Non-abelian Topological Approach to Non-locality of a Hypergraph State

We present a theoretical study of new families of stochastic complex information modules encoded in the hypergraph states which are defined by the fractional entropic descriptor. The essential connection between the Lyapunov exponents and d-regular hypergraph fractal set is elucidated. To further resolve the divergence in the complexity of classical and quantum representation of a hypergraph, we have investigated the notion of non-amenability and its relation to combinatorics of dynamical self-organization for the case of fractal system of free group on finite generators. The exact relation between notion of hypergraph non-locality and quantum encoding through system sets of specified non-Abelian fractal geometric structures is presented. Obtained results give important impetus towards designing of approximation algorithms for chip imprinted circuits in scalable quantum information systems.


Introduction
The field of algebraic topology has passed an essential evolution in attaching algebraic objects to topological spaces starting from the simple graph models associated to vertices and edges in terms of Laplacian matrices, leading to higher-order dimensional structures modified through simplices or cliques, finalizing with an elegant mature representation of non-local hypergraph states [1].From the aspect of the algebraic topology, the dimension of the specific measure for the hypergraph (set system) OPEN ACCESS is a function of the corresponding metric [2].As a natural extension former implicates that for algebraic dynamical system the entropy of the measure depends on set system evolution process upon its metric spaces [3], i.e., the metric space represents the natural constituent assembly of the set system dimension in dynamical framework.Moreover, for algebraic dynamical systems the dimension directly relates to entropy via geometric concept of the Lyapunov exponents [4,5], which can be successfully applied to quantify the set system complexity in a sense of the topological [6,7] as well as the metric entropy [8].
In general sense the topological entropy represents quantitative measure of complexity for continuous map defined on compact metric spaces of dynamical system [2].While the metric entropy quantifies the number of typical orbits, the topological entropy represents the exponential growth rate of the number of orbit segments that are distinguishable with finite resolution.Hence, dynamical property of the maps is regulated by the topological entropy, i.e., between distinguishable orbits it measures growth rate of the number of different orbits of length n in infinite limit.Specifically, topological entropy as a noncausal measure of exponential increase of the number of maximal intervals of monotonicity is introduced for a piecewise interval monotone map [9,10].
Important feature of the set system topological entropy is its close relation to periodic orbits.For each periodic orbit P of continuous map f [11] given on the interval I, it can be assigned a corresponding "local pointwise" topological entropy h(P) [12] representing the connection map between distinct points of periodic orbit [13], and at the same time representing the infimum of the entropies of all corresponding maps which display orbits of the same configuration as P. On the other hand, for the piecewise monotone interval, the entropy of each continuous map f corresponds to the supremum of the number h(P), relating in that way to sum of all periodic orbits P of f [14].
In this study we introduce certain situations where the non-local correlations arise from the non-Abelian structure of topological algebraic set systems, addressing at the same time the divergence in dimension and complexity measures of classical and quantum hypergraph representation.In particular, connection between the periodic orbits and the topological entropy for continuous maps on specific fractal hypergraph sets, realized by the action of the free group on two generators 2 F [15,16], is analyzed in Section 2. Geometric measure of exponential divergence between nearby orbits of 2 F free group is represented via fractional entropic descriptor [17] where action of two generators on a free group corresponds to iteration on two Lyapunov scales.In Section 3, we further exploit generalization of the Shannon measure entropy in order to assess relation between the Lyapunov exponents and the Rényi entropy [18], addressing at the same time some of the unique properties of the presented fractal set.The Rényi entropy as a measure of complexity of dynamical system is analyzed in a sense of correlation of probability distribution function to its local expansion rate, defined by the Lyapunov exponents.Dynamical features of trajectory based metric are further compared to topological entropy equivalent in order to fully quantify the parameters of complexity and characterize combinatorics of dynamical self-organization for the case of fractal system of free group on finite generators.Finally, in Sections 4 and 5, using the underlying fractal structure of the free group on two generators we construct the quantum hypergraph state [1] and demonstrate essential relation between its non-local correlations and the system complexity.Using the stabilizer formalism, the hypergraph states are defined as a class of multiqubit quantum states which generalizes graph states [19].Obtained results allow efficient mapping of the quantum hypergraph states into corresponding logic circuits.Namely, in scope of the stabilizer formalism [20], in terms of a group theory, it is possible to efficiently implement topological logic gates [21], using operations, such as controlled-NOT, Phase, Hadamard gates.In particular, a stabilizer code on n qubits (a quantum error-correcting code [22]) uses Pauli operators as stabilizer generators.For instance, in order to stabilize a subspace of 2 s dimension which belongs to the 2n dimensional Hilbert space, in total s stabilizer generators are required.In that case, -k n s = logical qubits will be encoded into - 2 n s dimensional logical subspace.The space stabilized by the generators does not necessarily have to form an Abelian subgroup of the Pauli group over qubits [23].In particular case of generators, where non-Abelian group is of size 2 s e n k + = − ( k is notation of the number of qubits, codeword length is denoted as n ; e denotes ebits and s denotes the code ancilla qubit) the non-Abelian subgroup can be decomposed into two subgroups: the commuting isotropic group and the entanglement subgroup with anticommuting pairs [24].

