Complete Toy Models of Holographic Duality

Complete Toy Models of Holographic Duality

Tamara Kohler Department of Computer Science, University College London, UK Toby Cubitt Department of Computer Science, University College London, UK
Abstract

Tensor networks have previously been used to construct holographic quantum error correcting codes (HQECC), which are toy models of the AdS/CFT correspondence. HQECC have been able to exhibit many of the interesting features of the duality, such as complementary recovery and redundant encoding. However, in previous HQECC the boundary Hamiltonian which results from mapping a local bulk Hamiltonian to the boundary is a non-local Hamiltonian, with global terms that act over the entire boundary. In this work, we combine HQECC with recently developed Hamiltonian simulation theory to construct a bulk-boundary mapping in which local bulk Hamiltonians map to local boundary Hamiltonians. This allows us to construct a toy model of a complete holographic duality between models, not just states and observables. This mapping of local Hamiltonians extends the toy models to encompass the relationship between bulk and boundary energy scales, and in particular the mapping between bulk and boundary time dynamics. We discuss what insight this gives into toy models of black hole formation.

1 Introduction

The AdS/CFT correspondence is a conjectured duality between quantum gravity in dimensional asymptotically AdS space, and a conformal field theory defined on its boundary [28]. It has provided insight into theories of quantum gravity, and has also been used as a tool for studying strongly-interacting quantum field theories. Recently it has been shown that important insight into the emergence of bulk locality in AdS/CFT can be gained through the theory of quantum error correcting codes [3]. This idea has been used to construct holographic quantum error correcting codes (HQECC) [31, 39, 38, 43, 21], which realise many of the interesting structural features of AdS/CFT.

Holographic quantum codes give a map from bulk to boundary Hilbert space, hence also from observables in the bulk to corresponding boundary observables. But the AdS/CFT correspondence is also a mapping between models, not just between states and observables; it relates quantum theories of gravity in the bulk to conformal field theories in one dimension lower on the boundary. For holographic code models, this means realising a mapping between local Hamiltonians in the bulk and local Hamiltonians on the boundary.

Since holographic quantum codes give a mapping from any bulk operator to the boundary, one can certainly map any local bulk Hamiltonian to the boundary. But this gives a completely non-local boundary Hamiltonian, with global interactions that act on the whole boundary Hilbert space at once. Local observables deep in the bulk are expected to map under AdS/CFT duality to non-local boundary observables, so this is fine – indeed, expected – for observables. But a global Hamiltonian acting on the entire boundary Hilbert space has lost all relation to the boundary geometry; there is no meaningful sense in which it acts in one dimension lower. Indeed, for toy spin models, any Hamiltonian whatsoever can be realised using a global operator. For the correspondence between bulk and boundary models to be meaningful, the local Hamiltonian describing the bulk physics needs to map to a local Hamiltonian on the boundary. For this reason, [31] study the mapping of observables and states in their construction, and do not apply it local Hamiltonians.

By standing on the shoulders of the holographic quantum code results, in particular the HaPPY code [31], and combining stabilizer code techniques with the recent mathematical theory of Hamiltonian simulation [9], we build on these previous results to construct a complete holographic duality between 3D hyperbolic space and its 2D boundary. This allows us to extend the toy models of holographic duality in previous HQECC to encompass Hamiltonians, and in doing so enables us to say something about how energy scales and dynamics in the bulk are reflected in the boundary.

The remainder of the paper is set out as follows. In Section 2 we present our main result, and give an overview of the proof. In Section 3 we introduce the background required to understand the construction, and prove a number of lemmas which are used in the main result. In Section 4.2 we set out the general procedure for constructing a HQECC in , and prove our main result: that there exists a complete holographic duality between and its 2D boundary. The proof does not rely on the properties of any particular HQECC, and holds provided that there exists a HQECC in . In Section 4.3 and Section 4.4 we provide two examples of HQECC in . Finally in Section 5 we discuss the implications of our results, including a toy model of black hole formation within these HQECC.

2 Main results

In this paper we construct a complete duality between and its 2D boundary. Our main results are encapsulated in the following theorem:

Theorem 2.1.

