Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces
HHOLOGRAPHIC CODES ON BRUHAT–TITS BUILDINGS ANDDRINFELD SYMMETRIC SPACES
MATILDE MARCOLLI
Abstract.
This paper is based on the author’s talk at the Arbeitstagung 2017.It discusses some general approaches to the construction of classical and quantumholographic codes on Bruhat–Tits trees and buildings and on Drinfeld symmetricspaces, in the context of the p -adic AdS/CFT correspondence. Dedicated to Yuri Manin on the occasion of his 80th birthday Introduction
This paper is based on the talk given by the author at the Arbeitstagung 2017“Physical Mathematics” in honor of Yuri Manin’s 80th birthday. It is an introductionto an ongoing joint project with Matthew Heydeman, Sarthak Parikh and IngmarSaberi, on the construction of holographic classical and quantum codes on Bruhat–Tits trees and higher rank Bruhat–Tits buildings and on Drinfeld symmetric spaces,and associated entanglement entropy formulae. A discussion of the entanglemententropy and the relation to other holographic codes constructions, such as [42], willbe presented in a forthcoming joint paper in preparation. The present paper shouldbe regarded as covering some background material on the question of constructingholographic codes on p -adic symmetric spaces, based on algebro-geometric properties.In [32], [33], Manin gave a compelling view of the idea of “Arithmetical Physics”,according to which physics in the usual Archimedean setting or real and complexnumbers would cast non-Archimedean shadows that live over the finite primes andarithmetic properties associated to these non-Archimedean models can be used tobetter understand the physics that we experience at the Archimedean “prime atinfinity”. According to this general philosophy Spec( Z ) is the “arithmetic coordinate”of physics and geometry. A famous example where this principle manifests itself isgiven by the description of the Polyakov measure for the bosonic string in terms ofthe Faltings height function at algebraic points of the moduli space of curves, whichleads naturally to the question of whether the Polyakov measure is in fact an adelicobject and whether there is an overall arithmetic expression for the string partitionfunction, [34], [35]. More generally, one can ask to what extent are the fundamentallaws of physics adelic. Does physics in the Archimedean setting (partition functions,action functionals, real and complex variables) have p -adic manifestations? Can thesebe used to provide convenient “discretized models? of physics, powerful enough todetermine their Archimedean counterpart? a r X i v : . [ m a t h - ph ] J a n MATILDE MARCOLLI
Various forms of p -adic and adelic phenomena in physics and their relation to theusual Archimedean formulation were developed over the years. We refer the readersto [7], [10], [46], [47] for some references relevant to the point of view discussed inthis paper.Here we focus in particular on the holographic AdS/CFT correspondence and onthe recent viewpoint relating information (entanglement entropy) of quantum stateson the boundary to geometry (classical gravity) on the bulk, [41] and the tensornetworks and holographic codes approach of [42]. The existence of a p -adic versionof the holographic AdS/CFT correspondence was already proposed in [40], basedon earlier results of Manin [36], [37] expressing the Green function on a compactRiemann surface with Schottky uniformization to configurations of geodesics in thebulk hyperbolic handlebody (which are higher genus generalizations of EuclideanBTZ black holes [28]) and results of Drinfeld and Manin [39] on periods of Mumfordcurves uniformized by p -adic Schottly groups.In [22] we developed a non-Archimedean version of AdS/CFT holography, basedon the approach originally proposed in [40], which would be compatible with themore recent viewpoint on the holographic correspondence based on the ideas of ten-sor networks and holographic codes and the correspondence between entanglemententropy and bulk geometry. Versions of p -adic AdS/CFT correspondence were alsodeveloped in [18], and in subsequent work [3], [19], [16], [15], [17] and others. Thetheme of non-archimedean versions of holography has clearly become a very activearea of current research.In this paper, we return to the point of view of tensor networks and holographiccodes discussed in [22] and we present some new constructions which are based onthe geometry of Bruhat–Tits trees and buildings and of Drinfeld symmetric spaces.The main difference between the approach we propose here and other constructionsof holographic codes such as [42], or for instance [3], [5], [21], lies in the fact thatwe rely on well known techniques for the construction of classical codes associated toalgebro-geometric objects [45] and on algorithms relating classical to quantum codes[6]. The construction of algebro-geometric codes played a crucial role in the study ofasymptotic problems in coding theory, as shown by Manin in [38].We first present here a construction of holographic codes that is based on thegeometry of the Bruhat-Tits trees and algebro-geometric Reed–Solomon codes asso-ciated to projective lines over a finite field, together with an application of the CRSSalgorithm that associates quantum codes to classical q -ary codes.We then revisit the approach to holographic codes via tessellations of the hyperbolicplane, as in [42]. Instead of relating such constructions to the Bruhat–Tits trees viaa non-canonical planar embedding of the tree, as in [22], we use here a purely p -adicviewpoint, working with the Drinfeld p -adic upper half plane as a replacement of thereal hyperbolic plane, and its (canonical) map to the Bruhat–Tits tree. Instead oftessellations of the real hyperbolic plane we use actions of p -adic Fuchsian groupson the Drinfeld plane and associated surface codes. We show that this approach is OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 3 restricted by the strong constraints that exist on p -adic Fuchsian groups. For example,we show that a p -adic analog of the holographic pentagon code of [42] constructedwith this method can only exist when p = 2.We then propose an extension of this approach via holographic codes to higherrank buildings, based on algebro-geometric codes associated to higher dimensionalalgebraic varieties, as constructed in [45].2. Algebro-Geometric Codes on the Bruhat–Tits tree
In this section we describe a construction of holographic codes on the Bruhat–Titstrees that are obtained via Reed–Solomon algebro-geometric codes on projective linesover finite fields.2.1.
Reed–Solomon codes and classical codes on the Bruhat–Tits tree.
Theset of algebraic points X ( F q ) of a curve X over a finite field F q can be used toconstruct algebro-geometric error-correcting codes, see [44]. Algebro-geometric codesassociated to a curve X over a finite field F q consists of a choice of a set A of algebraicpoints A ⊂ X ( F q ) and a divisor D on X with support disjoint from A . The linearcode C = C X ( A, D ) is obtained by considering rational functions f ∈ F q ( X ) withpoles at D and evaluating them at the points of A . A bound on the order of pole of f at D determines the dimension of the linear code.We are interested here in the simplest case of algebro-geometric codes, the Reed–Solomon codes constructed using the points of P ( F q ). Given a set of points A ⊂ P ( F q ) with A = n ≤ q + 1 we consider two types of Reed–Solomon codes, one con-structed using the point ∞ ∈ P ( F q ) as divisor, that is, using polynomials f ∈ F q [ x ],and using a set A of n ≤ q points in A ( F q ) = F q for evaluation. The corre-sponding Reed–Solomon code C = { ( f ( x ) , · · · , f ( x n )) : f ∈ F q [ x ] , deg( f ) < k } gives an [ n, k, n − k + 1] q classical code, where n ≤ q . The other type of Reed–Solomon codes are obtained using homogeneous polynomials and a set A of n ≤ q + 1 points in P ( F q ). The resulting code ˆ C = { ( f ( u , v ) , . . . , f ( u n , v n )) : f ∈ F q [ u, v ] , homogeneous with deg( f ) < k } , with x i = ( u i : v i ) ∈ P ( F q ). We alsoconsider generalized Reed-Solomon codes of these two types, where for a vector w = ( w , . . . , w n ) ∈ F nq one defines C w,k = { ( w f ( x ) , · · · , w n f ( x n )) : f ∈ F q [ x ] , deg( f ) < k } ˆ C w,k = { ( w f ( u , v ) , . . . , w n f ( u n , v n )) : f ∈ F q [ u, v ] , homogeneous , deg( f ) < k } . For K a finite extension of Q p with residue field F q , with q = p r , the Bruhat–Tits tree T K is a homogeneous tree with valence q + 1 = P ( F q ) and with ends ∂ T K = P ( K ). The choice of a projective coordinate on P ( K ) fixes three points { , , ∞} ∈ P ( K ), hence it fixes a unique root vertex ν ∈ V ( T K ). The star ofvertices surrounding ν can then be identified with a copy of P ( F q ), which in algebro-geometric terms corresponds to the reduction modulo the maximal ideal m in O K .The root vertex ν is therefore associated to the reduction curve P . We canconstruct a holographic classical code to the Bruhat–Tits tree by assigning to the root MATILDE MARCOLLI vertex ν and its star of q + 1 edges a Reed–Solomon code with an assigned number k of logical inputs ( q -ary bits) located at ν and outputs at each of the q + 1 legs. Thiscan be done by a (generalized) Reed-Solomon code ˆ C w,k of maximal length n = q + 1,seen as an encoding ˆ C w,k : F kq → F q +1 q , which inputs a k -tuple of q -ary bits a =( a , . . . , a k − ) ∈ F kq , uses the homogeneous polynomial f a ( u, v ) = (cid:80) ki =0 a i u i v k − − i ,and outputs a q -ary bit f ( u j , v j ) ∈ F q at each point x j = ( u j : v j ) ∈ P ( F q ) identifiedwith a leg of the vertex ν in the Bruhat–Tits tree.The choice of the projective coordinate on P ( K ), hence of the root vertex ν in T K ,determines a choice of a leg at each other vertex ν (cid:54) = ν , given by the unique directionout of ν towards the root ν . We can identify this choice with a choice of the point {∞} in each copy of P ( F q ) at each vertex ν (cid:54) = ν of the tree. Proceeding from thecenter, if we assign a Reed–Solomon code at each vertex of T K , and by homogeneitywe expect all of them to have the same number k of inputs, we see that at eachsuccessive steps the leg of the star of edges at ν has already one value assigned at theleg labelled by the point ∞ ∈ P ( F q ), which corresponds to the output coming fromthe matching leg in the star of the previous vertex coming from the root ν . Thus, inprojective coordinates ( u : v ) where (0 : 1) is the point at infinity, the Reed–Solomoncode ˆ C w,k associated to the vertex ν takes k − a = ( a , . . . , a k − ) ∈ F k − q and one additional input a given by the value at ∞ assigned by the previous code,and deposits a new q -ary bit f a ( u, v ) = (cid:80) k − i =0 a i u i v k − − i at each of the remaining legsat the vertex v pointing away from the root, labelled by the points x ∈ A ( F q ) = F q .Note how the construction considered here has one root vertex play a special rolewith an F kq logical input and a Reed–Solomon code of length q + 1, while all the othervertices have a further logical input of F k − q . This asymmetry is inevitable if we wantto use the algebro-geometric structure underlying the Bruhat–Tits tree to constructa classical code, since the root vertex plays the special role of the algebraic curvegiven by the reduction modulo m , while the sets of vertices in the tree at distance m from the root correspond to reducing modulo powers m m . Thus, the asymmetric roleof the root vertex and the other vertices is built into the relation between P ( K ) andits reduction curves.The construction described here determines a classical code associated to theBruhat–Tits tree with logical inputs at the vertices and outputs at the forward point-ing legs. In the limit where one considers the whole tree, the outputs consist of a q -ary bit deposited at each point of the boundary P ( K ). We want to transformthis classical code built using the algebro-geometric properties of the Bruhat–Titstree, into a quantum error correcting code that generates a holographic code for theBruhat–Tits tree and its boundary at infinity.2.2. Classical Algebro-Geometric Codes for Mumford Curves.
