*1.3. Observables*

Many observables have been proposed to better understand quantum gravity. One possible consequence of quantum gravity that can arise in different ways and was discussed at length in our workshop is non-locality. However, the meaning of non-locality depends heavily on the context it is discussed in, and it is not completely understood how these notions are related. Even in classical general relativity, locality is a subtle concept. This is due to diffeomorphism invariance—the fundamental symmetry of general relativity, which encodes the independence of physics under the choice of coordinate systems. As a result, only diffeomorphism invariant quantities are physically relevant. For example, this condition severely complicates the definition of local subsystems in general relativity. Indeed, splitting systems into subsystems, e.g., to compute the entanglement entropy between them, is highly topical, ye<sup>t</sup> in gravity it must be defined in a diffeomorphism invariant way. Similar to the situation in (lattice) gauge theories, this can be achieved by introducing new degrees of freedom and symmetries on the boundary separating the subsystems [67].

### 1.3.1. Non-Locality in Quantum Gravity

One facet of non-locality discussed at the workshop was in the context of effective quantum field theories. The essential idea put forward by Knorr and Saueressig is to define an effective continuum theory for CDT [49], where the terms and couplings in the effective field theory are chosen by comparing expectation values of the 3-volume covariance in both theories (for one specific value of parameters in CDT). The theory they define contains non-local terms, in the sense that the associated operator is a product of the field (and its derivative) evaluated at different points in space-time2.

It remains an open question whether similar relations hold once more observables are considered or when the parameters in CDT are changed. Nevertheless, the potential implications of such non-local terms are intriguing and it will be interesting to explore whether similar effective quantum field theories can be derived from other discrete quantum gravity approaches. Non-locality can also arise in discrete theories. Spin foam models and CDT can be regarded as (at least initially) local theories, since they assign amplitudes to each building block of the triangulation, where these amplitudes only depend on the variables attached to said building block. A priori only neighbouring building blocks are "interacting" via the variables they are sharing. However, under coarse graining/renormalization, generically non-local interactions will arise involving building blocks beyond nearest neighbours.

Conversely, in causal set theory, non-locality is built into the theory from the beginning. A causal set element is connected to all causally related elements, with nearest neighbours corresponding to elements close to the light-cone. Since the light-cone in a generic space-time is non-compact, a causal set element, in an infinite causal set, would have infinitely many nearest neighbours. Additional non-locality also arises through a regularization parameter in the d'Alembertian for a scalar field [68–71]. This parameter is introduced to dampen fluctuations in the discrete theory, in effect smearing the derivative operator over several layers. This non-locality of a scalar field on a causal set, and of the causal set itself gives rise to phenomenological predictions which can be tested [72–74]. On the other hand, we do not have any current observational evidence of non-locality; hence, any non-local effects need to remain weak enough to not conflict with this. For example, when modelling the motion of a point particle through a causal set as traversing along the longest path, this introduces momentum diffusion above these stringent limits [75].

<sup>2</sup> More precisely, the operator is the inverse d'Alembertian squared sandwiched by two Ricci scalars.

### 1.3.2. Summing over Topology

Another point of contention between different theories is the question: If we sum over different geometries, should we hold their topology fixed, or should we sum over all possible topologies? The first time this problem arose was in non-critical string theory, where the theory of strings requires a complete sum over all possible topologies of 2D geometries. If this is naively implemented, it leads to a dominance by topologies with many handles and a divergent sum [76]. However, modern models, and a suitable renormalization, make it possible to calculate the sum over topologies. In CDT, the topology of space and time individually are fixed, the simulations restrict spatial topology to either be a sphere or a torus, and, for numerical reasons, time is treated as periodic in most simulations (With the exception of the results in [77], which broadly agree with the results found using periodic time.). Causal set theory, on the other hand, does not restrict the path integral in this way, it does not even require all partial orders in the path integral to be geometries.