Let denote 3D hyperbolic space, and let denote a ball of radius centred at . Consider any arrangement of qudits in such that, for some fixed , at most qudits and at least one qudit are contained within any . Let denote the minimum radius ball containing all the qudits (which wlog we can take to be centred at the origin). Let be any local Hamiltonian on these qudits, where each acts only on qudits contained within some .

Then we can construct a Hamiltonian on a 2D boundary manifold with the following properties:

1. surrounds all the qudits, has diameter , and is homeomorphic to the Euclidean 2-sphere.

2. The Hilbert space of the boundary consists of a triangulation of by triangles of area, with a qubit at the centre of each triangle, and a total of triangles/qubits.

3. Any local observable/measurement in the bulk has a set of corresponding observables/measurements on the boundary with the same outcome. A local bulk operator can be reconstructed on a boundary region if acts within the greedy entanglement wedge of , denoted .111The entanglement wedge, is a bulk region constructed from the minimal area surface used in the Ryu-Takayanagi formula. It has been suggested that on a given boundary region, , it should be possible to reconstruct all operators which lie in [27]. The greedy entanglement wedge is a discretised version defined in [31, Definition 8]

4. consists of 2-local, nearest-neighbour interactions between the boundary qubits. Furthermore, can be chosen to have full local symmetry; i.e. the local interactions can be chosen to all be Heisenberg interactions: .

5. is a -simulation of in the rigorous sense of [9, Definition 23], with , and .

This result allows us to extend toy models of holographic duality such as [31] to include a mapping between Hamiltonians. In doing so we show that the expected relationship between bulk and boundary energy scales can be realised by local boundary models. In particular, in our construction toy models of static black holes (as proposed in [31]) correspond to high-energy states of the local boundary model, as would be expected in AdS/CFT.

Moreover, in our toy model we can say something about how dynamics in the bulk correspond to dynamics on the boundary. Even without writing down a specific bulk Hamiltonian we are able to demonstrate that the formation of a (toy model) static black hole in the bulk corresponds to the boundary unitarily evolving to a state outside of the code space of the HQECC, as expected in AdS/CFT (see Section 5.2 for details).

2.1 Proof overview

To construct the bulk/boundary map between Hilbert spaces, observables and local Hamiltonians described by Theorem 4.9, we combine new tensor network constructions of HQECC inspired by [31], with perturbation gadget techniques originally developed in Hamiltonian complexity theory. We then use the recently developed theoretical framework of analogue Hamiltonian simulation [9] to show that this gives a complete duality between the bulk and boundary physics.

[31] constructs a HQECC by building a tensor network composed out of perfect tensors, arranged in a tessellation of hyperbolic 2-space by pentagons. This gives a map from 2D bulk to 1D boundary. However, the Hamiltonian simulation constructions of [9] only work in 2D or higher, which means we require at least a 3D bulk and 2D boundary. We must therefore generalise the holographic tensor network codes to 3D as a first step. When working in it is possible to use the Poincare disc model to visualise the tessellations and determine their properties. However, in this is more difficult, and generalising the HQECC to 3D and higher requires a more systematic approach.

We use hyperbolic Coxeter groups to analyse honeycombings (higher-dimensional tessellations) of . (The techniques also generalise beyond 3D.) A Coxeter system is a pair , where is a group, generated by a set of involutions, subject only to relations of the form where , for . Coxeter groups admit a geometric representation as groups generated by reflections. Associated to every hyperbolic Coxeter system is a Coxeter polytope , where tessellates . All of the properties of the tessellation can be determined directly from the Coxeter system using combinatorics of Coxeter groups. For example, we use the Coxeter relations to prove that the boundary of the HQECC is homeomorphic to the Euclidean 2-sphere.

Generalising the method in [31], we construct tensor networks by taking a Coxeter system with Coxeter polytope , and placing perfect tensors in each polyhedral cell of (a finite portion of) the tessellation of by . Each perfect tensor in the interior of the tessellation has one free index, corresponding to a bulk qudit; the other indices are contracted with neighbouring tensors. Tensors at the outer edge can be shown, again using the Coxeter relations, to have between and additional free indices (where the perfect tensor has a total of indices), which correspond to qudits on the boundary. We can show that if the tessellation of associated to a Coxeter system has the properties required for a HQECC, then the associated Coxeter polytope has at least 7 faces, which means we require perfect tensors with at least 8 indices. There are no qubit perfect tensors with indices [14, 33, 25], so we must use qudit perfect tensors.