The construc-tion above can be generalized in the case of Mumford curves. Let Γ be a p -adicSchottky group and Ω Γ = P ( K ) (cid:114) Λ Γ the domain of discontinuity of Γ acting on theboundary P ( K ), the complement of the limit set Λ Γ . The quotient X = Ω Γ / Γ is aMumford curve of genus g equal to the number of generators of the Schottky group. OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 5
Unlike complex Riemann surfaces, which always admit a Schottky uniformization,only very special p -adic curves admit a Mumford curve uniformization. Indeed, thesecurves must have the property that their reduction mod m is totally split: as a curveover F q it consists of a collection of P ’s with incidence relations described by thedual graph G . This is the finite graph at the center of the quotient T K / Γ, obtainedas the quotient G = T Γ / Γ, where T Γ is the subtree if T K spanned by the geodesic axesof the hyperbolic elements γ (cid:54) = 1 of Γ, with ∂ T Γ = Λ Γ .Using the identification between the finite graph G and the dual graph of thereduction curve, we can again associate to each vertex in G a copy of P ( F q ) (thecorresponding component in the curve), and to each of these projective lines a Reed–Solomon code as in the previous construction. Now, however, we need to imposecompatibility conditions between these codes at the incidence points between differentcomponents of the curve, that is, along the edges of the finite graph G . Thus, weassociate to the finite graph G a classical code C ( G ) constructed as follows. Start witha code ˆ C w,k associated to each vertex v , which inputs a = ( a , . . . , a k − ) and outputs f a ( u, v ) = (cid:80) i a i u i v k − − i at each point x = ( u : v ) of the associated P ( F q ). Considerthe set F of functions f = ( f , . . . , f N ), with N = V ( G ), and f i a homogeneouspolynomial of degree deg( f i ) < k on the i -th component P ( F q ), with the propertythat if x = ( u i : v i ) = ( u j : v j ) is an intersection point between the i -th and the j -thcomponents of the reduction curve, then f i ( u i : v i ) = f j ( u j : v j ). Thus, each edges e ∈ E ( G ) imposes a relation between f i and f j which requires the value that thecodes ˆ C w,k at the vertices ν i and ν j deposit at the point x to be the same. Thus, theresulting code C ( G ) is an F q -linear code with input F kN − Mq where N = V ( G ) and M = E ( G ). We have kN − M = ( k − N + 1 − b ( G ) hence we need to assume k > b ( G ) − /N .The free legs of the graph G are all the legs that point towards the infinite treesin T K / Γ that extend from the vertices of G to the boundary Mumford curve X ( K ) = ∂ T K / Γ. At each vertex along these trees we consider Reed-Solomon codes as in thecase of P ( K ), with one input coming from the previous vertex closer to G and k − q forward pointing legs. This determines a classicalcode associated to the infinite graph T K / Γ, with logical inputs at the vertices andoutputs at the points of the Mumford curve X ( K ). The finite graph G and the infinitegraph T K / Γ containing it are a genus g generalization of the p -adic BTZ black hole,which corresponds to the g = 1 case of Mumford–Tate elliptic curves.2.3. Reed–Solomon Codes and Quantum Algebro-Geometric Codes.
Thereis a general procedure for passing from classical codes to quantum codes, based onthe Calderbank–Rains–Shor–Sloane algorithm [6], see also [1]. It can be appliedto certain classes of algebro-geometric codes and in particular to generalized Reed–Solomon codes.Let H = C q be the Hilbert space of a single q -ary qubit and H n = ( C q ) ⊗ n the spaceof n q -ary qubits. We label an orthonormal basis of H by | a (cid:105) with a ∈ F q . Thus, a q -ary qubit is a vector ψ = (cid:80) a ∈ F q λ a | a (cid:105) with λ a ∈ C , and an n -tuple of q -ary qubits MATILDE MARCOLLI is given by a vector ψ = (cid:80) a =( a ...a n ) ∈ F nq λ a | a (cid:105) where | a (cid:105) = | a (cid:105) ⊗ · · · ⊗ | a n (cid:105) . Quantumerror correcting codes are subspaces C of H n that are error correcting for a certainnumber of “q-ary bit flip” and “phase flip” errors. More precisely, an error operator E is detectable by a quantum code C if P C EP C = λ E P C , where P C is the orthogonalprojection onto the code subspace and λ E is a scalar. In particular, one considers erroroperators that affect up to a certain number of qubits in an n -qubits state, namelyerror operators of the form E = E ⊗ · · · ⊗ E n , of weight ω ( E ) = { i : E i (cid:54) = I } . Theminimum distance d Q ( C ) of the quantum code is the largest d such that all errorswith ω ( E ) < d are detectable.The bit and phase flip error operators are defined on a single q -ary qubit as T b | a (cid:105) = | a + b (cid:105) , R b | a (cid:105) = ξ Tr( (cid:104) a,b (cid:105) ) | a (cid:105) , where ξ is a p -th primitive root of unity, and Tr : F q → F p is the trace function,Tr( a ) = (cid:80) r − i =0 a p i , with (cid:104) a, b (cid:105) = (cid:80) ri =1 a i b i and with R b i | a j (cid:105) = ξ Tr( a j b i ) | a j (cid:105) . Let { γ i } ri =1 be a basis of F q as an F p -vector space, so that a = (cid:80) i a i γ i and b = (cid:80) i b i γ i . Then theerror operators T b and R b can be written respectively as T b = T b ⊗ · · · ⊗ T b r , R b = R b ⊗ · · · ⊗ R b r with T and R given by the operators acting on C p of matrix form T = · · ·
00 0 1 · · · · · ·
11 0 0 · · · R = ξ ξ . . . ξ p − satisfying the commutation relation T R = ξRT . The operators T a R b with a, b ∈ F q form an orthonormal basis for M q × q ( C ) under the inner product (cid:104) A, B (cid:105) = q − Tr( A ∗ B ),hence these operators generate all possible quantum errors on the space C q of a single q -ary qubit. The action of error operators on a state of n q -ary qubits can similarlybe written in terms of operators T a R b with E a,b = T a R b = ( T a ⊗ · · · ⊗ T a n )( R b ⊗ · · · ⊗ R b n ) , for a = ( a , . . . , a n ) , b = ( b . . . , b n ) ∈ F nq . The operators E a,b satisfy E pa,b = I and thecommutation and composition rules E a,b E a (cid:48) ,b (cid:48) = ξ (cid:104) a,b (cid:48) (cid:105)−(cid:104) b,a (cid:48) (cid:105) E a (cid:48) ,b (cid:48) E a,b , E a,b E a (cid:48) ,b (cid:48) = ξ −(cid:104) b,a (cid:48) (cid:105) E a + a (cid:48) ,b + b (cid:48) , where (cid:104) a, b (cid:105) = (cid:80) i (cid:104) a i , b i (cid:105) = (cid:80) i,j a i,j b i,j , with a i , b i ∈ F q , written as a i = (cid:80) j a i,j γ j and b i = (cid:80) j b i,j γ j , after identifying F q as a vector space with F rp . Thus, we can considerthe group G n = { ξ i E a,b , a, b ∈ F nq , ≤ i ≤ p − } of order pq n . A quantum stabilizererror-correcting code C is a subspace C ⊂ H n that is a joint eigenspace of operators E a,b in an abelian subgroup S ⊂ G n . OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 7