Non-Abelian Statistics over Hypergraph Fractal Set
As we introduce some of the basic notations related to the hypergraph based fractal set, without loss of generality we assume topology embedded in 3   , where a hypergraph [25] represents a compact connected Hausdorff space represented by a subset ( ) . Considering a point , x V ∈ the number of edges which contain x will be denoted as the valence, v , of x.In case of a tree configuration where 2 v ≥ , we have the set of branching points ( ) B G which coincide with the spatial hypergraph coordinates.
We shall first introduce some basic definitions related to group-theoretic construction of the specific graph sets.Thus, important property relating to the topological entropy of the special class of hypepergraphs, introduced as a generalization of graphs in which infinite number of infinite clusters appears, is presented.Such geometric structure is the Cayley fractal set − .Uniqueness of the infinite cluster only appears when 1 p = and this property is connected to notion of the non-amenability [27], which directly relates to the non-Abelian statistics [28].Hence, amenability in terms of a group structure directly influences power to determine a specific probability measure on G that is left invariant on subsets of G .Latter gives a clear relation between non-amenability and uniqueness of the fractal set, seen as an infinite cluster, which can be quantitatively assessed from the line of topological entropy and geometric concept of Lyapunov.

Example
where ⋅ denotes the related set cardinality, p Γ ⊂ Γ are finite subsets on a finitely generated group G , and p ∂Γ is the edge-set representing the boundary of p Γ in reference to a given set of generators acting on G.
Definition 3-Dirichlet's principle [31]: For each eventual partition of a set X (consisting of n elements, associated to positive integers) there exists a subset , x X x N ⊆ = , where N is a positive integer which corresponds to n , such that the mapping : П X X → restricted to the subset x is either a constant or a 1-1 correspondence.
The above definition addresses existence and important relation of topological conjugacy between the set-system partitions and the elements of underlying subset.

Example: Let ( )
, T X E be a tree with vertex set ( ) X T and edge set ( ) E T .Partitioning of ( ) . The order of p Γ is x N = ; it is associated to the number of i x elements, which conform with the length-N Cayley permutation (C-permutation) p [34] of ordered set , where 1 2 ... N x x x < < < .Then, for every : x p x N = it follows that: is the weight probability distribution of each edge-set: two successive elements from x appear in N-th level C-permutation for which: if = x N (where x is associated to the number of i x elements), then the probability (the sum is implicitly over all possible i x ) and as a result the denominator converges to a positive limit with probability one, 1 . Likewise, the numerator : . We focus next on a connection between uniqueness of the , G H Γ free group fractal set and the topological entropy, introducing a key role of non-amenability notion which is a prime characteristic of non-Abelian group topology [36].Assuming a continuous map f of a compact metric space  , the topological entropy represents the supremum of the metric entropy, where supremum is taken over all finvariant Borel probability measures μ [37,38] on a topological space.This property of the topological entropy can be easily extended for an arbitrary function ( ) where supremum is taken over all localization parameters represented by the probability measures μ [39].Now, for N-level Cayley tree ,