In order to later generate a local boundary model using perturbation gadgets, we need the tensor network to preserve the Pauli rank of operators. As we are working with qudits rather than qubits, we mean generalised Pauli operators on qudits, rather than qubit Paulis, and we choose prime-dimensional qudits. We use perfect tensors which describe qudit stabilizer absolutely maximally entangled states (AMES), constructed via the method in [22] from classical Reed-Solomon codes. Using properties of stabilizer groups, we show that tensor networks composed of these qudit stabilizer perfect tensors preserve the generalised Pauli rank of operators.

This Coxeter polytope qudit perfect tensor network gives a HQECC in . The non-local boundary Hamiltonian is given by , where is the projector onto the code-subspace of the HQECC, is the encoding isometry of the HQECC and satisfies .222 is not unique, as expected in AdS/CFT Comparing with the classification of Hamiltonian simulations in [9] it is clear that this mapping is an example of a simulation. (In fact, a perfect simulation in the terminology of [9].)

In order to construct a local boundary Hamiltonian we first determine the distribution of Pauli weights of the terms in from the properties of the Coxeter system. We then use perturbation gadgets to reduce the boundary Hamiltonian to a 2-local planar Hamiltonian. The techniques we use follow the methods from [30], however the perturbation gadgets derived in [30] can’t be used in our construction as the generalised Pauli operators aren’t Hermitian. We therefore generalise those to qudit perturbation gadgets which act on operators of the form , where . These gadgets meet the requirements in [9, 6] to be perturbative simulations. Finally we use simulation techniques from [9] to simulate the planar 2-local qudit Hamiltonian with a qubit Hamiltonian on a triangular lattice with full local SU(2) symmetry.

See Section 4.2.5 for the full proof.

3 Preliminaries

3.1 Perfect tensors and pseudo-perfect tensors

Perfect tensors were first introduced in [31], where they were used in the construction of HQECC from a 2D bulk to a 1D boundary.

Definition 1 (Perfect tensors, definition 2 from [31]).

A -index tensor is a perfect tensor if, for any bipartition of its indices into a set and a complementary set with , is proportional to an isometric tensor from to .

This definition is equivalent to requiring that the tensor is a unitary from any set of legs to the complementary set.

For one of the holographic error correcting code constructions in this work we will introduce a generalisation of perfect tensors, which we refer to as pseudo-perfect tensors.

Definition 2 (Pseudo-perfect tensors).

A -index tensor is a pseudo-perfect tensor if, for any bipartition of its indices into a set and a complementary set with , is proportional to an isometric tensor from to .

3.1.1 (Pseudo-)perfect tensors and absolutely maximally entangled states

Perfect and pseudo-perfect tensors are closely related to the concept of absolutely maximally entangled (AME) states, where AME states are states which are maximally entangled across all bipartitions. More formally:

Definition 3 (Absolutely maximally entangled states, definition 1 from [23]).

An AME state is a pure state, shared among parties , each having a system of dimension . Hence, , with the following equivalent properties:

1. is maximally entangled for any possible bipartition. This means that for any bipartition of into disjoint sets and with , and without loss of generality , the state can be written in the form:

 |Φ⟩=1√dm∑k∈Zmq|k1⟩B1|k2⟩B2...|km⟩Bm|ϕ(k)⟩A (1)

with

2. The reduced density matrix of every subset of parties with is maximally mixed, .

3. The reduced density matrix of every subset of parties with is maximally mixed.

4. The von Neumann entropy of every subset of parties with is maximal, .

5. The von Neumann entropy of every subset of parties with is maximal, .

These are all necessary and sufficient conditions for a state to be absolutely maximally entangled. We denote such state as an AME(n,q) state.

The connection between perfect tensors and AME states was noted in [31], and separately in [15] (where perfect tensors are referred to as multi-unitary matrices). Here we generalise the arguments from [15] to encompass the case of pseudo-perfect tensors.