Let ϕ ∈ Aut F p ( F rp ) be an automorphism. In particular, we consider ϕ given by thetrace as in [1], so that we the associated pairing is (cid:104) ( a, b ) , ( a (cid:48) , b (cid:48) ) (cid:105) = (cid:104) a, ϕ ( b (cid:48) ) (cid:105) − (cid:104) a (cid:48) , ϕ ( b ) (cid:105) = Tr( (cid:104) a, b (cid:48) (cid:105) ∗ − (cid:104) a (cid:48) , b (cid:105) ∗ ) , where, for a, b ∈ F nq , the inner product (cid:104) a, b (cid:105) (cid:54) = (cid:104) a, b (cid:105) ∗ , since (cid:104) a, b (cid:105) ∗ = (cid:80) ni =1 a i b i , while (cid:104) a, b (cid:105) = (cid:80) ni =1 (cid:104) a i , b i (cid:105) = (cid:80) ni =1 (cid:80) rj =1 a i,j b i,j . If C ⊂ F nq is a classical self-orthogonalcode with respect to this pairing, then the subgroup S ⊂ G n given by the elements ξ i E a,ϕ ( b ) with ( a, b ) ∈ C is an abelian subgroup of G n , because of the commuta-tion rule above. This construction is the CRSS algorithm that associates to a self-orthogonal classical [2 n, k, d ] q code a stabilizer quantum [[ n, n − k, d Q ]] q -code, where d Q = min { ω ( a, b ) : ( a, b ) ∈ C ⊥ (cid:114) C } , where the weight ω ( a, b ) = { i : a i (cid:54) = 0 or b i (cid:54) =0 } , and C ⊥ = { ( v, w ) ∈ F nq : (cid:104) ( a, b ) , ( v, w ) (cid:105) = 0 , ∀ ( a, b ) ∈ C } .We can view the CRSS algorithm assigning the quantum stabilizer code C to theclassical code C as an encoding process that takes the q k input vectors ( v, w ) ∈ F nq of the classical code C and encodes the states | ( v, w ) (cid:105) using the vectors ψ ∈ ( C q ) ⊗ n satisfying E v,ϕ ( w ) ψ = λψ in a common eigenspace of the E v,ϕ ( w ) .A slightly more general version of the CRSS algorithm starts with two classicallinear q -ary codes C ⊆ C of length n and dimensions k and k and associatesto them a quantum code C = C ( C , C ) with parameters [[ n, k − k , min { d ( C (cid:114) C ) , d ( C ⊥ (cid:114) C ) } ]] q , see [13], [26]. The procedure for the construction of the quantumcode is similar to the version of the CRSS algorithm recalled above. One constructsa code C = γC + ¯ γC in F nq with γ a primitive element of F q and { γ, ¯ γ } a linearbasis of F q as an F q -vector space. By identifying F nq as an F q -vector space with F nq ,we obtain a self orthogonal C ⊂ F nq , to which the CRSS algorithm discussed beforecan be applied.Conditions under which Reed-Solomon codes satisfy a self-dual condition, and thecorresponding quantum Reed-Solomon codes obtained via a CRSS type algorithmare analyzed, for instance, in [12] and [14]. We use here a construction of [1], whichshows that, if C is a q -ary classical [ n, k, d ] q -code, which is Hermitian-self-dual, thenthere exists an associated q -ary [[ n, n − k, d Q ]] q -quantum code, with d Q ≥ d . HereHermitian-self-dual means that the classical code C is self dual with respect to the“Hermitian” pairing (cid:104) v, w (cid:105) H = n (cid:88) i =1 v i w qi , for v, w ∈ F nq . This is a variant of the CRSS algorithm described above, where a Hermitian-self-dualcode of length n over the field extension F q is used to construct a self-dual code ˜ C oflength 2 n over F q to which the CRSS algorithm can be applied, obtained by expandingthe code words v ∈ C using a basis { , γ } of F q as a F q -vector space, where γ is anelement in F q (cid:114)F q satisfying γ q = − γ + γ for some fixed γ ∈ F q . Using this approach,it suffices to construct generalized Reed–Solomon codes ˆ C w,k of length n < q + 1 over F q that are Hermitian self-dual, in order to obtain associated quantum codes ˆ C w, n − k MATILDE MARCOLLI as a code subspace of the n q -ary qubits space H n = ( C q ) ⊗ n . It is possible to ensurethe Hermitian self-duality condition for generalized Reed–Solomon codes by takingthe weights vector w = ( w , . . . , w n ) ∈ ( F ∗ q ) n to satisfy (cid:80) ni =1 w q +1 i x qj + (cid:96)i = 0 for all0 ≤ j, (cid:96) ≤ k −
1, where x = ( x , . . . , x n ) ∈ F nq are the n chosen points (excluding ∞ ) of P ( F q ). Using this method, it is proved in [29] that the choice w i = 1, with n = q = F q and k = q , produces a Reed-Solomon code C = C ,q that is Hermitian-self-dual, and an associated [[ q + 1 , q − q + 1 , q + 1]] q -quantum Reed-Solomon codeˆ C . Moreover, it is also shown in [29] that for w i satisfying w q +1 i = ( (cid:81) j (cid:54) = i ( x i − x j )) − ,with n ≤ q and x = ( x , . . . , x n ) ∈ F q , and for k ≤ (cid:98) n/ (cid:99) , the generalized Reed-Solomon codes satisfy C w,k ⊆ C w,n − k and C w,k is hermitian self dual to C w,n − k . Thus,hence the CRSS algorithm can be applied to obtain an [[ n, n − k, k + 1]] q -quantumReed-Solomon code C w,k = C ( C w,k , C w,n − k ).2.4. The case of the perfect tensors.
In particular, from the construction de-scribed above we see that we obtain the case of perfect tensors as the special casewhere n = q and k = ( q − /
2. We obtain this using the generalized Reed–Solomoncodes as in Theorem 6 of [29], for the case n ≤ q and k ≤ (cid:98) n (cid:99) , with a choice of theweights w i satisfying w q +1 i = ( (cid:81) j (cid:54) = i ( x i − x j )) − , with n = q and x i ∈ F q . As shownin Theorem 6 of [29], this produces two classical generalized Reed-Solomon codes C w, q − ⊆ C w, q +12 that are hermitian self-dual. The associated quantum generalizedReed–Solomon code is then obtained via the general construction of Ashikhmin–Knill(Theorem 4 and Corollary 1 of [1]) that associates to a classical [ n, k, d ] q code con-tained in its hermitian dual a quantum [[ n, n − k, d ]] q code. One can see directlythat, in the case of perfect tensors when n = q , the weights are constant and givenby w q +1 i = p − i = 1 , . . . , q .Thus, we can regard the construction described above with generalized Reed–Solomon codes as a generalization of the usual construction of perfect tensors, whichrecovers the perfect tensor case for a particular choice of (constant) weights of theclassical Reed–Solomon codes.The more general cases with non-constant weights assign different weights to dif-ferent directions in the Bruhat–Tits tree. These may be useful in view of holographicmodels where the bulk geometry is dynamical, as in [15], and also described by dif-ferent weights in different directions in the tree.2.5. Holographic quantum codes on Bruhat–Tits trees and Mumford curves.