G H
Γ , where p Γ ⊂ Γ is its underlying graph structure, we define an f -invariant Borel probability measure μ on the finite generating set : and x represents the number of i x elements, which give raise to the length-N Cayley permutation), as: where 0 0 / Δ > and free group, periodic orbits are defined on a hyperbolic 0 ε = space (due to a tree structure).In particular, geometric measure of the exponential divergence between nearby orbits of given sets is represented by the Lyapunov exponents, which can be successfully embedded into probabilistic framework in the following way.
Action of two generators on a free group corresponds to iteration of two scales which form the fractal set [40,41].Correspondingly, two scales produce a spectrum of Lyapunov exponents: in this case for the symmetric map, ( ) ( )  varying from 0 to 1 in the range: where ( ) and ( ) In presence of the infinite sequences of intervals, which are defined by two scales , .Then, the entropic descriptor [17], i.e., a measure of complexity associated to the fractal dimension ( ) quantifies the number of unique sequences produced by the ratio = m x n . Thus iterations) determines a periodic point [14] for the shift σ , where for each 1 ≥ n , there are 2 n points per period n for σ ; this specific property and its relation to topological entropy we discuss in the next section.Now, the entropy n H of the probability measure μ , which is defined on the finite (or countable)  is the probability operator acting over the specified generating set and λ k is the Lyapunov coefficient .

Topological Entropy and Periodic Orbit Growth for Hypergraph Fractal Set
Without loss of generality, topological entropy is defined as a measure of maximal complexity of dynamical system.Definition 4: Let  be the finite set of n elements (alphabet), describing the discrete topology [42].The dynamical system consisting of the sets of all bi-infinite symbol sequences where the shift map : x ∈  to the left, ( ) Theorem 1: (Hasselblatt and Katok [43]) Topological entropy and periodic orbit growth coincide for shifts.

Proof
where for 1 n → straightforwardly topological entropy for bilateral k-shift coincides with the logarithm of k sequences: stems from the restriction of the bilateral k-shift over invariant subset of k sequences, with a property of symbolic system [44]. The above can be easily explained using the following coin experiment.
→ be the twofold-switch operator acting over all spatial elements of 0 M ∞ { } , H T which denote the symbolic system (subshift) [44], where this operator determines the outcomes of an infinite series of trials (measurements).Then, considering that the sum of all possible outcomes in n trials is given by 2 n n N =  , the topological entropy is defined as Thus, specifying the outcomes as ( )  V where cylinder set of length n is defined on space [ ] , 0 1 ,..., 0 1 and the number of length-n cylinders that overlap space As a result, the connection between the topological entropy and the transition matrix [15] is given by: ( ) ( ) where the topological entropy equals to the logarithm of the largest eigenvalue of the transition matrix (spectral radius) [46].

Relations between Lyapunov Spectrum, System Dimension and Measure Entropy for Hypergraph Fractal Set
In case of the systems with infinite degrees of freedom (e.g., the presented fractal system of the free group on two generators, where applies notion of the non-amenability given by Equation ( 1)) assuming d -dimensional system of linear size d L , it is possible to establish relation between Lyapunov spectrum, dimension of the system and measure entropy.The first step towards such relation is to resolve the case of 'thermodynamic' limit, L → ∞ , for the Lyapunov spectrum, which can be assessed by analyzing whether ratio [ ] , converges to a density function Starting from the Pesin formula [47], where θ is the Heaviside function: ( ) which infers that the Kolmogorov-Sinai entropy [48,49] is in this case proportional to the maximum Lyapunov exponent 1 p i i h λ = =  .Thus, there is only one positive Lyapunov exponent, as the system size approaches thermodynamic limit L → ∞ .
Local entropy l h follows from Equation ( 14); it is defined for each degree of freedom for the system of linear size d L , as: where after setting the existence of bound: as L → ∞ , total entropy h is directly related to the dimension of the system.Indeed, using the Kaplan and Yorke formula [50], the fractal dimension D can be estimated from the Lyapunov spectrum as:  ( ) ( ) , Here 1 M − is a diagonal matrix with corresponding elements 1/ .
H − is also a Hermitian matrix where ( ) ( ) ( ) leading to: where , i k δ is the Kronecker delta function, ensuing that generating function satisfies: , where , j l M are elements of the diagonal matrix: M acts as a diagonal matrix by extending these eigenvectors to the complex basis of d C .Consequently, as the normalization for each , i j ρ element is given by n d it follows that for each , i j the coefficients , i j ρ represent eigenvectors of the adjacency matrix A with eigenvalue 0 if i j ≠ , and eigenvalue Now we can easily generalize the concept of the Cayley