A -index tensor, where each index ranges over values, describes a pure quantum state of -dimensional qudits:

 |ψ⟩=∑a1a2...at∈ZtqTa1a2...at|a1a2...at⟩ (2)

A necessary and sufficient condition for to be an AME state is that the reduced density matrix of any set of particles such that is maximally mixed. The reduced density matrix can be calculated as , where is a matrix formed by reshaping . Therefore, the state is an AME state if and only if the tensor is an isometry from any set of indices to the complementary set of indices with .

If is even (odd) this implies that is a perfect (pseudo-perfect) tensor. Therefore an AME state containing an even (odd) number of qudits can be described by a perfect (pseudo-perfect) tensor, and every perfect (pseudo-perfect) tensor describes an AME state on an even (odd) number of qudits.

3.1.2 (Pseudo-)perfect tensors and quantum error correcting codes

An quantum error correcting code (QECC) encodes -dimensional qudits into -dimensional qudits, such that located errors (or unlocated errors) can be corrected. The quantum Singleton bound states that . A QECC that saturates the quantum Singleton bound is known as a quantum maximum distance separable (MDS) code.

Previous work has established that every AME state is the purification of a quantum MDS code [23, 24].333The original proof actually demonstrates that AME states are the purification of a threshold quantum secret sharing (QSS) scheme, however every QSS scheme is equivalent to a quantum MDS code [7] so the result follows immediately. Furthermore, viewing the perfect tensor which describes an AME state as a linear map from 1 leg to legs, it is the encoding isometry of the quantum MDS code encoding one logical qudit [31].

We can generalise the proof in [24] to further characterise the connection between (pseudo-)perfect tensors and QECC:

Theorem 3.1.

Every AME state is the purification of a QECC for 444The proof of this theorem is a straightforward generalisation of [24, Theorem 2].

Proof.

Let be an AME state. For any partition of the state into disjoint sets and such that and we can write:

 |Φ⟩=√1qm∑i∈Zmq|i1...im⟩A|ϕ(i)⟩B (3)

where . If we fix qudits to be ‘logical’ qudits, this becomes

 |Φ⟩=√1qm∑i∈Zkq,j∈Zm−kq|i1...ik⟩L|j1...jm−k⟩A|ϕ(i,j)⟩B (4)

where we now have that is a fixed set of qudits such that , and and are any sets of qudits disjoint from and each other such that , . The set are the physical qudits. Define the basis states of a QECC as:

 (5)

Encode a logical state in the physical qudits as:

 |a⟩=∑i∈Zkqai|i⟩↪∑i∈Zkqai|Φi⟩ (6)

Now consider throwing away of the physical qudits. Since the sets and in Eq. 6 are arbitrary, we can always choose that the qudits we throw away are in the set . The qudits we are left with are then in the state:

 |ψ⟩=∑i,jai|ϕ(i,j)⟩B (7)

We can recover the logical state by performing the unitary operation:

 UB|ϕ(i,j)⟩B={|i1⟩B1...|ik⟩Bm−k|j1⟩Bm−k+1...|jm−k⟩Bmfor t even|i1⟩B1...|ik⟩Bm−k|j1⟩Bm−k+1...|jm−k⟩Bm|0⟩Bm+1for t odd (8)

Therefore, any set of qudits contains all the information about the logical state. By the no-cloning theorem, any set of qudits contains no information about the logical state, so the QECC can correct exactly erasure errors. This gives . ∎

Therefore, an AME state is the purification of a quantum MDS code with parameters ; while an AME state is the purification of a QECC with parameters . The parameters in the AME case do not saturate the Singleton bound, so it is not an MDS code, but it is an optimal QECC.555The terms MDS quantum code and optimal quantum code are sometimes used interchangeably. Here, by an optimal quantum code we mean either an MDS code, or a code for which is odd so which cannot saturate the Singleton bound, but for which the distance is maximal given this constraint.

If we consider the (pseudo-)perfect tensor, , which describes an AME state we have:

 |Φ⟩=∑i∈Zkq,j∈Zt−kqTi,j|i⟩L|j⟩P (9)

where and are the sets of logical and physical qudits in the corresponding QECC, , . The basis states for the QECC are then:

 |Φi⟩=∑j∈Zt−kqTi,j|j⟩P (10)

and the encoding isometry is:

 V=∑i∈Zkq,j∈Zt−kqTi,j|j⟩⟨i| (11)

So, viewed as an isometry from legs to legs a (pseudo-)perfect tensor is the encoding isometry of a QECC.