We use the procedure described above to pass from classical algebro-geometric codes,in particular generalized Reed-Solomon codes, to associated quantum stabilizer codes,to construct a holographic code on the Bruhat–Tits tree T K associated to the classicalcodes constructed above.We have seen above that, in order to apply the CRSS algorithm, we pass to aquadratic extension F q of F q and consider Reed-Solomon codes over F q . In termsof the Bruhat–Tits tree, we can pass to an unramified quadratic extension L of thefield K , so that the Bruhat–Tits tree T L is obtained from the Bruhat–Tits tree T K OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 9 simply by adding new branches at each vertex, so as to obtain a homogeneous treeof valence q + 1. Since the extension is unramified, it is not necessary to insert newvertices along the edges, and we can view the tree T K as a subtree of T L . We thenproceed to construct a classical code associated to T L using Reed-Solomon codes ˆ C w,k placed at the vertices according to the procedure described in the previous sections.Using the construction above, with w i = 1 and k = q , we associate to each vertex aquantum Reed-Solomon code ˆ C with code parameters [[ q + 1 , q − q + 1 , q + 1]] q .This corresponds to considering a state space H q +1 = ( C q ) ⊗ q +1 associated to eachvertex, which we can think of as a state with a q -ary qubit sitting at each of the q + 1 points of P ( F q ), or equivalently at each of the legs surrounding that vertexin T L . The quantum code ˆ C detects quantum errors of weight up to q + 1 = P ( F q ).Thus, by identifying T K ⊂ T L and P ( F q ) ⊂ P ( F q ) as the set of directions along thesubtree T K , we can arrange that the code ˆ C corrects quantum errors along the T K directions. One can also use a bipartition A ∪ A c of the edges at each vertex of T L ,with A = k and associate to the bipartition a pair of codes ˆ C w,k and ˆ C w,n − k withassociated quantum Reed-Solomon codes ˆ C w,k as above at the vertices of T K .One can think of the classical codes ˆ C associated to the vertices of the Bruhat–Titstree in this way as performing an encoding of q + 1 classical q -ary bits associated tothe points of P ( K ) into q + 1 classical q -ary bits associated to the points of P ( L ).Thus, the whole classical code associated to this quadratic extension can be seen asa way of encoding a state consisting of classical q -ary bits associated to the edges of T K into a set of classical q -ary bits associated to the edges of T L , and the letter into astate of q -ary bits associated to the set of boundary points P ( L ). The correspondingCRSS quantum codes ˆ C at the vertices of the Bruhat–Tits tree T L encode the inputgiven by the common eigenspace of the error operators associated to the code wordsof the classical code ˆ C into a state consisting of a q -ary qubit placed at each legaround the vertex.In order to combine these quantum codes placed at the vertices of the Bruhat–Titstree T L into a holographic code over the whole tree, with logical inputs in the bulkand physical outputs at the boundary P ( L ), notice that at each vertex v we have thesame subspace H ν given by the common eigenspace H ν = { ψ : E v,ϕ ( w ) ψ = λψ } for allwords ( v, w ) in the classical code. We encode states ψ ν ∈ H ν as ψ ν = ( ψ ν,x ) x ∈ P ( F q ) ,where the points x ∈ P ( F q ) label the legs around the vertex ν , so that we think of ψ ν,x ∈ C q as the q -ary qubit deposited on the leg x by the quantum code ˆ C sittingat the vertex ν . Starting at the root vertex and proceeding towards the outside ofthe tree, at each next step, the leg ∞ ∈ P ( F q ) around the new vertex ν is theone connected to a leg x i ∈ F q ⊂ P ( F q ) of the previous vertex ν (cid:48) , which receivesan output ψ ν (cid:48) ,i . Thus, the q -ary qubit ψ ν, ∞ is determined as it has to match theoutput ψ ν (cid:48) ,i of the previous code, while the remaining possible inputs correspond tothe choices of ψ ∈ H v with that fixed ψ v, ∞ component. Proceeding towards theboundary of the tree determines a holographic code on T L that outputs q -ary qubitsat the points of P ( L ). As mentioned above, the quantum code detects errors along the subtree T K . As in the case of the classical codes, there is an asymmetry in thisconstruction of the holographic code between the roles of the root vertex and of theremaining vertices of the Bruhat–Tits tree.3. Discrete and continuous bulk spaces: Bruhat-Tits buildings andDrinfeld symmetric spaces
Unlike its Archimedean counterparts, either Euclidean AdS /CFT with bulk H and boundary P ( R ) or Euclidean AdS /CFT with bulk H and boundary P ( C ), the p -adic AdS/CFT correspondence has two different choices of bulk spaces (one discreteand one continuous) which share the same conformal boundary at infinity. Thediscrete version of the bulk space is given by the Bruhat–Tits tree T K of PGL(2 , K ),with K a finite extension of Q p , while the continuous form of the bulk space is givenby Drinfeld’s p -adic upper half plane Ω. Both have the same boundary P ( K ). Weargue here that the full picture of the p -adic AdS/CFT correspondence should takeinto account both of these bulk spaces and the relation between them induced by thenorm map.The rank-two case can be generalized to higher rank, with the Bruhat–Tits build-ings of PGL( n, K ) generalizing the Bruhat–Tits tree and the higher dimensional Drin-feld symmetric spaces generalizing the Drinfeld upper half plane, see [23]. The geometry of the Drinfeld plane.
We review quickly the geometry of theDrinfeld upper half plane, see [4]. We denote by K a finite extension of Q p and by C p the completion of the algebraic closure of K . Drinfeld’s p -adic upper half plane isthe space Ω = P ( C p ) (cid:114) P ( K ) . We also denote by T K the Bruhat–Tits tree of PGL(2 , K ), with boundary at infinity P ( K ). It is convenient to think of P ( C p ) as the set of classes, up to homotheties in C ∗ p , of non-zero K -linear maps ϕ : K → C p , with P ( K ) the set of classes of mapsas above with K -rank equal to one. This can be seen by identifying points ( α : β )of P with homogeneous ideals (cid:104) yα − xβ (cid:105) in the polynomial ring in the variables( x, y ). The K -linear map ϕ : K → C p given by ϕ ( x, y ) = yα − xβ has a non-trivial kernel when α/β ∈ K (assuming β (cid:54) = 0) and is invertible if α/β ∈ C p (cid:114) K .Thus, P ( C p ) (cid:114) P ( K ) can be identified with the set of homothety classes of invertible K -linear maps ϕ : K → C p .Given such an injective linear map, one can then compose it with the norm on C p . Recall that the Bruhat–Tits tree can be defined in terms of equivalence classes ofnorms. Namely, vertices of the Bruhat–Tits tree correspond to classes of lattices M in K up to similarity, namely M ∼ M if M = λM for some λ ∈ K ∗ . To a lattice M one associates a norm | · | M , namely a real valued function on K which is positive onnon-zero elements, satisfies | a · x | M = | a |·| x | M for all a ∈ K and x ∈ K , with | a | the p -adic norm on K , and | x + y | M ≤ max {| x | M , | y | M } . The norm | x | M is defined as follows.Let π be a uniformizer in O K such that k = O K /π O K is the residue field k = F q . Thefractional ideal { λ ∈ K : λx ∈ M } is generated by a power π m . The norm is then OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 11 defined as | x | M = q m on non-zero vectors. Equivalent norms |·| M = γ |·| M for γ ∈ R ∗ + correspond to equivalent lattices. Two vertices in the Bruhat–Tits tree are adjacentiff the corresponding equivalence classes of lattices have representatives satisfying πM ⊂ M (cid:48) ⊂ M . To see this in terms of norms, we can choose an O K -basis { e , e } for M and { e , πe } for M (cid:48) . Then | xe + ye | M = max {| x | , | y |} and | xe + ye | M (cid:48) =max {| x | , | π | − · | y |} . The edge e between the vertices v = [ M ] and v (cid:48) = [ M (cid:48) ] is thenparameterized by the classes of norms | xe + ye | t = max {| x | , | π | − t · | y |} for 0 ≤ t ≤ K determines a map from the Drinfeld uppar half plane to the Bruhat–Titstree, directly induced by the norm. Namely, given a point in Ω, which we identify asabove with an invertible K -linear map ϕ : K → C p , we obtain a surjective mapΥ : Ω → T K by setting Υ( ϕ ) = | · | ϕ , where | · | ϕ is the norm on K defined by | x | ϕ = | ϕ ( x ) | , wherethe norm on the right-hand-side if the p -adic norm on C p . The explicit form of thismap is discussed in [4]. We identify a point ( ζ : ζ ) ∈ P ( C p ) (cid:114) P ( K ) with the map ϕ : K → C p that maps xe + ye (cid:55)→ xζ + yζ ∈ P ( C p ). In an affine patch (say with ζ (cid:54) = 0) we can write the homotethy class of ϕ as xe + ye (cid:55)→ xζ + y ∈ C p (cid:114) K . Thenthe preimages under the map Υ of two adjacent vertices v, v (cid:48) of T K and the edge e connecting them are given, respectively, byΥ − ( v ) = { ζ ∈ C p : | ζ | ≤ } (cid:114) (cid:91) a ∈O K /π O K { ζ ∈ C p : | ζ − a | < } Υ − ( v (cid:48) ) = { ζ ∈ C p : | ζ | ≤ q − } (cid:114) (cid:91) b ∈ π O K /π O K { ζ ∈ C p : | ζ − b | < q − } where v = [ M ], v (cid:48) = [ M (cid:48) ] with πM ⊂ M (cid:48) ⊂ M , and for e t = (1 − t ) v + tv (cid:48) , for0 < t <
1, along the edge e Υ − ( e t ) = { ζ ∈ C p : | ζ | ≤ q − t } , while Υ − ( e ) = { ζ ∈ C p : | ζ | ≤ } (cid:114) (cid:91) a ∈ ( O K (cid:114) π O K ) /π O K { ζ ∈ C p : | ζ − a | < } (cid:114) (cid:91) b ∈ π O K /π O K { ζ ∈ C p : | ζ − b | < q − } . For a detailed proof of this fact we refer to § − ( v ), Υ − ( v (cid:48) ) and Υ − ( e ) with ∂e = { v, v (cid:48) } in theBruhat–Tits tree can be illustrated as follows (from [4]): where the light colored region is Υ − ( v ), the striped shaded region is Υ − ( v (cid:48) ), and thedark shaded cylinder connecting them is Υ − ( e ). Thus, one can visualize the Drinfeldplane as a continuum that is a “tubular neighborhood” of the discrete Bruhat–Titstree, with the regions Υ − ( v ) viewed as the p -adic analog of pair-of-pants decomposi-tions for complex Riemann surfaces. A lift of the projection map Υ to the Bruhat-Titstree realizes the tree as a skeleton of the Drinfeld plane. Higher rank buildings and Drinfeld symmetric spaces.
