be decomposed as

  • …within a specified error tolerance (in some distance measure between and ).

Starting with [5, 6, 7], the idea of computational complexity has begun to see rising interest from the AdS/CFT community. A tentative entry to the holographic dictionary called the volume proposal was proposed and motivated in [6, 7, 8, 9, 10]. According to this, the complexity of a field theory state with a smooth holographic dual geometry should be measured by the volumes of certain spacelike extremal co-dimension one bulk hypersurfaces, i.e.

(2)

wherein a length scale has to be introduced into equation (2) for dimensional reasons which is usually picked to be the AdS scale [9, 10, 11, 12].

Interestingly, computational complexity is not the only field theory quantity that has been proposed to be holographically dual to the volumes of extremal co-dimension one hypersurfaces in the bulk. In [13], it was argued that the volume of an extremal spacelike co-dimension one hypersurface is approximately dual to a quantity called fidelity susceptibility [14, 15] according to the formula

(3)

where is an order one factor, is the AdS radius and determines the dimension such that the AdS space is dimensional. Given two normalised states and depending on one parameter , is defined as444The name fidelity susceptibility derives from the fact that is called the fidelity.

(4)

and measures the distance between the two states in a sense, hence it may also be reffered to as the quantum information metric [16]. The derivation of (3) in [13] (see also [17, 18]) assumed that two states and are the ground states of a theory allowing for a holographic dual, and that the difference is the result of a perturbation of the Hamiltonian by with an exactly marginal operator . The bulk spacetime dual to this field theory problem is a Janus solution [19, 20], but as shown in [13] this geometry can be approximated by a simpler spacetime with a probe defect brane embedded into it, leading to (3). This proposal has been utilised for holographic calculations in [21, 22, 23, 24], and our results concerning changes induced by infinitesimal conformal transformations, to be derived in section 3, may also have an interesting physical interpretation from the perspective of fidelity susceptibility, however in this paper we will focus on bulk volumes as a holographic dual to computational complexity.

The necessity to include a lengthscale in the definition (2) of holographic complexity was considered unsatisfactory by some, and so [11, 12] proposed the competing action proposal

(5)

wherein is the bulk action over a certain (co-dimension zero) bulk region, the Wheeler-DeWitt patch.

Both the volume- and action proposals for holographic complexity where subsequently used for a number of holographic investigations. Specific topics of interest where the boundary terms required to calculate the action in (5) correctly [25], the time dependence of complexity [26, 27, 28] (especially with respect to the so called Lloyd’s bound, see however [29, 30]), generalisations of holographic complexity to mixed states [31, 32, 33, 34, 4], RG-flows [24, 35], and many others. Interesting connections have been made between holographic complexity and kinematic space approaches [36, 34] as well as Liouville theory in two dimensions [37, 38, 39, 40].555Although a comparably young topic, the literature on holographic complexity has indeed grown to a considerable size by know. We apologize to everyone who feels they where unjustly left out above.

This amount of progress on the holographic side has also led to increased efforts to provide better and more concrete definitions of quantum computational complexity in quantum mechanical or even quantum field theory contexts [41, 3, 37, 38, 42, 43, 44, 45, 46]. However, in the proposals (2) and (5) for holographic complexity calculations it is not clear what the relevant reference state, gate set and error tolerance are to be. If one or both of these proposals for holographic complexity are to be correct, then these choices need to be somehow implicit in the holographic dictionary.

This is the main motivation of our present work: We will, focusing on the volume proposal (2) for now, calculate how the complexity of a given state of a two dimensional conformal field theory (CFT) with smooth holographic dual changes under a conformal transformation. Such conformal transformations can of course be applied to any two dimensional CFT, irrespectively of whether the central charge is large or not, or whether the CFT satisfies any other requirement for having a holographic dual. Hence we believe these results will be useful in the future for comparisons between the holographic and field theory proposals for complexity, and this may help to elucidate the choices of gateset, reference state and error tolerance that are implicit in the holographic proposals.666 See also [44, 47] for recent papers comparing results from the action- and volume-proposal and discussing what they may imply for a possible field theory definition of complexity.

The structure of our paper is as follows: In section 2, we will review how to implement a (small) conformal transformation in AdS/CFT. This will provide the general setup of our work and fix the notation. Then, based on the volume proposal (2), we will calculate in section 3 how holographic complexity changes under small conformal transformations. A few illustrative examples will be discussed in section 4, and we close in section 5 with a conclusion and outlook. Some technical details about the implementation of conformal transformations on the field theory side will be presented in appendix A.

Note: While in the final stages of preparing this draft, we became aware of the upcoming paper [48] that seems to share some common ideas with ours, however approaching the topic from the field theory side.