3.1.3 Existence of (pseudo-)perfect tensors

It is possible to construct -index (pseudo-)perfect tensors for arbitrarily large by increasing [2, 16, 23]. In our constructions we are specifically interested in (pseudo-)perfect tensors which describe stabilizer states / codes. These can be constructed for arbitrary provided is chosen appropriately [22]. A brief outline of the construction is given in Section 3.3.2.

3.2 Qudit stabilizer codes and states

We restrict our attention to qudits of dimension where is an odd prime.

3.2.1 Generalised Pauli group

The generalised Pauli operators on dimensional qudits are defined as:

 X=p−1∑j=0|j+1⟩⟨j| (12)
 Z=p−1∑j=0ωj|j⟩⟨j| (13)

where . The generalised Pauli operators obey the relations and .

The Pauli group on qudits is given by where , . Two elements and commute if and only if , where all addition is mod .

3.2.2 Qudit stabilizer codes

A stabilizer code on qudits is a -dimensional subspace of the Hilbert space given by:

 C={|ψ⟩∣M|ψ⟩=|ψ⟩∀M∈S} (14)

where is an Abelian subgroup of that does not contain .

The projector onto is given by [12]:

 Π=1|S|∑M∈SM (15)

where . is an elementary Abelian -group, so this implies that a minimal generating set for contains elements.

The minimum weight of a logical operator in an stabilizer code is . This is also the minimum weight of any operator that is not in the stabilizer, but which commutes with every element of the stabilizer.

3.2.3 Qudit stabilizer codes and (pseudo-)perfect tensors

In Section 3.3 we will see that we can construct (pseudo-)perfect tensors for arbitrary such that the AME state described by the tensor is a stabilizer state.666A stabilizer state is a stabilizer code with . This implies, using the method in [13] for generating short qubit stabilizer codes from longer ones, that the QECCs described by the tensors are stabilizer codes:

Theorem 3.2.

If a (pseudo-)perfect tensor, , with legs describes a stabilizer AME state, then the QECCs described by the tensor are stabilizer codes. The stabilizers of the code are given by the stabilizers of the AME state which start with , restricted to the last qudits.

Proof.

Consider an AME stabilizer state with stabilizer :

 |Φ⟩=√1pm∑i∈Zmp|i1...im⟩A|ϕ(i)⟩B (16)

where , .

We have that for all , where so a minimal generating set for contains elements. We can always pick a generating set for so that and begin with and respectively, and to begin with . Define to be restricted to the last qudits, where .

Consider the codespace, , for a error correcting code described by :

 |ψ⟩=∑i1∈Zai1|Φi1⟩ (17)

where and . If we act on with we find for , where for .

The group generated by for contains elements, and it stabilizes the code. We can carry out a similar procedure starting from the code, and removing more stabilizer generators to obtain the other QECC.

We refer to (pseudo-)perfect tensors which describe stabilizer AME states (and therefore stabilizer QECC) as stabilizer (pseudo-)perfect tensors.

We also require that all the QECC used in our construction map logical Pauli operators to physical Pauli operators. It is known that for qubit stabilizer codes a basis can always be chosen so that this is true [13], and the same group-theoretic proof applies to qudit stabilizer codes.777The discussion in [13] actually shows that there is an automorphism between and , where is the normalizer of in . As this is sufficient. The discussion in [13] can be extended to qudits of prime dimension by replacing phase factors of 4 with factors of , and dimension factors of 2 with factors of . The physical Pauli operators we obtain using this method are not given by acting on the logical Pauli operators with the encoding isometry, but they have the same action in the code subspace. So, we have that for qudit stabilizer codes it is always possible to pick a basis where where is a -qudit Pauli operator, is an -qudit Pauli operator, and is the encoding isometry of the QECC.

In our holographic QECC we do not have complete freedom to pick a basis, so we also need to show that we can pick this basis consistently. In order to show this we will require two lemmas about qudit stabilizer codes.

Lemma 3.3.

The smallest subgroup, , of the Pauli group such that , , where is the entire Pauli group.

Proof.