In loop quantum gravity and other canonical formulations of quantum gravity, the topology of space-time is usually fixed, since space-time is assumed to be globally hyperbolic in order to define the (3 + 1) split. Indeed, describing time evolution in a non-globally hyperbolic space-time is rather cumbersome. In spin foams, the issue is more subtle: any 2-complex that is compatible with given boundary data is in principle allowed. This concerns non-trivial 4-dimensional topology but also includes the possibility to change spatial topology between initial and final state. Whether one should sum over different topologies is debated in the literature [29] and depends on the interpretation of the spin foam. In the refinement approach [20], where the goal is to define a consistent theory across discretisations, one usually does not consider topology change. Since the goal is to identify the same physical process across discretisations, it is natural to fix the topology of the boundary states (If the boundary consists of several parts, e.g., initial and final state, their respective topologies can differ, but are kept fixed. Moreover, it is not clear how to embed states of differing topologies into a common discretisation.). The topology in the bulk is usually kept fixed as well, mostly for convenience. On the other hand, it is frequently argued that one should sum over all possible spin foams, i.e., all 2-complexes including all topologies. The most suitable framework to consider this are group field theories [27], in which the sum over spin foams appears as a perturbative expansion of Feynman diagrams generated by the action of the theory. Whether this theory is well-defined depends on whether it is renormalizable as a quantum field theory [28]. In the related tensor models, the sum over topologies is well defined. In particular, they can identify which topologies dominate in the perturbative expansion in the so-called large-*N* limit, where *N* is the dimension of the tensor indices [7,9].

## 1.3.3. Quantum Cosmology

One of the most promising routes for quantum gravity to make contact with experiments is cosmology. Quantum gravity effects may be revealed by future high precision experiments, e.g., the dynamics of the early universe might have left imprints in the cosmic microwave background. Indeed, it is an exciting prospect to see how quantum gravity can reshape our understanding of the origin of the universe, and whether it can augment, replace, or derive the current paradigm of inflation [78], which successfully explains the (almost) homogeneity, isotropy and flatness of our universe.

However, contact to this cosmological sector is difficult for non-perturbative theories of quantum gravity. While it is challenging for many approaches to define or model such a subsector of the theory, it is even more so to show how such a sector (plus fluctuations around it) could emerge dynamically. This difficulty is exemplified by the difference between loop quantum cosmology and cosmology in loop quantum gravity: in loop quantum cosmology, the system is symmetry reduced, e.g., to a homogeneous and isotropic universe, at the classical level before quantization. In loop quantum gravity, this symmetry reduction is to be implemented at the quantum level and explored in different directions. The early symmetry reduction in loop quantum cosmology simplifies calculations

considerably and allows for interesting tests. For example, loop quantum cosmology (LQC) with an inflationary phase after the bounce predicts changes in the cosmic microwave background power spectrum compared to other inflationary models [79]. However, there are strong arguments that the early reduction in symmetry might remove crucial information from the theory; hence, to confirm the results of loop quantum cosmology, it is vitally important to derive symmetry reduced models from the full theory.

Antonia Zipfel gave a nice overview of the current status of the relation between LQC and LQG: In loop quantum gravity, this can be tackled directly by looking for suitably defined cosmological subsectors [80,81], e.g., by translating homogeneity and isotropy conditions on the phase space of general relativity to loop quantum gravity [82]. While this procedure is mathematically robust and relates well to the full theory, it is hard to implement in a given model and only approximately recreates the symmetry. A different idea is to study the evolution of coherent states, e.g., peaked on homogeneous and isotropic space-times [83,84]. Using these states, one can derive an effective Hamiltonian, as the expectation value of the constraint with respect to these semi-classical states. However, it is a priori not clear whether these coherent states are preserved under evolution. Another attempt to connect loop quantum gravity and loop quantum cosmology is called quantum reduced loop gravity [85], which relies on the kinematical construction of full loop quantum gravity. Then, a gauge fixing that restricts the spatial metric (and triads) to be diagonal is implemented. The symmetry reduction happens at the quantum level, where one only considers a dynamics which preserves the diagonal metric condition. Yet another perspective on the difference between imposing symmetry reduction before or after quantization is given in the context of general relativity in radial gauge [86]. In [87], the two methods are closely compared, beginning at the level of the phase space in order to identify the variables in the reduced theory with suitable phase space functions in the full theory. This analysis is continued at the quantum level, where the subsectors of the theories and the properties of operators can be compared. While a qualitative match between both theories is achieved at the kinematical level, one finds quantitative and state-dependent differences in the scaling behaviour of operators and mismatches in their commutators. This suggests that the identification of subsectors needs to be improved further.