2 Gravity setup

We consider the vacuum state of a two-dimensional CFT with a classical holographic dual. The bulk geometry is given by AdS space which in the Poincaré-patch is written as

(6)

where is our radial coordinate and are the light-cone coordinates of the field theory. In these coordinates, the boundary of AdS is at and the Poincaré horizon at . Placing a cutoff at (), it is possible to holographically calculate the expectation value of the energy-momentum tensor of the boundary CFT by the method of [49], which gives a vanishing result as expected.

Although Einstein-Hilbert gravity in three dimensions is trivial in the sense of having no propoagating degrees of freedom, there is a surprising number of vacuum solutions which were derived in [50] (see also [51]). The reason is that these solutions, called Bañados geometries, can be derived from (6) by diffeomorphisms which are global in the sense that they act nontrivially near the boundary, such that while the resulting metric is still asymptotically AdS in the appropriate sense, the holographic energy-momentum tensor has changed. Despite all being locally isometric to AdS in the bulk, these geometries are hence dual to different states of the dual field theory. In [51], these transformations were hence termed solution generating diffeomorphisms (SGDs), and in this section we will explain these transformations in detail, following the outline and notation of [51].

In order to apply a SGD to the geometry described by (6), we perform a coordinate transformation777One virtue of the SGDs formulated in [50] was that they preserved a specific form of the metric tensor. As can be seen from equation (8), this is not the case for the SGDs as used in [51] and herein. As can be read-off from the holographic energy-momentum tensor (10), both types of SGD implement conformal transformations on the CFT-state. The apparent difference between the SGDs comes merely from the fact that the SGDs of [50] and the corresponding SGDs of [51] differ by a bulk-diffeomorphism that acts trivially at the boundary. We mostly follow the convention and notation of [51] as therein the authors also explicitly discuss how to implement such SGDs on two-sided black holes and thereby construct an infinite family of purifications of, e.g., the thermal state. We hope to investigate this class of solutions from the viewpoint of holographic complexity in the future.

(7a)
(7b)
(7c)

The metric in the tilted coordinates is [51]

(8a)
(8b)

In principle, general relativity is invariant under coordinate transformations. However, these SGDs fall off slowly at the boundary and are asymptotically non-trivial gauge transformation. The new cutoff is at

(9)

with the field theory UV-cutoff . This shows the non-triviality of the coordinate transformation: in terms of the old coordinates, the cutoff surface is wrapped and different to the one which describes the vacuum state (i.e. ). This is shown in Figure 1.

Cutoff surfaces
Figure 1: A conformal diagram of the Poincaré-patch of AdS. The vertical line is the asymptotic boundary while the two diagonal lines are the two Poincaré-horizons where . The two cutoff surfaces and are shown as dashed (red) and dotted (blue) lines, respectively.

We started with a CFT state with vanishing energy-momentum tensor. In the deformed state, with the cutoff defined as in (9), we have [51]

(10a)
(10b)
(10c)

which is precisely what we expect for the change of the energy-momentum tensor after applying a conformal transformation to the groundstate, see appendix A. The SGDs hence implement conformal transformations on the boundary theory, which is also evident from the change of the induced metric of the cutoff surface under these transformations. In field theory terms, the conformal transformation with functions hence maps the ground state to a new state

(11)

with known (commuting) unitary operators and , see [51] and appendix A for an explicit construction in the case of a small conformal transformation.

In the following, we consider a small SGD, i.e.

(12a)
(12b)

with the expansion parameter . The metric defined by the line element (8) can then be expanded in orders of and we find

(13)

where we have switched from lightcone coordinates to standard coordinates on the boundary. The components of the energy-momentum tensor read

(14a)
(14b)
(14c)

For later, it will be convenient to introduce the Fourier transforms of the functions :

(15a)
(15b)
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3 Complexity = Volume

In this section we calculate the complexity of states (11) for small conformal transformations (12) using the Complexity=Volume (CV) proposal (2). For this, we have to calculate the maximal volume of a co-dimension one spacelike slice with fixed boundary conditions, i.e. we have to maximize

(18)

with the determinant of the induced metric depending on the embedding function . This spacelike slice is anchored at the boundary at a constant time slice of the coordinates . For a small conformal transformation, we can expand the embedding888For , the metric (13) is just the Poincaré metric, and the appropriate embedding for a maximal volume slice anchored to a constant time-slice on the boundary is just given by .

(19)

Just as the metric functions (15), we can write the embedding function as an (inverse) Fourier transform backgroundcolor=Gray!30!White,bordercolor=Gray!30!White,caption=Sign convention for ,inlinebackgroundcolor=Gray!30!White,bordercolor=Gray!30!White,caption=Sign convention for ,inlinetodo: backgroundcolor=Gray!30!White,bordercolor=Gray!30!White,caption=Sign convention for ,inlineIt is always or . Since the sign convention is different for the embedding, goes together with .