Rényi Topological Entropy and Lyapunov Exponents for Non-Abelian Fractal Set
Assume that the fractal set is partitioned into distinguishable clusters of size r.In order to localize a point with a precision r one must determine the partition which contains the point and quantify average data on particular partition.For this one can use the Shannon information formula, and also the topological Rényi entropy of order q , which is directly related to the generalized dimension [52]: where i P is the probability measure of the ith partition.Now, let П be a partition assigned by a free group G on rank two generators H G ⊆ where ( ) { } π n i are the elements of partitions n П obtained under measure preserving transformation [53]: ( ) The Rényi entropy of order q is the supremum over all distinguishable partitions n П : Next step establishes relation between the topological Rényi entropy of order q and the generalized Lyapunov exponents Λ .By assigning the probability distribution function ( ) occurrence of distinguishable partitions n П that are specified in time steps: for which: and using the generalized measure ( ) Λ k i which represents the average expansion rate of information given by the building block k i of dimensionε which constituents each partition i П ; the initial building block 0 i is defined by the corresponding Lyapunov spectrum ( ) After the initialization is performed, the Lyapunov spectrum is given by: ( ) ( ) ( ) from which we obtain the probability rate for the expansion of the spatial partitions building blocks: Substitution of Equation ( 26) into Equation (25) gives: where α n is obtained from the normalization: ( ) and represents the number of various distinguishable clusters (blocks) of length n.Now, from Equation ( 27) follows the measure: which establishes the correlation between the topological Rényi entropy of order q and the generalized Lyapunov exponents ( ) Λ k i , for the limit n → ∞ this correlation reads: Example: Let ( ) , G V H be a hypergraph defined on a free group G on finite generators H , where vertex set: , ,..., , and topological order coefficients: ( ) , ,..., , ,..., ,..., , ,..., , ,..., , ,..., and defined on a metric space x M with probability measure , μ where a nonnegative function : R is associated to each generator set i H .Then, under constraint of integrability [38]: and according to Equation (24) where the Rényi entropy of order q is the supremum over all distinguishable hypergraph partitions, it follows the expression for the fractional Rényi entropy of order q for a hypergraph defined on a free group G with respect to the generator set i H G ⊆ : ( ) In particular case of two generators, the Rényi's entropy dependence from the probability measure x μ , and the probability rate for the expansion of the spatial partitions (Equation ( 31)), for different values of order q , is presented in Figure 1 and Figure 2, respectively.Figure 3 shows the maximum allowed correlation between the topological Rényi entropy of order q and the generalized Lyapunov exponents ( ) Λ k i for the partitioned hypergraph fractal set (Equation ( 24) obtained by action of the rank -2 free group on finite generators [26].hypergraph fractal set of free group on finite generators, according to equation (31), and order: q = 0.4 (dash-dotted), q = 5 (dashed) and q = 1 (full line).31)) for different input order parameters (designated by solid lines): q = 0.2 (blue), q = 0.4 (yellow), q = 0.5 (dark cyan), q = 0.8 (dark yellow).

