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In quantum gravity, several approaches have been proposed until now for the quantum description of discrete geometries. These theoretical frameworks include loop quantum gravity, causal dynamical triangulations, causal sets, quantum graphity, and energetic spin networks. Most of these approaches describe discrete spaces as homogeneous network manifolds. Here we define Complex Quantum Network Manifolds (CQNM) describing the evolution of quantum network states, and constructed from growing simplicial complexes of dimension . We show that in d = 2 CQNM are homogeneous networks while for d > 2 they are scale-free i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the self-organized evolution of CQNM quantum statistics emerge spontaneously. Here we define the generalized degrees associated with the -faces of the -dimensional CQNMs, and we show that the statistics of these generalized degrees can either follow Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimension of the -faces.


I. INTRODUCTION
In this supplementary information we give the details of the derivation discussed in the main text. In Sec. II we define Complex Quantum Network Manifolds (CQNMs); in Sec. III we discuss the relation between the CQNM and the evolution of quantum network states; finally in Sec. IV we define the generalized degrees, and we derive the generalized degree distribution in the case β = 0 and β > 0.

II. COMPLEX QUANTUM NETWORK MANIFOLDS
Here we present the non-equilibrium dynamics of Complex Quantum Network Manifolds (CQNMs). This dynamics is inspired by biological evolution and self-organized models and generates discrete manifolds formed by simplicial complexes of dimension d. In particular CQNMs are formed by gluing d-simplices along (d − 1)-faces, in order that each (d − 1)-face belongs at most to two d-dimensional simplices.
Let us indicate with S d,δ the set of all δ-faces with δ < d belonging to the d-dimensional CQNMs. A (d − 1)-face α ∈ S d,d−1 is "saturated" if it belongs to two simplices of dimension d, whereas it is "unsaturated" if it belongs only to a single d-dimensional simplex. We will assign a variable ξ α = 0, 1 to each face α ∈ S d,d−1 , indicating either that the face is unsaturated (ξ α = 1) or that the face is saturated (ξ α = 0). Moreover, to each node i we assign an energy i drawn from a distribution g( ) and quenched during the evolution of the network. To every δ-face α ∈ S d,δ we associate an energy α given by the sum of the energy of the nodes that belong to α, At time t = 1 the CQNM is formed by a single d-dimensional simplex. At each time t > 1 we add a simplex of dimension d to an unsaturated (d − 1)-face α ∈ S d,d−1 chosen with probability Π α given by where β is a parameter of the model called inverse temperature and Z is a normalization sum given by Having chosen the (d − 1)-face α, we glue to it a new d-dimensional complex containing all the nodes of the face α plus the new node i. It follows that the new node i is linked to each node j belonging to α.
Since at time t = 1 the number of nodes in the CQNM is N (1) = d + 1, and at each time we add a new additional node, the total number of nodes is N (t) = t + d. The CQNM evolution up to time t is fully determined by the sequences the energy of the node added to the CQNM at time t > 1, i with i ≤ d + 1 indicates the energy of an initial node i of the CQNM, and α t indicates the (d − 1)-face to which the new d-dimensional complex is added at time t . A similar dynamics for simplicial complexes of dimension d = 2 has been proposed in [1].

III. QUANTUM NETWORK STATES
A. The network Hilbert space Following an approach similar to the one used in "Quantum Graphity" and related models [2][3][4][5], in this section we associate an Hilbert space H tot to a simplicial complex of N nodes formed by gluing together d-dimensional simplices along (d − 1)-faces. The Hilbert space H tot is given by indicating the maximum number of (d − 1)-faces in a network of N nodes.
Here an Hilbert space H node is associated to each possible node i of the simplicial complex, with a α = 0, 1. We indicate with c † α , c α respectively the fermionic creation and annihilation operators acting on this space. Finally the Hilbert spaceH d,d−1 associated to a (d−1)-face α is the Hilbert space of a fermionic oscillator with basis {|n α }, with n α = 0, 1. We indicate with h † α , h α respectively the fermionic creation and annihilation operators acting on this space.