(20)

With (19), we can now expand the integrand of (18) in orders of , and the lowest nontrivial order leads to equations of motion for of the form

(21a)
(21b)

Therefore, we have a second order partial differential equation for the Fourier coefficients . The boundary conditions are the fixed behaviour at the asymptotic boundary (i.e. ) and at the Poincaré-horizon at we demand that does not diverge to keep the perturbative expansion meaningful. This would mean that the embedding function does not leave the Poincaré-patch through one of the null-segments of the Poincaré-horizon in figure 1, but instead goes into the corner on the left hand side of the figure.999One can avoid the problem of having to specify boundary conditions at the Poincaré-horizon altogether by mapping the problem to global AdS and solving the equations for the embedding there. Then, it would suffice to give Dirichlet boundary conditions at the full boundary circle of global AdS.

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Using these conditions to solve (21b), we obtain a piece-wise smooth result101010In all our calculations, we implicitly assume that integrals are sufficiently well behaved to interchange integration order or integration and differentiation where necessary. Note that for all the specific examples to be discussed in section 4, it is possible to analytically calculate the embedding and confirm that the equation (21a) as well as the boundary conditions are satisfied.

(23)

Now, let us turn to the volume. Expanding (18) up to second order in , we obtain111111 The first-order term is where the first term vanishes because of extremality of the -order embedding in the Poincaré-metric. It directly follows from this that the second-order term only depends on the first order terms in the changes of embedding and metric.

(24)

with

(25a)
(25b)
(25c)
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We can already point out the following observations: First of all, we see that the change in complexity (see (2)) due to the operators being applied to the groundstate is always independent of the cutoff , i.e. UV finite.121212In the terms of [52], the complexity of formation is finite. Secondly, we find

(28)

for any . So any (with small ) applied on the groundstate (described by the Poincaré-metric) will only increase the complexity with respect to the reference state. We hence see that among the geometries (13) in a neighbourhood around the groundstate, the groundstate is the least complex with respect to the reference state . This result has an interesting physical interpretation: The operators only map states with smooth dual geometry to other states with smooth dual geometry. However, it is commonly assumed that the reference state will be a state without spatial entanglement, and such states cannot have a smooth dual geometry [53]. So in the total space of states, the states of the form in a neighbourhood of form a set in which locally minimises the complexity with respect to the reference state. See figure 2 for an illustrative sketch of the space of states.

Figure 2: The space of states with the reference state , the groundstate and the state for the generator of a small conformal transformation. The red dashed lines signify the complexities of the states and with respect to the reference state .

The way in which complexity is defined as a distance measure between states can be very abstract and does not need to employ a Riemannian metric, for example it may also be based on Finslerian geometry [1, 2]. In fact, although one would commonly assume that for a distance measure between states and the symmetry property has to be satisfied [10], it was suggested in [45] this requirement might have to be abandoned for definitions of complexity. E.g. one could imagine defining the complexity with respect to a gate-set that includes a given gate, but not its inverse. Then, operators will in general not have the same complexity as their inverse operators. However, from the result (25) it is evident that is invariant under , which to first order in corresponds to , see appendix A. So when applied to the vacuum state , the two operators and lead (to leading order in ) to a change of complexity by the same amount. This is not true when these operators are applied to generic states.

One of the few kinds of geometric intuition that we can definitely rely on when dealing with complexities is the triangle inequality [1, 2]

(29)

hence

(30)

So if the proportionality factor in equation (2) could be fixed, our results would lead to quantitative lower bounds on the complexities of the field theory operators . However, as we see from (24), the righthand-side of the above bound will be of order , while for small the change of the state due to the action of will be of order , and hence we would intuitively assume that and will be of order also. If this is true, the bound of (30) is not very strict. It would be interesting to extend our studies to operators being applied to general Bañados geometries without the need of taking small, however we leave this for the future.

The result (25) can be rewritten suggestively as two time-independent pieces, which only depend on or , and a time-dependent mixed term

(31a)
(31b)
(31c)

The reason why this is interesting is that the energy-momentum tensor of the two-dimensional CFT can be understood in terms of left- and right-moving modes. Setting either or will hence result in a configuration with translational invariance in one of the light-cone coordinates . As we integrate over the spatial direction to obtain the holographic complexity, the result for the change in complexity will then be time-independent and given by either or in (31a). If both , we get a time-dependent result.

The result (31) allows for some general insights into the behaviour of under simple manipulations of the functions . For example, we find that under rescalings of arguments, ,

(32a)
(32b)
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Another interesting thing to look at is what happens under addition of functions , as at first order in this corresponds to carrying out two small conformal transformations after each other (see appendix A.3). We obtain

(36a)
(36b)
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In the next section, we will proceed to pick some illustrative examples of which allow for particularly simple analytical expressions to be derived for and .

4 Examples

Example 1

To begin, let us choose131313For simplicity of notation, we will drop the tildes over the coordinates in this section.