An analogous descrip-tion holds relating the Bruhat–Tits buildings T n, K of PGL n +1 ( K ), with K a finiteextension of Q p and the associated Drinfeld symmetric spaceΩ n = P n ( C p ) (cid:114) ∪ H ∈H K H, where H K is the set of all K -rational hyperplanes in P n ( C p ). There is again a mapΥ n : Ω n → T n, K where the preimages of simplices in the Bruhat-Tits building isdescribed in terms of norm conditions, [23].The Bruhat–Tits building T n, K of PGL n +1 ( K ) is a simplicial complex with vertexset V ( T n, K ) = T n, K given by the similarity classes M ∼ M if M = λM for λ ∈ K ∗ of lattices in an n + 1 dimensional vector space V over K . A set { [ M ] , . . . , [ M (cid:96) ] } of such classes defines an (cid:96) -simplex in T (cid:96)n, K in the Bruhat–Tits building iff M (cid:41) M (cid:41) M (cid:41) · · · (cid:41) M (cid:96) (cid:41) πM , with π ∈ O K a prime element with F q = O K /π O K the residue field. Such a sequence determines a flag ¯ M (cid:41) ¯ M (cid:41) · · · (cid:41) ¯ M (cid:96) ⊇ M i = M i /πM i of an n + 1-dimensional F q -vector space. The (cid:96) -simplices in T (cid:96)n, K containing a given vertex [ M ] are in one-to-one correspondence with such flagswith [ M ] = [ M ]. As before, we consider norms on V (cid:39) K n +1 and similarity classesof norms. There is a PGL n +1 ( K )-equivariant homeomorphism between the resultingspace of equivalence classes of norms and the geometric realization of the simplicialcomplex T n, K . OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 13
Consider then points ζ = ( ζ : · · · : ζ n ) ∈ P n ( C p ) and the map ϕ : V → C p given by (cid:80) ni =0 a i e i (cid:55)→ (cid:80) ni =0 a i ζ i . The map | (cid:80) ni =0 a i e i | ϕ = | (cid:80) ni =0 a i ζ i | determines anequivalence class of norms iff the point ζ ∈ P ( C p ) does not lie in any K -rationalhyperplane. This determines the map Υ : Ω n → T n, K that generalizes in higher rankthe map from the Drinfeld plane to the Bruhat-Tits tree. As in the previous case, onecan describe the preimages under this map. For example, the preimage of a vertex v = [ M ] is given byΥ − ( v ) = {| ζ | = · · · = | ζ n | = 1 } (cid:114) ∪ H { ζ mod π ∈ H } with the union over hyperplanes andΥ − ( e t ) = {| ζ | = · · · = | ζ n − | = 1 , | ζ n | = q − t } for e t point along an edge e , with 0 < t <
1, see § Tensor networks on the Drinfeld plane
Because the p -adic AdS/CFT correspondence has two different choices of bulkspace, in addition to considering classical and quantum codes associated to theBruhat–Tits tree in constructing a version of tensor networks, we can also workwith the Drinfeld p -adic upper half plane. Because this is a continuous rather thana discrete space, the type of construction we can consider there will be more similarto the type of construction of tensor networks on the ordinary upper half plane (the2-dimensional real hyperbolic plane H ) described in [42]. The map Υ from the Drin-feld upper half plane to the Bruhat–Tits tree will then make it possible to relate theconstruction of tensor networks on the first to the latter. To this purpose, we startby reviewing the construction of the pentagon holographic code from [42].4.1. Pentagon Code on the Real Hyperbolic Plane.
In [42] a holographic codeis constructed using a tessellation of the real hyperbolic plane H by pentagons, withquantum codes given by a six leg perfect tensor placed at each tile. Unlike the codesdiscussed in the previous section on Bruhat–Tits trees, this code has no preferredbase point in the tiling and all tiles are treated equally, and the codes are symmetricwith respect to permutations of the five legs places across the edges of the tiles, thuspreserving the full symmetry group of the tiling. We discuss briefly some aspects ofthis pentagon code here before turning to analogous constructions on the Drinfeld p -adic upper half plane.The real hyperbolic plane H (which we can conveniently represent as the Poincar´edisk) has a regular periodic tessellation by right-angle pentagons. The corresponding symmetry group is the Fuchsian group Γ ⊂ PSL(2 , R ) of signature(2 , , , ,
2) generated by the reflections about the sides of a single right-angled hy-perbolic pentagon. An interesting property of this Fuchsian group, from the algebro-geometric perspective is the fact that, if one subdivides an equilateral right-angledhyperbolic pentagon into 10 triangles with angles π/ , π/ , π/
5, then one can realizethe group Γ as a finite index subgroup of a triangle Fuchsian group Γ (cid:48) of signature(2 , , H / Γ = X is arithmetic as an algebraic curve (that is, it is defined over a numberfield), [8].The construction of the pentagon holographic code in [42] places over each tile ofthis right-hangled pentagon tiling a quantum code given by the six leg perfect tensordetermined by a 5-qubit [[5 , , -quantum code C ⊂ H ⊗ , C = { ψ ∈ H ⊗ : S j ψ = ψ } where S = X ⊗ Z ⊗ Z ⊗ X ⊗ I , with X, Y, Z the Pauli gates and S , S , S , S = S S S S the cyclic permutations of S , and with H = C the 1-qbit Hilbert space.This is visualized as a code over H that has one logical input at each tiles of thepentagon tessellation and physical outputs across each edge of the tile, which arecontracted with the legs of the nearby tiles, so that the resulting holographic codehas one logical input at each tiles and outputs at the points at the boundary P ( R )that correspond to infinite sequences of tiles.4.2. Triangle Fuchsian Groups and Holographic Codes.