A quantum network state can therefore be decomposed as As already proposed in the literature [2,3], here we assume that the quantum network state follows a Markovian evolution. In particular we assume that at time t = 1 the state is given by where Z(1) is fixed by the normalization condition ψ(1)|ψ(1) = 1. The quantum network state is updated at each time t > 1 according to the unitary transformation with the unitary operator U t given by where F(i, α) indicates the set of all the (d − 1)-faces α formed by the node i and a subset of the nodes in α ∈ Q d,d−1 (N ) and Z(t) is fixed by the normalization condition C. Path integral characterizing the quantum network state evolution The quantity Z(t) is a path integral over CQNM evolutions determined by the sequences In fact, using the normalization condition in Eq. (9) and the evolution of the quantum network state given by Eqs. (7), (8) we get where the terms a α (t) and n α (t) that appear in Eq. (11) can be expressed in terms of the We note that Z(t) can also be interpreted as the partition function of the statistical mechanics problem over possible evolutions of CQNM. In fact, the CQNM evolution is of a given evolution is given by where is given by Eq. (11) and Z(t) is fixed by the condition Eq.
For example, in a CQNM of dimension d = 2, the generalized degree k 2,1 (α) is the number of triangles incident to a link α while the generalized degree k 2,0 (α) indicates the number of triangles incident to a node α. Similarly in a CQNM of dimension d = 3, the generalized degrees k 3,2 , k 3,1 and k 3,0 indicate the number of tetrahedra incident respectively to a triangular face, a link or a node.

B. Distribution of Generalized Degrees for β = 0
Let us define the probability π d,δ (α) that a new d-dimensional simplex is attached to a δface α. Since each d-dimensional simplex is attached to a random unsaturated (d − 1)−face, and the number of such faces is (d − 1)t, we have that for δ = d − 1 Let us now observe that each δ-face, with δ < d−1, which has generalized degree k d,δ (α) = k, In fact, is is easy to check that a δ−face with generalized degree k d,δ = k = 1 is incident to simplex, a number d − δ − 1 of unsaturated (d − 1)−faces are added to the δ-face while a previously unsaturated (d − 1)-face incident to it becomes saturated. Therefore the number of (d − 1)-unsaturated faces incident to a δ-face of generalized degree k d,δ = k follows Eq.
Moreover, the average number n d,δ (k) of δ-faces of generalized degree k d,δ = k that increases their generalized degree by one at a generic time t > 1, is given by where δ x,y indicates the Kronecker delta, while for δ < d − 1 it is given by Using Eqs. (18) − (19) and the master equation approach [11], it is possible to derive the exact distribution for the generalized degrees. We indicate with N t d,δ (k) the average number of δ-faces that at time t have generalized degree k d,δ = k during the temporal evolution of a d-dimensional CQNM. The master equation [11] for N t d,δ (k) reads For δ = d − 2 instead, we find an exponential distribution, i.e.
Finally for 0 ≤ δ < d − 2 we have the distribution From Eq. (22) it follows that for 0 ≤ δ < d−2 and k 1 the generalized degree distribution follows a power-law with exponent γ d,δ , i.e. and Therefore the generalized degree distribution P d,δ (k) given by Eq. (22) is scale-free, i.e.
it has diverging second moment (k d,δ ) 2 , as long as γ d,δ ∈ (2, 3]. This implies that the generalized degree distribution is scale-free for C. Distribution of Generalized Degrees for β > 0 In the case β > 0 the distribution of the generalized degrees P d,δ (k) are convolutions of the conditional probabilities P d,δ (k| ) that δ-faces with energy have given generalized degree k d,δ = k. Here we derive the distribution of generalized degrees P d,δ (k) for different values of δ and d as a function of the inverse temperature β. The procedure for finding these distributions is similar for every value of δ.
First we will determine the master equations [11] for the average number N t d,δ (k| ) of δfaces of energy that have generalized degree k at time t. Then we will solve these equations, imposing the scaling, valid in general for growing networks, given by N d,δ (k| ) = m d,δ ρ d,δ ( )tP d,δ (k| ), where ρ d,δ ( ) is the probability that a δ-face has energy and m d,δ are the number of δ-faces added at each time to the CQNM. The master equations will also depend on self-consistent parameters µ d,δ called chemical potentials that need to satisfy selfconsistent equations for the derivation to hold.
Let us consider first the case δ = d − 1. The average number N t d,δ (k| ) of δ = (d − 1)-faces of energy that at time t have generalized degrees k d,δ = k follows the master equation given where n F ( , µ d,d−1 ) is the Fermi-Dirac occupation number with chemical potential µ d,d−1 , i.e.
Using Eqs. (28), and performing the average k d,d−1 | of the generalized degree k d,d−1 over (d − 1)-faces of energy , one can easily find that This result shows that the average of generalized degrees of (d − 1)-faces of energy is determined by the Fermi-Dirac statistics with chemical potential µ d,d−1 .