This problem of how and where symmetry should be imposed arises in all non-perturbative approaches to quantum gravity and is dealt with in different ways. Reducing the symmetry classically and then quantizing leads to interesting toy models; however, it is important to test results obtained thus against results arising in the full non-perturbative regime. In particular, it would be fascinating if a non-perturbative path integral might give rise to a ground state that has some cosmological features. This is the case in causal dynamical triangulations, where the ground state of simulations in one phase shares some characteristics with Euclidean de Sitter space. The average volume profile of the 3-volumes, centered in time, measured in simulations assuming a spherical topology of space, matches the volume profile of Euclidean de Sitter [88]. In addition to the 3-volume, the authors also studied the covariance between 3-volumes at different time steps, which is highly peaked for the same time and drops off quickly for larger time steps. The spectral dimension in this phase of the simulations also points at 4-dimensional behavior [89]. This work has been extended to toroidal topology, where the volume profile becomes constant [90]. It can be argued that this creation of a de Sitter volume profile is a non-perturbative emergence of cosmology [91].

In group field theory, the emergence of a homogeneous state is tackled by considering condensate states [92], presented in detail by Steffen Gielen: the excitations, e.g., above a Fock vacuum, are interpreted as discrete "atoms of space-time". The heuristic idea of how a smooth, continuous space-time can emerge from this microscopic description is a hydrodynamic one. A large collection of these space-time atoms undergo a phase transition and condense similar to Bose Einstein condensates [93], such that a macroscopic, effective dynamics emerges from their collective behaviour. In the context of cosmology, one considers a gas of equilateral, uncorrelated building blocks that describe weakly interacting Bose Einstein condensates. In this setup, one can compute expectation values of observables, e.g., the volume of these building blocks. The dynamics is truncated to the classical equations of motion of the mean field of the condensate, analogous to the Gross–Pitaevskii equation of a Bose Einstein condensate. Remarkably, in this setting, the expectation values of observables satisfy effective Friedmann equations [94].

Another possible effect of non-perturbative quantum gravity on cosmology are discreteness effects. In theories where the discreteness is considered as fundamental, such as causal set theory, effects of the discreteness can lead to observable effects and explain certain phenomena. For example, the randomness inherent in the discrete causal sets can give rise to a cosmological constant of the correct order of magnitude [4]. Since this cosmological constant is no longer constant, it can vary over the age of the universe. This idea has given rise to phenomenological models that can match the standard model of cosmology and agrees with many of the observables known therein [19].

Another fascinating possibility of cosmological characteristics arising from non-perturbative dynamics was hinted at in the model system of the 2D orders in causal set theory. The closest causal set equivalent to the Hartle–Hawking wave function for the early universe is to simulate 2D orders that are fixed to begin with a single element and to end in an *n* element anti-chain, the closest causal set equivalent to a spatial hyper surface of fixed volume. In this model, the configurations with the highest likelihood are those that expand rapidly and are very homogeneous [95]. While this is a highly simplified model, it shows the possibility to generate features similar to those that have been observed in our universe from non-perturbative dynamics.

### 1.3.4. Measuring Dimension

The dimension of space-time is a familiar concept in general relativity. For each point of a *d*-dimensional manifold, we can find a small open region, which we can smoothly map to an open region of R(1,(*d*−<sup>1</sup>)) (for Lorentzian signature). As a property of the (topological) manifold, we will refer to this as the topological dimension. While this notion is intuitive in continuum gravity, it is not obvious how to define a dimension in (discrete) quantum gravity. In CDT or spin foam gravity, it is natural to regard the dimension of the fundamental building blocks as the topological dimension. However, whether this "dimension" also emerges on large scales is unclear: 4D hypercubes arranged in one long line appear one dimensional on large scales or some building blocks might be degenerate, i.e., possess vanishing 4-volume. Furthermore, in causal set theory, one cannot associate a dimension to discrete space-time events. These difficulties have motivated the definition and investigation of effective dimension measures that allow us to infer the dimension of space-time, e.g., via simulations, and potential physical consequences. Indeed, it is an important first test for any approach to quantum gravity, whether these generalized notions of dimension agree with our expectation of four space-time dimensions on large scales.

Moreover, this measured dimension may change with the scale at which space-time is probed, which is further motivation to study such observables. In general, there are several ways to define measures of dimension and all of them have different implications. One example is the Hausdorff dimension [96]. This notion of dimension can be assigned to all metric spaces, via the so-called Hausdorff measure. It is usually defined for a positive, real parameter *d* and considers all possible open coverings of the metric space, such that the diameter of each open subset is smaller than . The Hausdorff measure with respect to *d* and is then given by the infimum of the sum of all the diameters of the subsets to the power *d*. To find the Hausdorff dimension, we send → 0 and find the infimum *d* for which the Hausdorff measure vanishes, which is directly related to how quickly volumes of sets shrink with decreasing diameter. In quantum gravity, but also random geometries, this notion of dimension is frequently inferred from the exponential growth of volumes with respect to the radius. Then, the Hausdorff dimension is defined as the logarithmic derivative of the volume with respect to the radius, which can change as a function of the radius.