Underlying Non-Locality of 2 F -Hypergraph State
In this section we present a more structural understanding of non-locality [54] of a hypergraph state which is obtained by action of the rank two free group, called 2 F .In particular, we show that intrinsic non-local character of the non-Abelian actions [55] of the free group on two generators is closely tied to notion of the non-amenability.We consider the algebraic topological structure obtained via generating set where a ; b denote basis of the free group 2 which realize an infinite 4-regular tree represented as the undirected Cayley set.In this case, the infinite 4-regular structure gives a hypergraph based covering space for the wedge of two circles 1 1 Ĥ H , ( )   where ( ) ( ) ( ) Definition 5: Group G is amenable if and only if it does not produce paradoxical decomposition [43].
From the constraint of amenability and Equation (1) it follows that a rank two free group is non-amenable and as a result it is non-Abelian [27] (see fundamental theorem of finitely generated Abelian groups), and there exists a countable subset H on the sphere S such that the decomposition − S H is F -paradoxical [56,57]; a straight repercussion is that ( ) is paradoxical too.Thus, the elements of the free group of two generators ( ) distance preserving as operators on 3  , where they represent a group of nontrivial, independent rotations of the sphere about axis that passes through the sphere center.and the generalized Lyapunov exponents ( ) Λ k i obtained for the supremum output entropy conjecture over all distinguishable partitions (Equation ( 24)).The x-y axes are the average expansion rate of information given by the building block k i of dimension ε on N intervals and input order parameters q, respectively.
Likewise, there are rotations R a , R b of the unit sphere in 3   that generate exactly the free group on two generators.To prove that let us define a group of rotations ( ) ( ) In this case elements of a group: Proof: The natural decomposition of ( )
The following relations establish correlation between information strings and the set of free group on two generators: But, another decomposition yields the following property: where ( ) ω ⋅ is the complement of the corresponding set ( ) ω ⋅ , and where actual multiplying ( ) ( ) Correspondingly, the third decomposition of 2 F is given by: ( ) ( ) ( ) ( ) Now, in order to demonstrate notion of the non-amenability which directly implicates non-locality of the hypergraph quantum states based on the free group 2 F , one can start by assigning the mean probability distribution ( ) ( ) likewise: Then, from Equation (33) we have: ( ) Comparison of previous relation with Equations (37) and (38) directly gives: finally, by inserting into Equation ( 43) the relations which address the mean probability distribution of where N is the number of sequences of intervals corresponding to the fractal dimension (see Figure 4.) where uniqueness of the infinite cluster only appears when 1 p = .Analogue inequality relation can be obtained for all corresponding decomposition elements of ( ) 2

. F SO
∈ Former contraintuitive inequality [54,58] directly provides the resource for analysis of the non-local properties of the hypergraph quantum states [1], defined in this case on the 4-regular subset (Equation ( 44)) establishes demarcation line between the complexity of the classical (local realism) and the quantum counterpart [59] of system state.
The main prerequisite for encoding is to perform the initialization of the system with probabilities { } ( )  where action of the measurement is determined by the projective measurement operator X Pi X X = over a given generating set (see Equation ( 46)), producing in that way the state ρ which possesses nonlocal correlations, representing at the same time the topological state.In particular, the symmetry of the underlying topology is efficiently captured and corresponds exactly to the symmetry of the joint probabilities for each measurement, see Figure 5.In this case the tensor algebra ( ) x x x ω ω ω in V .
obtained with operator λ denote rotation and a phase shift, respectively.For the case when ( ) ( ) .This result is in agreement with Equation ( 44) for p = 0, with direct repercussion to notion of the non-local correlations [55] based on non-Abelian set of rank two free group 2 F .

Conclusions
A major advance towards the characterization of complexity of dynamical systems has affected communication complexity as well, offering new paths towards successful implementation of quantum information processing, especially in the field of topological insulators.In particular, the topological Rényi entropy has qualified as a good probe of the topological order, where the amount of fractal distribution present in the system and its scaling are essential for distinguishing between different phases of matter.In addition to former, wide-range implications of here presented results include the fundamental result about distribution and presence of quantum correlation and non-locality in hypegraph state based on free group on two generators, proceeding to direct implementations of underling topology of the free group 2 F fractal sets in chip integrated circuits for quantum computing.We have defined the fractional Rényi entropy of order q for the hypergraph fractal sets based on a (non-Abelian) free group on finite generators and shown that intractability of the fractal dynamical processes can be efficiently bypassed using the geometrical concept of Lyapunov, which has proved to be the most viable method for the investigation of the complexity evolution, having its intrinsic relation to the topological and the metric (information) entropy.

E
are the probability distributions which denote the occurrences of the set system finite spatial partitions.are the linear subspaces corresponding to exponents k λ so that the dimension D of k E corresponds to multiplicity of k λ .For a hypergraph based fractal set of rank two directly to Lyapunov coefficients λ k .For the rank two generators { }


are the probabilities for occurrence of the partitions:

:
Let : k k σ →   be a bilateral k-shift, acting on sequence{ } 1,..., k , and [ ] [ ] { } 1 ,..., k α = be a topological generator which transfers the spatial partitions of k  into closed cylinders of length 1. of spatial partition of k  into n k cylinders of length n.As a result we have: case when the output is biased ({ } , H T represents the full shift), the number of typical outcomes in n trials is