Let us now consider the case δ = d − 2. In this case we assume that asymptotically in time we can define the chemical potential µ d,d−2 as In this assumption, the master equations [11] for the average number N t d,d−2 (k| ) of (d − 2)faces with energy and generalized degree k ≥ 1, read where m d,d−2 = d(d − 1)/2 is the number of (d − 2)−faces added at each time t to the CQNM, ρ d,d−2 ( ) is the probability that such faces have energy , and δ x,y indicates the Kronecker delta. In the large network limit t 1 we observe that N t d,d−2 (k| ) tm d,d−2 ρ d,d−2 ( )P d,d−2 (k| ) where P d,d−2 (k| ) indicates the probability that a (d − 2)−face of energy has generalized degree k. Solving Eq. (32) we get, for k ≥ 1. Therefore, summing over all the values of the energy of the nodes we get the full degree distribution P (k) for k ≥ 1. Using Eqs. (34), and performing the average k d,d−2 | of the generalized degree k d,d−2 over (d − 2)-faces of energy , one can easily find that where we have indicated with n Z ( , µ d,d−2 ) the Boltzmann distribution This result shows that the average of generalized degrees of (d − 2)-faces of energy is determined by the Boltzmann statistics with chemical potential µ d,d−2 .
Let us finally consider the case δ < d − 2. In this case we assume that asymptotically in time we can define the chemical potential µ d,δ given by Assuming that the chemical potential µ d,δ exists, the master equations [11] for the average number N t d,δ (k| ) of δ-faces with energy and generalized degree k ≥ 1 read the probability that such faces have energy , and δ x,y indicates the Kronecker delta. In the large network limit t 1 we observe that N t d,δ (k| ) tm d,δ ρ d,δ ( )P d,δ (k| ),where P k,δ (k| ) is the probability that a δ−face of energy has generalized degree k. Solving Eq. (38) we get, Using Eqs. (39), and performing the average k d,δ | of the generalized degree k d,δ over δ-faces of energy , one can easily find that This result shows that the average of generalized degrees of δ-faces of energy is determined by the Bose-Einstein statistics with chemical potential µ d,δ . Finally the chemical potentials µ d,δ , if they exist, can be found self-consistently by imposing the condition For small values of β, these equations have a solution that converges for β → 0 to the β = 0 solution discussed in the previous subsection. As the value of β increases it is possible that the chemical potentials µ d,δ become ill-defined and do not exist. In this case different phase transitions can occur. For the case d = 2 these transitions have been discussed in detail in [2]. For d > 2 we observe that the network might undergo a Bose-Einstein phase transition for values of the inverse temperature for which Eq. (43) cannot be solved in order to find the chemical potential µ d,δ . The detailed discussion of the possible phase transitions in CQNM is beyond the scope of this work and will be the subject of a separate publication.
D. Mean-field treatment of the case β > 0 It is interesting to characterize the evolution in time of the generalized degrees using the mean-field approach [11]. This approach reveals other aspects of the model that are responsible for the emergence of the statistics determining the distribution of the generalized degrees. Let us consider separately the mean-field equations determining the evolution of the generalized degrees of δ−faces with δ = d − 1, d − 2 or with δ < d − 2.
The (d − 1)-faces can have generalized degree k d,d−1 that can take only two values k d,d−1 = 1, 2. The indicator ξ α of a (d − 1)−face α with generalized degree k d,d−1 (α) = k is given by In fact for k = 2 the face is saturated and ξ α = 0 while for k = 1 the face is unsaturated, therefore ξ α = 1. The mean-field approach consists in neglecting fluctuations, and identifying the variable ξ α (evaluated at time t for a δ−face α arrived in the CQNM at time t α ) with its average ξ α =ξ α (t, t α ) over all the CQNM realizations. The mean-field equation forξ α is given by with initial conditionξ α (t α , t α ) = 1 where t α is the time at which the (d − 1)−face is added to the CQNM. The dynamical Eq. (45) is derived from the dynamical rules of the CQNM evolution. In fact, at each time one (d − 1)−face is chosen with probability Π α given by Eq.
(2). This face becomes unsaturated and glued to the new d-dimensional simplex. Therefore at each timeξ α indicating the average of ξ α decreases in time by an amount given by Π α .
Assuming that for large time t we have Z e −βµ .
The average ofξ α over all (d − 1)−faces α with energy , i.e. ξ α | is given by Therefore we obtain also in the mean field approximation, that the average generalized degree of (d − 1)−faces with energy , for t ).
Therefore we obtain also in the mean field approximation, that the average generalized degree of δ−faces with energy , and δ < d − 2, for t 1 satisfies, where n B ( , µ d,δ ) is proportional to the Bose-Einstein distribution at temperature T = 1/β.
As a final remark we note that the mean-field Eqs.