One definition of a scale dependent dimension prevalent in quantum gravity is the spectral dimension. After first rising to prominence in 4D simulations of CDT [89,97], it was also explored in many

other theories, e.g., asymptotic safety [98], Hoˇrava-Lifshitz gravity [99], causal set theory [100,101], loop quantum gravity [102,103], spin foams [64] and noncommutative geometry [104,105]. This dimension measure is related to studying the heat equation/a diffusion process on space-time. It crucially depends on the Laplace operator and its spectral properties. More precisely, this dimension measure is defined as the logarithmic derivative of the heat kernel. For calculations in discrete theories, the heat kernel can also be considered as the return probability of a random walker and thus calculated as an average over a sample of random walks. As a result, the spectral dimension encodes how space-time is ordered and thus might reveal interesting consequences for how matter propagates on this geometry; however, it is not obvious how to find this connection. Indeed, Giulia Gubitosi pointed out that the spectral dimension is problematic as a quantity of interest, since it cannot be measured experimentally. In most approaches, it is implemented purely on the geometry, since most computer simulations currently do not include matter. However, all currently conceived experimental measurements of space-time need test particles/test fields. Hence, to define practically observable quantities, we will need to work with matter. As an alternative, she suggested the thermal dimension, which tries to define a temperature based on the scaling of thermodynamic properties of matter [106]. This proposal is based on the dimension dependence in the Stefan Boltzmann law, describing the thermal radiation of a theory. While this is interesting in principle, and they show how it works in Hoˇrava–Lifshitz gravity, where a preferred frame is available, the implementation for a non-perturbative theory is more challenging. Defining a temperature and other thermodynamic quantities in a background independent way can be complicated, and a nice discussion of these problems in the context of GFT is given in [107].

### 1.3.5. Other Observables

In addition to these larger overarching themes that were discussed at lengths and from the perspective of different theories, there were also some interesting observables discussed that are not ye<sup>t</sup> explored in many theories. One such promising observable is the so-called quantum Ricci curvature [108]; the idea underlying this observable is the following: consider two points in a *d*-dimensional manifold with geodesic distance *δ* and imagine each of them to be surrounded by a sphere of radius . The points on the sphere are parametrized by a vector from the center to the sphere itself. Points on the two spheres are related by parallel transporting a vector from one sphere to the other along the geodesic connecting the centers. The average distance of points on these two spheres depends on the Ricci curvature 2-form (evaluated for the tangent vector of the geodesic connecting the centers), e.g., if the Ricci curvature is positive the average distance is smaller than *δ*. Since this concept is based on parallel transport, it is not straightforwardly applicable to the simplicial geometries underlying (C)DT. Instead, one considers the average distance between all points on the spheres, allowing the authors to identify the sign of curvature in constantly curved geometries. Moreover, they have tested it for 2D-(E)DT with spherical topology revealing a positively curved geometry modeled as a 5D sphere emphasizing the highly non-classical and fractal geometries in this model [109]. It will be interesting to see the behaviour of this observable in 4D CDT and whether it can be translated to other approaches of quantum gravity.

Another interesting route to explore is holography, more precisely the deep relation between a theory in the bulk and the theory on the boundary. This is most prominently represented in the continuum by the infamous AdS/CFT correspondence [110]. Naturally, it is an interesting question to ask whether these ideas can be generalized to non-perturbative approaches to quantum gravity and what the corresponding boundary theories might be. A very interesting calculation has been performed for the Ponzano–Regge model of 3D spin foams, studying the partition function and dual boundary theory of the twisted solid torus [111,112] (see also a similar calculation for linearized Regge calculus [113]). Strikingly, the results are consistent with results from perturbative quantum field theory in the continuum [114,115] and the characters of the BMS group are recovered. In addition, there have been several derivations for holographic entanglement entropy, more precisely the Ryu–Takayanagi

formula for Renyi entropy, where the entropy (of the boundary theory) associated with a boundary subsystem is proportional to the minimal bulk area attached to this section of the boundary [116,117].

### *1.4. Numerical Methods in Quantum Gravity*

Using physical intuition to develop our algorithms can lead to massive improvements in speed. At our workshop, we were introduced to two algorithms employing this, the chimera algorithm for numerical loop quantum cosmology and the transfer matrix algorithm for CDT.