In view of adaptingthis construction to the p -adic setting, it is better to first consider a modification thatwill allow us to work directly with the triangle Fuchsian group Γ(2 , ,
5) rather thanwith its index 10 subgroup Γ of signature (2 , , , ,
2) which is the symmetry groupof the regular right-angled pentagon tiling.This means replacing each pentagons in the tiling with its subdivision into a tri-angulation of 10 hyperbolic triangles with a vertex at the center of the pentagon tile
OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 15 and the other vertices in the middle of the edges and at the original vertices of thepentagon.We then consider holographic codes constructed by quantum codes associated to thetriangle tiles. To this purpose, we do not necessarily require the group to be Γ(2 , , a, b, c ) ⊂ PSL ( R ) of hyperbolic type, a − + b − + c − < | (cid:105) (cid:55)→ | (cid:105) + | (cid:105) + | (cid:105)| (cid:105) (cid:55)→ | (cid:105) + | (cid:105) + | (cid:105)| (cid:105) (cid:55)→ | (cid:105) + | (cid:105) + | (cid:105) . This code can be represented as a perfect tensor | a (cid:105) (cid:55)→ T abcd | bcd (cid:105) in the sense of [42].By placing a copy of this code (thought of as a copy of the tensor T abcd at each triangletile of the tiling specified by the Fuchsian triangle group, one obtains a holographiccode with a logical input qutritt at each tile and physical output qutritts at points ofthe boundary P ( R ) corresponding to limit points of infinite sequences of tiles, froma specified base point in the bulk.A possible drawback of this simple construction is the fact that the quantum codewe are using does not contain any information about the specific triangle group thatdetermines the tessellation. This should be corrected by taking into consideration thestabilizer subgroups of edges and vertices, and incorporating them into the structureof the quantum code.This can be done by considering quantum codes placed at the vertices, ratherthan at the faces, of the tessellation of a hyperbolic triangle group Γ( a, b, c ). Thisrequires using perfect tensors of different valences, depending on the cardinality ofthe stabilizer group G v ⊂ Γ( a, b, c ) of the vertex v . A triangle Fuchsian group Γ( a, b, c ) in PSL ( R ) is generated by elements γ = σ σ , γ = σ σ and γ = σ σ , where the σ i with σ i = 1 are the reflections about the sidesof the fundamental domain triangle in H . The generators γ i satisfy the relations γ a = γ b = γ c = γ γ γ = 1, that correspond to rotations by angles 2 π/a , 2 π/b and2 π/c , respectively, with stabilizer groups Z /a Z , Z /b Z , Z /c Z associated to the verticesof the tessellation. Let (cid:96) = lcm { a, b, c } and consider the embedding Z /a Z (cid:44) → Z /(cid:96) Z by identifying Z /(cid:96) Z with (cid:96) -th roots of unity and mapping the generator of Z /a Z to ζ (cid:96)/a , where ζ is a primitive (cid:96) -th root. Similarly, for the other two groups. We canthen consider a construction like the quantum codes described in [22]. At a vertexlabelled by a stabilizer Z /a Z we consider the polynomial code | α (cid:105) (cid:55)→ (cid:88) α ,...,α a − ∈ Z /(cid:96) Z ⊗ x ∈ Z /a Z | f α ( x (cid:96)/a ) (cid:105) where f α ( t ) = α + α t + · · · + α a − t a − + αt a ∈ Z /(cid:96) Z [ t ]. This encodes an input in (cid:96) ( Z /(cid:96) Z ) into an output in (cid:96) ( Z /(cid:96) Z ) ⊗ (cid:96) ( Z /a Z ), which we think of as an (cid:96) -ary qubitdeposited at each side of the tessellation around the vertex. We can express this asa tensor T i ...i a with a + 1 legs. By contracting legs along the matching edges of thetessellations we obtain a holographic code that inputs an (cid:96) -ary qubit at each vertex ofthe tessellation and outputs at the points in the boundary P ( R ) that are endpointsof geodesic lines consisting of edges of the tessellation.4.3. Surface Quantum Codes.
There is another interesting construction of quan-tum stabilizer codes associated to tessellations of the hyperbolic planes, which wasdeveloped in [48]. These codes are constructed in general for a tiling defining a 2-dimensional surface (possibly with boundary). In particular, as shown in [48], theconstruction applies to the case of hyperbolic triangle Fuchsian groups, through theassociated Cayley graph and the tessellation defined by it. In particular it applies tothe triangle group Γ(2 , ,
5) which we use here as a replacement for the right-angledpentagon tile of [42]. The construction of surface codes in [48] arises as a naturalgeneralization of Kitaev’s toric code of [27]. They have the advantage that they relyagain on the CRSS algorithm that coverts classical into quantum codes, hence theycan be investigated in terms of classical coding theory techniques.Consider a tessellation R of a complex Riemann surface Σ and its dual R ∗ that hasa vertex for each face of R with two vertices being adjacent in R ∗ if the correspondingfaces in R share a common boundary edge. Let E = ( (cid:15) v,e ) be the vertex-edge incidencematrix of R and let E ∗ be the vertex-edge incidence matrix of the dual graph R ∗ .Let V and V ∗ be the F q -vector spaces spanned by the rows of E and E ∗ , respectively.The rows of E are orthogonal to V ∗ and the rows of E ∗ are orthogonal to V , withrespect to the standard pairing (cid:104) v, v (cid:48) (cid:105) = (cid:80) i v i v (cid:48) i . The first homology groups of R and R ∗ can be identified with the quotients V ⊥ /V ∗ and V ∗⊥ /V . A quantum codecan be associated to these data by a version of the CRSS algorithm, using the pair ofmatrices E and E ∗ . The construction of the quantum code follows the same procedureillustrated above: to pairs ( v, w ) of vectors v ∈ V , w ∈ V ∗ , one associates an erroroperator E ( v,w ) . The condition that the spaces V and V ∗ are mutually orthogonal OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 17 implies that the bilinear pairing (cid:104) ( v, w ) , ( v (cid:48) , w (cid:48) ) (cid:105) = (cid:104) v, w (cid:48) (cid:105) − (cid:104) v (cid:48) , w (cid:105) vanishes, hencethe group S formed by these E ( v,w ) and the ξ j , 0 ≤ j ≤ p − E ( v,w ) . This imposes dim V + dim V ∗ stabilizer conditions on n q -ary qubits, where n is the number of columns of E and E ∗ (number of edges of the graph R ), hence theparameters of the resulting quantum code are [[ n, k, d ]] q , where k = n − dim V − dim V ∗ and d = min { d V ⊥ (cid:114) V ∗ , d V ∗⊥ (cid:114) V } with d V ⊥ (cid:114) V ∗ = min { ω ( v ) : v (cid:54) = 0 , v ∈ V ⊥ (cid:114) V ∗ } with ω ( v ) = { i : v i (cid:54) = 0 } and similarly for d V ∗⊥ (cid:114) V .The Kitaev toric code consists of this construction applied to a graph R obtainedby a tessellation of a torus into squares. Generalizations to other Riemann surfacesand other tessellations were described in [48]. The main idea is to associate quantumsurface codes to increasingly large portions of a given tessellation of the hyperbolicplane or to suitable quotients of such regions.In our case, we can start with the right-angled pentagon tessellation R and its dualgraph R ∗ . After choosing a root vertex v of R ∗ (the center of a chosen face in thetiling) we denote by R N and R ∗ N the finite tessellations obtained by considering onlythe points that are up to N steps away from v (that is, such that the hyperbolicgeodesic to v passes through at most N tiles. Let V N and E N be the number ofvertices and edges in R N and let V ∗ N be the number of vertices in R ∗ N . The region R N has boundary, so in the construction of the dual graph R ∗ N we assume that thedual graph has E N = E ∗ N where the edges of R ∗ N include an edge cutting througheach boundary edge of R N and number of vertices V ∗ N given by the number of facesof R N plus one additional vertex for each boundary edge of R N . This will correctlyproduce, in the limit when N → ∞ boundary vertices on P ( R ) at the endpoints ofall geodesics of the dual graph R ∗ of the tessellation R , which should be the physicaloutputs of a holographic quantum code. Note that, starting from the central pentagonas zeroth step, at the first step one adds 10 new pentagons, five of which share anedge with the initial one and five that share a vertex. At the second step, one adds40 new pentagons, where each of the 5 pentagons of the first step that shared an edgewith the central pentagon (we call these tiles of the first kind) will be adjacent to 2new tiles of the first kind (sharing an edge) and 1 tile of the second kind (sharing avertex), while each of the 5 tiles of the second kind will be adjacent to 3 new tiles ofthe first kind and 2 new tiles of the second kind. Thus, if we let F N be the numberof new tiles (faces) added to the tessellation at the N -th step, with F N = m N + n N ,where m N and n N are, respectively, the number of tiles of the first and second kind,namely those that share a full edge or just a vertex with a tile of the ( N − m N +1 = 2 m N + 3 n N , n N +1 = m N + 2 n N with initial condition m = n = 5. This gives( m , n ) = (5 , , ( m , n ) = (25 , , ( m , n ) = (95 , , ( m , n ) = (355 , , ( m , n ) = (1325 , , ( m , n ) = (4945 , , . . . which corresponds to F = 10, F = 40, F = 150, F = 560, F = 2090, F = 7800 . . . Similarly, let V N denote the number of vertices added to the tessellation at the N -th step in the construction. We count as before the numbers m N and n N of facesadded at the N -th step, and for each face we count new vertices counterclockwise,counting the leftmost vertex (common to the next adjacent face) and not countingthe rightmost vertex (which we include in the counting for the next tile). This gives anumber of new vertices equal to W N = 2 m N + 3 n N = m N +1 , which is again computedin terms of the recursion above. We have V N = (cid:80) Nk =0 W k and V ∗ N = (cid:80) Nk =0 F k + E ∂,N ,where E ∂,N is the number of boundary edges at the N -th stage in the construction.This number is also equal to E ∂,N = 2 m N + 3 n N = m N +1 .One can also consider closed surfaces (without boundary) and associated quantumcodes by passing to Cayley graphs of quotient groups of the triangle Fuchsian groupassociated to the tessellation. In particular, the case that corresponds to the right-angled pentagon tile of the pentagon code of [42] is m = 4 and (cid:96) = 5, for which weuse the presentation Γ(2 , ,
5) = (cid:104) a, b | a = 1 , b = 1 , ( ab ) = 1 (cid:105) . The 2-complex used for the construction of the surface code in [48] is built by consid-ering 2-cycles of length (cid:96) = 5 and 2 m = 8 of the form { x, xb, xb , xb , xb , xb = x } and { x, xa, xab, xaba, x ( ab ) , x ( ab ) a, x ( ab ) , a ( ab ) a, x ( ab ) = x } , at every vertex x ,where all vertices have valence 3, with two edges { x, xb } and { x, xb − } along an (cid:96) = 5-face and the remaining edge { x, xa } along a 2 m = 8-face. By constructingan explicit matrix representation of Γ(2 , m, (cid:96) ) in the matrix group SL ( Z [ ξ ]), with ξ = 2 cos( π/m(cid:96) ), and taking reduction of the matrix entries modulo a prime p ,one obtains a finite quotient group G , as the image of Γ(2 , m, (cid:96) ) (as a subgroup ofSL ( Z [ ξ ])) in the quotient SL ( F p [ X ] / ( h ( X ))) where h ( X ) is a function of the 2 m(cid:96) -thnormalized Chebyshev polynomial. It is shown in [48] that this finite quotient group G has the property that any word in the generators that is the identity in G withoutbeing the identity in Γ(2 , m, (cid:96) ) must be of length at least log p . This condition onthe finite quotient group ensures that the finite graph given by the Cayley graph of G can be identified with a portion of the infinite Cayley graph of Γ(2 , m, (cid:96) ), given bythe neighborhood of size log p of a vertex. Provided that log p is sufficiently large, the2-cycles will then correspond to the (cid:96) -cycles and 2 m -cycles in this region, as the onlywords within that length that are equal to the identity in G are those already equal tothe identity in the triangle group. The quantum code associated to the Cayley graphof G and its dual graph then has code parameters [[ n, k, d ]] q with n = E , dimension k ≥ E (1 − (cid:96) + m ), where E and V are the number of edges and vertices in theCayley graph of G . Thus, the dimension grows linearly in the length of the code,while as shown in § p .The advantage of thinking in terms of triangle groups rather than pentagon codesis that there is a parallel theory of p -adic hyperbolic triangle groups in PGL ( K ), for K a (sufficiently large) finite extension of Q p , see [24], [25]. These are much more OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 19 severely constrained than the Fuchsian triangle groups in PSL ( R ) and only exist forsmall values of p .4.4. Triangle Groups on the Bruhat-Tits trees.