) Lemma 1 :
Extending relation(10) to a hypergraph subset ( ) , G V H for a finite type subshift[44]: where M is irreducible n n × transition matrix with entries in { } 0, 1 , under condition that the largest eigenvalue is always real by the Perron-Frobenius theorem[45], results that the topological entropy ( ) top h σ is equal to the largest positive eigenvalue of matrix M. cylinders of length n, where σ is a subshift of finite type, and resulting topological entropy coincides with the number of cylinders of length n in ,

2 L 2 LProposition 2 :
of the attractor, D λ , is proportional to d L leading to the existence of inherent dimension density of freedom.Now we are ready to elucidate the essential connection between the Lyapunov exponents i λ and dregular hypergraph fractal set.We use the set-theoretic approach, starting from the structure of the Cayley tree.The Cayley tree on a free group G generated by the finite set H G ⊆ is represented by the graph with vertex set V G ∈ and the edge set ( ) G is the function space on G , then ( ) G can be decomposed into subspaces: meaning that it has the value 1 on the i-th vertex of a set G and 0 otherwise.Assume that the Cayley set satisfies relation:1 H H − = .This condition as a result infers existence of orthogonal set of eigenvectors and eigenvalues belonging to the adjacency matrix A.Under constraint that the eigenvalues of A, i.e., 1 ,..., i d λ λ are bounded by corresponding vector subspaces i E , the norm of the F -uniform, d-regular (each vertex has degree d) Cayley hypergraph on n vertices is defined as

Proposition 3 : 2 L
The eigenvalues of the d-regular Cayley hypergraph can be determined from the , G H Γ = Γ subgraph and from the ( ) G decomposition.Proof: Let H be the symmetric set of generators with the vertex set assigned on a finite group G , then the d-regular Cayley hypergraph on G and H is composed of edges tree character, d-regular Cayley hypergraph has intrinsic Markov property and it can be analyzed in terms of a piecewise linear Markov map f[51] according to the spectral radius of matrix M (whose elements are {0, 1}).Direct consequence is that the topological entropy corresponds to the maximal eigenvalue of the matrix M as it is already shown by Equation(13). Moreover, taking into account the bound Kolmogorov-Sinai entropy in case of d-regular Cayley hypergraph is proportional to:

Figure 1 .
Figure 1.Topological Rényi entropy as a function of the probability measure x μ over

Figure 3 .
Figure 3. Maximum allowed correlation between the topological Rényi entropy of order q

y 2 F
and z as integers, 0, ≠ y and n is the length of the information string.for a generating set{ }: , H a b we can define a rank 2 b b a b H , which determines paradoxical decomposition of − S H , manifesting in this way intrinsic non-local feature of the hypergraph fractal state based on free five parts where only four are needed to establish two exact copies of the original sphere S , i.e., to perform the paradoxical decomposition.
we obtain the characteristic inequality relation for the free group 2 F set corresponding to CHSH inequality[58]:

Figure 4 .
Figure 4. Maximum violation of ( ) 2 CHSH m F inequalities obtained for max 2.82 S ≈ and 0 p ≥ versus number of interval sequences N (Equation (44)), corresponding to CHSH

Figure 5 .
Figure 5. (a) Geometrical-representation of a rank-2 free group 1 1 2 , , , , F a a b b a b H − − = ∈ , where H G ⊆ is a generating set.(b) Scheme of the hypergraph state quantum correlations [1,60,62] that reflect exact symmetry of the underlying 1 1 2 , , , , − − = ∈ F a a b b a b H topology,

2 F
. Consequently, for the d-regular hypergraph state on n vertices, the maximal value for I is bounded with the maximal eigenvalue max , all Ф k ± states eigenvalues are zero, 0 λ = .Consequently, the maximal violation of CHSH inequality is obtained for combinations of k ± Ψ states which yield also maximal value for ( ) Tr I ρ =  [62] equivalent to the maximal eigenvalue ,max

Proposition 1: In case of a free group on two generators obtained tree structure ,
let A and F be the adjacency matrix and generating function of the Cayley hypergraph, respectively.A and F are bounded on the vector subspace i E corresponding to the d dimensional state ρ (where the coefficients of { } − .As a result, a 4-valent hypergraph is generated by action H G × .Qubit states are assigned as: set elements, which is an associative k-algebra spanned by