### 1.4.1. The Chimera Algorithm

Parampreet Singh told us about the chimera algorithm developed in loop quantum cosmology. One of the key features of loop quantum cosmology is the resolution of the Big Bang singularity at the origin of the universe via a Big Bounce [15]. The vital dynamics responsible for this result is encoded in the quantum Hamiltonian constraint, which is a difference equation with uniform discretisation in volume. Indeed, for small volumes and large space-time curvature, these dynamics significantly deviate from the classical dynamics given by a Wheeler–DeWitt differential equation. However, for large space-time volume and small curvature, the quantum and classical dynamics agree very well. This is the fundamental idea underlying the chimera algorithm [118].

Difference equations, which describe the evolution in the deep quantum regime, are much more costly to compute compared to ordinary differential equations. This issue is emphasized as soon as the quantum states, whose evolution is studied, are not sharply peaked on classical configurations. Thus, the chimera algorithm introduces a hybrid lattice, where quantum evolution is only performed at small volumes and classical dynamics take over for large volumes. The intermediate region is carefully chosen for the results to match. That way, the numerical costs are drastically reduced and can be spent instead on studying the evolution of more general quantum states [118]. It will be interesting to explore whether this idea of a hybrid algorithm can be adapted in other approaches as well, e.g., spin foam models.

### 1.4.2. Transfer Matrix Approach

The transfer matrix approach is well known as a method to solve e.g., the Ising model in 2D. It is an analytic method based on splitting a problem into layers, e.g., time slices, calculating the dynamics of a single layer and then combining consecutive layers by convolution. This method also works to analytically solve CDT in two dimensions. In higher dimensions, CDT can only be explored using computer simulations; however, the foliated structure still makes it a prime candidate for the transfer matrix approach. Andrzej Görlich explained how this insight and a clever numerical implementation of the transfer matrix approach were used in [119]. In his algorithm, he measures the transfer matrix between slices of fixed size, such that he only has to simulate two slices of geometry, instead of the entire universe. This allows for more focused measurements, in particular improving the precision in measuring off-diagonal elements of the transition amplitude immensely. The transfer matrix approach also had another, unexpected, payoff in showing that what was before considered a single de Sitter like phase, called phase C, splits into two different phases: one that is de Sitter like, and one with alternating large and small spatial slices, called the bifurcation phase [119].

### 1.4.3. Markov Chain Monte Carlo Simulations

Other improvements in code are less about understanding the physical situation of the problem, and more about understanding the idiosyncrasies of a particular simulation. The most used tool to calculate a path integral using computer simulations are Markov Chain Monte Carlo (MCMC) simulations. How this algorithm is applied in their respective approaches to quantum gravity was explained by Andrzej Görlich, Jack Laiho, and Sumati Surya. In these, the ensemble of geometries is sampled with a frequency proportional to the weight of the configurations in the path integral, which makes it easy to calculate averages of observables directly from the simulations. A Markov chain is a chain in which the likelihood to transition between two states only depends on these two states. One algorithm to generate such a chain is the Metropolis–Hastings algorithm. This algorithm generates a Markov Chain by proposing a new state as a function of the old one. The function proposing these states depends on the theory used, e.g., in dynamical triangulations it is given by Pachner moves, which locally change the triangulation [120]. The probability to accept a proposed move then depends on the weight of the geometry in the path integral, given by *<sup>e</sup>*<sup>−</sup>*S*, with *S* the action of the theory. One important feature is that a new state will always be accepted if it has a higher weight, but even states with a lower weight can still be accepted with a probability proportional to *eSold*−*Snew* . This makes it possible to prove that, if the moves are ergodic, the Metropolis–Hastings algorithm will find a global minimum of the action, if run sufficiently long. Unfortunately, the convergence towards this can be very slow, particularly close to phase transitions, since most proposed moves will have a very low probability of being accepted. This is known as critical slowing down and is related to the divergence of the correlation length arising there.

### 1.4.4. Parallel Rejection

One algorithm to overcome critical slowing down is the parallel rejection algorithm, discussed in Andrzej Görlich's talk. In general, MCMC simulations are difficult to parallelize, particular in gravity systems, since changes in the value of the action are non-local, hence proposed moves are not independent and need to be calculated sequentially. In practice, this means that most simulations are "naively parallelized" by just starting the simulations for several different points in the phase diagram, different parameter values, at the same time on different cores. Parallel rejection is an algorithm that does actual parallelization, for at least some regions of the phase diagram, where it can substantially speed up the algorithm. In regions of the parameter space where the acceptance rate of moves is particularly low, parallel rejection proposes and calculates multiple moves at the same time, on different cores. Once one of them is accepted (which can be ∼1% or less of proposed moves), the geometry is updated and the parallel rejection restarted. This can drastically reduce the time in which the code remains in a given configuration in these regions of the phase diagram [55].