In order to consider analogousconstructions in the Drinfeld p -adic upper half plane Ω = P ( C p ) (cid:114) P ( K ), we firstneed to consider possible tilings of Ω. As in the case of the real hyperbolic plane H , we can think of a tessellation of the Drinfeld plane Ω as a fundamental domain F for the action of a subgroup Γ ⊂ GL ( K ) and its translates γ ( F ), γ ∈ Γ, withthe property that Ω / Γ is compact. Using the reduction map Υ : Ω → T K from theDrinfeld plane to the Bruhat–Tits tree, the property that Ω / Γ is compact translatesinto the property that T K / Γ is a finite graph.Unlike what happens in the case of Fuchsian groups acting on the real hyperbolicplane, the existence of p -adic triangle graphs is much more severely constrained. Oneis particularly interested in triangle groups of Mumford type. These are trianglegroups Γ ⊂ GL ( Q p ) such that ( P ( C p ) (cid:114) Λ Γ ) / Γ (cid:39) P ( C p ) and the uniformizationmap π : P ( C p ) (cid:114) Λ Γ → P ( C p ) is ramified at three points. In particular, by theclassification result of [24], [25] a p -adic triangle group of Mumford type, of signature(2 , ,
5) exists only when p = 2. No hyperbolic triangle groups of Mumford type existfor p >
5. The complete list of hyperbolic p -adic triangle groups Γ( a, b, c ) of Mumfordtype that can exist in the cases p = 2, p = 3, and p = 5 is given in [25].Let F be a fundamental domain for the action of the triangle group Γ(2 , ,
5) onthe Drinfeld p -adic upper half plane Ω with p = 2 and let T be a fundamental domainfor the action of the same group Γ(2 , ,
5) on the Bruhat–Tits tree T K of a (sufficientlylarge) finite extension K of Q . Since the reduction map Υ : Ω → T K is equivariantwith respect to the action of GL ( K ), we can assume that T = Υ( F ). More generally,we can consider any choice of one of the possible hyperbolic p -adic triangle groupsΓ( a, b, c ) of Mumford type, with p ∈ { , , } , acting on the Bruhat–Tits tree of a(sufficiently large) finite extension K of Q p , for one of these three possible values of p , and we proceed in the same way.A good way of describing the fundamental domain of the action of a finitely gen-erated discrete subgroup Γ ⊂ PGL ( K ) on the Bruhat–Tits tree T K and the resultingquotient graph is in terms of graphs of groups , as shown in [24], [25]. The theory ofgraphs of groups was developed in [2], [43]. A graph of groups consists of a finitedirected graph with groups G v and G e associated to the vertices and edges of thegraph, with G ¯ e = G e , together with injective group homomorphisms ϕ s : G e → G s ( e ) and ϕ t : G e → G t ( e ) from the group associated to an edge to the groups associated tothe source and target vertices. The fundamental group of a graph of groups is con-structed choosing a spanning tree of the graph: it is generated by the vertex groups G v together with an element h e for each edge e , with relations h ¯ e = h − e and h − e ϕ s ( g ) h e = ϕ t ( g ) , ∀ g ∈ G e and with h e = 1 for all e in the chosen spanning tree. If one denotes by G the graphand by G • the collection of groups associated to the vertices and edges, one writes π ( G , G • ) = lim −→ ϕ, G G • for the resulting amalgam given by the fundamental group ofthe graph of groups. In the case where the graph consists of one edge and two vertices,this fundamental group is just the pushfoward in the category of groups, namely theamalgamated free product G s ( e ) (cid:63) G e G t ( e ) . The main idea (see [2], [43]) is to associateto the action of a discrete group on a tree a quotient given not just by a graph but bythe richer structure of a graph of groups, which keeps track of the information aboutthe stabilizers of vertices and edges. In the case of a discrete subgroup Γ ⊂ PGL ( K ),we consider the tree of groups given by the subtree T Γ of the Bruhat–Tits tree T K together with the stabilizers G v and G e of vertices and edges, and we obtains a graphof groups as the quotient graph T Γ / Γ. It is shown in [25] that p -adic triangle groupsof Mumford type are characterized by the property that the quotient graph T = T Γ / Γis a tree consisting of three lines meeting at a single root vertex v . Such trees arecalled tripods . This tree, decorated with the stabilizer groups of vertices and edges isa tree of groups. The ends of this tree are the three branch points, at 0, 1 and ∞ ,of the genus zero curve Ω Γ / Γ. The group Γ can be reconstructed from the tree ofgroups (
T, G • ) as the associated fundamental group, [24]. Indeed the possible p -adictriangle groups of Mumford type are explicitly constructed using this method. Forexample, the tripod associated to the p -adic triangle group Γ(2 , ,
5) with p = 2, seenas a tree of groups, is the case (cid:96) = m = 1 of the following family (from [25]):with subgroups D ⊂ D ⊂ S and D intersecting A ⊂ S trivially. In the case (cid:96) = m = 1 the resulting amalgam agrees with the pushout S (cid:63) A A .4.5. Tessellations of the Drinfeld Plane.
A general algorithm exists for comput-ing fundamental domains in Bruhat–Tits trees for the action of certain quaterniongroups, see [11]. In these cases the algorithm produces
OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 21 (1) a connected subtree D Γ of the Bruhat–Tits tree which is a fundamental domainfor the group action, in the sense that the edges of D Γ form a complete set ofcoset representatives for E ( T ) / Γ;(2) the edge and vertex stabilizer groups G e , G v for e ∈ E ( D Γ ) and v ∈ V ( D Γ );(3) an explicit form for the quotient map by identifications ( v, v (cid:48) , γ ) between pairsof boundary vertices v, v (cid:48) of the fundamental domain D Γ , with γ ∈ Γ suchthat v (cid:48) = γv .This algorithm can be used to produce corresponding tessellations of the Drinfeld p -adic upper half plane. Let Γ, D Γ , G e , G v , and { ( v, v (cid:48) , γ ) } be given as above, throughthe algorithm of [11]. Using the projection map Υ : Ω → T from the Drinfeld planeto the Bruhat–Tits tree, we can construct an associated tessellation of the Drinfeldplane, where the tiles are given by γT , with γ ∈ Γ and T = (cid:91) v ∈ V ( D Γ ) Υ − ( v ) ∪ (cid:91) e ∈ E ( D Γ ) Υ − ( e ) . The gluing rules for the tiles are prescribed by the data (2) and (3) associated to thefundamental domain on the Bruhat–Tits tree.4.6.
Lifting Holographic Codes from the Bruhat–Tits Tree.