### 1.4.5. Adopting Methods from Other Fields

Another interesting option is to start using tools from other areas of science, in particular from computer science. There are many techniques that are solidly established in other fields but have not been widely adapted in numerical quantum gravity yet. For example, in QCD, the default algorithm for simulations is not the Metropolis–Hastings algorithm; instead, the algorithms in use are hybrid Monte Carlo [121] explained in Jack Laiho's talk. In these, the step of proposing a new configuration is guided by a supplementary Hamiltonian function. This Hamiltonian function is defined with respect to the probability distribution we wish to sample from and introduces fictitious momenta. While the momenta are randomly updated, a step in the configuration variables is chosen via a Metropolis algorithm with respect to the Hamiltonian equations of motion, which results in a faster convergence of results. This "Hamiltonian" is not be confused with an energy functional or the Hamiltonian constraint in gravity and serves the purpose to optimize the updating of configurations. The drawback of this method is that it requires continuous configurations, which makes it unsuitable for many proposals in quantum gravity.

Parallel tempering, discussed by Andrzej Görlich, also known as replica exchange MCMC sampling, is very useful when the configurations generated, e.g., from a Metropolis algorithm, are highly auto-correlated, that is correlated with previously generated configurations [122]. Such correlated systems may suffer from critical slowing down, in which the system is unlikely to leave said configuration via the proposed updates. To avoid this, the principle idea of parallel tempering is to start several processes with different model parameters and exchange the configurations at some point. That way, regions in configuration space that are rarely explored for certain parameters become accessible, improving the accuracy of the simulation. Often, it is proposed that the parameters

only slightly vary. The probability to exchange the configurations has to satisfy the detailed balance condition. Crucially, this algorithm significantly reduces the auto-correlation time, i.e., the time it takes for configurations of the same Markov chain to become statistically independent.

In recent years, deep learning has emerged as a powerful method to analyze and search for patterns in large amounts of data. Image recognition is a particularly impressive example. Naturally, we would like to apply these methods to quantum gravity, e.g., to examine data generated in Monte Carlo simulations. In a nutshell, the idea of deep learning is to find an optimal function that quickly returns a desired output from a given large input. Deep neural networks are usually modelled to have an in- and output layer, chosen according to data and desired output. Between these layers, one implements several hidden layers, where each neuron in a hidden layer is connected to all neurons in the previous and following layer. These connections simply encode linear algebra operations on the data. Then, some of the data is used for training, i.e., these linear algebra operations ge<sup>t</sup> optimized to minimize a cost function. In supervised learning, where we know the desired result for a given sample, we would optimize the neural network to reproduce the already known answer. This is a particularly powerful approach when it comes to classification problems, e.g., recognizing handwriting or in quantum gravity it might help us to sort geometries with different properties. This approach bears an enormous potential, ye<sup>t</sup> comes with some obvious drawbacks. Indeed, it is not obvious how to design a deep neural network that can successfully analyze a given data set. Moreover, even once we have successfully trained a neural network, it might not be obvious what the computer has learned, limiting our interpretation and understanding of the problem. Another problem is that, at least for the easiest to apply algorithms with the clearest outcomes, called supervised learning, we need to label the data set beforehand. This was beautifully demonstrated by Will Cunningham in his talk: he uses causal sets of known dimension, either *d* = 2, 3 or 4, to train a neural network to determine the dimension of the causal set. This is an interesting toy model, which demonstrates the opportunity and the challenge of machine learning at the same time. The characterization of the causal sets he obtained through the algorithm could have been done equally well using many tools that have been developed in causal set theory, e.g., the Myrrheim–Meyer dimension [123] or the interval abundance [34], which are fast and simple to use. On the other hand, these tools took time to develop and relied on our deep understanding of the problem, while the computer was obviously not aware of these and still able to solve it.

In general, quantum gravity, in particular in approaches that heavily use Monte Carlo simulations, offers many opportunities to apply machine learning. It will not always be possible to label the large data sets generated by Monte Carlo simulations and unravel all of their "hidden" information. Hence, using machine learning to search for structure within, and to possibly identify new observables, is a worthwhile endeavour.