Another wayto obtain holographic codes on the Drinfeld plane is to lift the construction of theclassical and quantum codes on Bruhat-Tits trees described in § → T K . This means that the “tiles” to which we associate classical and quantumcodes in the Drinfeld plane are given, in this case, by the regions Υ − ( v ), the preim-ages in Ω of vertices of the Bruhat–Tits tree, and the outputs of each (classical orquantum) Reed–Solomon code is stored in the connecting regions Υ − ( e ). This canbe done by choosing a lift of the projection Υ, which realizes the Bruhat–Tits tree asa skeleton of Ω and constructing the holographic code over that skeleton. The choiceof a lift of the projection is non-canonical, hence this type of construction has thesame kind of drawback of the construction used in [22] to simulate the pentagon codevia a choice of a planar embedding of a tree along edges of the pentagon tiling of thereal hyperbolic plane. An advantage in this case, however, is that the projection Υ isequivariant with respect to the GL ( Q p ) symmetries so one maintains the symmetriesof the tree intact, unlike the case of the planar embedding used in [22].5. Holographic Codes on Higher Rank Bruhat–Tits Buildings
As above, we denote by T n, K the Bruhat–Tits building of GL n +1 ( K ) and by Ω n theDrinfeld symmetric space.Consider first the case of the Bruhat–Tits building of GL ( K ), with K a finiteextension of Q p with residue field F q , q = p r . The set of vertices adjacent to a givenvertex v ∈ V ( T , K ) is a bipartite set, consisting of the set of q + q + 1 F q -rationalpoints of the projective plane P over F q together with the set of q + q + 1 F q -rationallines of the projective plane P over F q . The surface X over F q obtained by blowingup all the F q -rational points of P contains an exceptional divisor (a line) for each F q -rational points of P and a proper transform (also a line) for each F q -rational linein P . Thus, to each vertex w adjacent to the given vertex v we associate a line (cid:96) w in the blowup surface X . Let u, w be vertices adjacent to v : the set { u, v, w } corresponds to a 2-simplex in the 2-dimensional simplicial complex T , K if and onlyif the lines (cid:96) u and (cid:96) w intersect nontrivially in X .In the case of Q one obtains the well known picture below, with the 7 points and7 lines of P ( F ) as vertices and with 21 edges, [9].In order to extend the construction of holographic codes to higher rank Bruhat–Tits buildings, in a way that reflects the associated geometries over finite fields thatdetermine the local structure of the building, we need to replace the classical Reed–Solomon codes with algebro-geometric codes associated to higher-dimensional alge-braic varieties.5.1. Codes on the Bruhat–Tits buildings of GL from algebro-geometriccodes on surfaces. A general procedure for constructing algebro-geometric codesover higher-dimensional algebraic varieties generalizing the Reed–Solomon codes isdescribed in [45], see also [20]. Given a smooth projective variety X over F q withan ample line bundle L , one obtains a linear code C ( X, L , P ), where P is a set of F q -algebraic points of X , as the image of the germ map α : Γ( X, L ) → ⊕ x ∈P L x (cid:39) F nq , which evaluates sections s ∈ Γ( X, L ) at points x ∈ P , with the last identificationgiven by a choice of an isomorphism L x (cid:39) F q of the fibers at x ∈ P , with n = P .For example, for X = P , with L = O ( m ), with 0 < m ≤ q , and P the setof all F q -rational points of P , one obtains a code C ( P , O ( m ) , P ( F q )) with length n = q + q + 1, dimension k = ( m + 1)( m + 2), and minimum distance bounded by d ≥ q + q + 1 − m ( q + 1), see [20].We focus here on the case of the Bruhat–Tits building of GL ( K ), with K a finiteextension of Q p with residue field F q , q = p r . As we mentioned above, the link of OLOGRAPHIC CODES ON BT BUILDINGS AND DRINFELD SPACES 23 a vertex in the Bruhat–Tits building is described in terms of the geometry of analgebraic surface X obtained by blowing up all the F q -algebraic points of P .We use the example above of algebro-geometric codes C ( P , L , P ( F q )) associatedto line bundles L over P to construct a classical holographic code on the Bruhat–Titsbuilding of GL ( K ). We fix a base vertex in the building and assign as logical inputthe datum of a divisor D on P so that L = L ( D ). Consider then the surface X over F q obtained by blowing up all the F q -rational points of P , and the pullback π ∗ L underthe projection map, and line bundles of the form ˆ L = π ∗ L ⊗ O ( − (cid:80) i k i E i ) where the E i are the exceptional divisors of the blowup. Assume that D and the k i are chosenso that ˆ L is represented by an effective divisor on X . We now consider the q + q + 1lines in X determined by the F q -lines of P and the q + q + 1 lines that correspondto the F q -points of P and the set P consisting of the q + 1 F q -rational points of eachof these lines, with P = 2( q + 1)( q + q + 1). The code C ( X, ˆ L , P ) can be viewed asa code that, given the logical input D at the base vertex v , deposits an output givenby a vector in F q +1 q at each adjacent vertex w in the Bruhat–Tits building. Theseoutputs are related by a consistency condition, which is determined by the edges and2-cells of the building. Namely, whenever w and u are vertices adjacent to v , suchthat { v, w, u } is a 2-cell in the building, we know the corresponding condition on X isthat the two lines (cid:96) w and (cid:96) u intersect. The presence of a point of intersection meansthat the corresponding vectors in F q +1 q must agree in one of the q + 1 coordinates.When one propagates the construction to nearby vertices in the Bruhat–Tits build-ing, part of the logical input is reserved for the output F q +1 q -vector of the nearbyvertices already reached by the previous steps from the chosen root vertex. As in thecase of the Bruhat–Tits tree, we identify the given F q +1 q -vector (computed as outputby the previous code) with assigned values at one of the lines in X that correspondsto one of the lines in P (which we can think of as the P at infinity in P ). There isa consistency condition for the output at a new vertex w that is adjacent to a 2-cellwhere the remaining two vertices v and v (cid:48) already have outputs x ( v ) , x ( v (cid:48) ) ∈ F q +1 q assigned by the previous codes: the outputs x ( v ) , x ( v (cid:48) ) at the two previous vertices v, v (cid:48) are two vectors in F q +1 q that agree in one coordinate, hence they fix the valuesof the sections at two intersecting lines in X . The resulting output x ( w ) at the newvertex w is then computed by the values at the q + 1 points of the line (cid:96) w of all sec-tions s that satisfy the constraints given by the assigned values at the points of (cid:96) v and (cid:96) v (cid:48) . The construction can in this way be propagated to the rest of the Bruhat–Titsbuilding of GL ( K ). This illustrates the general approach to constructing classicalholographic codes on higher rank Bruhat–Tits buildings.A construction of quantum holographic codes can be obtained from these classicalcodes using a version of the CRSS algorithm (possibly by allowing more general typesof weighted versions of the classical codes, as we discussed in the case of the Reed–Solomon codes). The details of the corresponding quantum codes for higher rankbuildings will be discussed in forthcoming work. Codes on Drinfeld symmetric spaces.
Another possible approach to theconstruction of holographic codes for higher-rank p -adic symmetric spaces consistsof working with Dirnfeld symmetric spaces instead of Bruhat–Tits buildings. Thisextends the approach discussed in § n → T n, K , from theDrinfeld space Ω n = P n ( C p ) (cid:114) ∪ H ∈H K H (the complement of the K -rational hyper-planes in P n ) to the Bruhat–Tits building of GL n ( K ). The idea here, as in § n , with logical inputs associatedto the regions Υ − ( v ), with v the vertices of T n, K and outputs and compatibility con-ditions along the edges, faces, and higher-dimensional cells. Since the projection mapΥ is equivariant with respect to the GL n ( K ) action, whatever symmetry the codesconstructed on T n, K exhibit will be inherited by the resulting codes on Ω n .The other possible approach consists of constructing a tensor network directly as-sociated to a given action of a discrete subgroup Γ of GL n ( K ) on the symmetric spaceΩ n . Roughly, the main idea in this case is to assign logical inputs to the fundamentaldomains of the action, while outputs should be associated to the generators of thediscrete group with compatibility conditions resulting from the relations. In this way,the codes assigned to each copy of the fundamental domain can be compatibly assem-bled into a global holographic code on Ω n , with logical inputs in the bulk and outputsat the boundary. The outputs should live on the points in the limit set of the groupaction on the rational hyperplanes H ∈ H K . We will discuss these constructions ofholographic codes on higher rank p -adic symmetric spaces in forthcoming work. Acknowledgment.
The author thanks Matthew Heydeman, Sarthak Parikh, andIngmar Saberi for many very useful discussions and an ongoing collaboration onseveral topics discussed in this paper, and especially Sarthak Parikh for suggestingseveral improvements to the paper. The author is partially supported by NSF grantDMS-1707882 and by the Perimeter Institute for Theoretical Physics.
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Department of Mathematics, University of Toronto, CanadaPerimeter Institute for Theoretical Physics, Waterloo, CanadaDivision of Physics, Mathematics, and Astronomy, Caltech, USA
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