Consider the following process for constructing a set element by element such that , , where :

1. Select an arbitrary element of , .

2. Pick an element such that . This ensures that does not commute with , and is our first element of .

3. Pick an arbitrary element, of which commutes with every element of and is not the identity.

4. Choose any element, of which does not commute with , and add it to .

5. Repeat steps (3) and (4) until , , such that .

When we construct , every element which we add to is independent from every element already in . To see this note that by assumption there is some which commutes with every element in , but does not commute with . If was not independent from the other elements of , and and where then and so , and would commute with , contradicting our initial assumption.

We need to determine the minimum number of elements in when this process terminates.

Suppose we have repeated steps (3) and (4) times, so that . If there is an element of which commutes with every element of then for all . is described by degrees of freedom: and , and there are homogeneous equations which needs to satisfy.888The equations are homogeneous as all constant terms are equal to zero. Homogeneity of the equations ensures that the equations are not inconsistent. Provided , the set of equations is underdetermined, and we can always choose a which commutes with every element of and is not the identity. If then the solution to the equations is uniquely determined, and is the identity. At this point we cannot continue with the process, so it terminates with .

Therefore, the smallest set of elements of such that , , where contains elements. At this stage is not a group because all the elements of are independent so the set isn’t closed. Any independent elements of generate the entire group, so the smallest group which contains every element of is itself. ∎

Lemma 3.4.

In an stabilizer code, the action of an encoded Pauli operator on any physical qudits can be chosen to be any element of .

Proof.

The encoded Pauli operator is not unique, and the different possible physical operators are related by elements of the stabilizer. We therefore need to show that the stabilizer restricted to any set of qudits is the entire Pauli group .

An stabilizer code can correct all Pauli errors of weight and less. A correctable error doesn’t commute with some element of the stabilizer, so for any Pauli operator of weight there such that . The result follows immediately from Lemma 3.3. ∎

Theorem 3.5.

If there exists a basis such that the QECC described by a (pseudo-)perfect tensor, , from qudit to qudits maps Pauli operators to Pauli operators, then all other QECC described by which include qudit in the logical set also map Pauli operators to Pauli operators in that basis.

Proof.

Let the AME state described by be given by:

 |Φ⟩=∑a∈Zp∑b∈Zk−1p∑c∈Zm−kp|a⟩l|b1...bk−1⟩L|c1...cm−k⟩A|ϕ(a,b,c)⟩B (18)

where , and .

The basis states of the QECC from qudit to other qudits are given by:

 |Φa⟩=∑b∈Zk−1p∑c∈Zm−kp|b1...bk−1⟩L|c1...cm−k⟩A|ϕ(a,b,c)⟩B (19)

and the encoding isometry is given by:

 V=∑a|Φa⟩⟨a| (20)

The basis states of a QECC from a set qudits (where , ) to qudits is given by:

 |Φa,b⟩=∑c∈Zm−kp|c1...cm−k⟩A|ϕ(a,b,c)⟩B (21)

and the encoding isometry is given by:

 V′=∑a|Φa,b⟩⟨a,b| (22)

By assumption we have:

 VP1V†=QVV† (23)

where and , .

Therefore:

 ∑aa′|Φa⟩⟨a|P1|a′⟩⟨Φa′|=Q∑a|Φa⟩⟨Φa| (24)

Consider the action of on :

 V′(P1⊗P2)V′†=∑a,a′,b,b′|Φa,b⟩⟨a,b|(P1⊗P2)|a,b⟩⟨Φa,b|=∑a,a′,b,b′⟨b||Φa⟩⟨a|P1|a′⟩⟨b|P2|b′⟩⟨Φa′||b′⟩=∑a,a′,b,b′⟨b|P2|b′⟩⟨b|(|Φa⟩⟨a|P1|a′⟩⟨Φa′|)|b′⟩=∑a,b,b′⟨b|P2|b′⟩⟨b|(Q|Φa⟩⟨Φa|)|b′⟩=∑a,b,b′,b′′⟨b|P2|b′⟩⟨b|Q′2|b′′⟩Q′|Φa,b′′⟩⟨Φa,b′|=∑a,bQ′|Φa,b⟩⟨Φa,b| if P2=Q′2=V′V′†P′(n′) (25)

where and and indicate restricted to the first and remaining qudits respectively ().

Therefore, if acts as on the first qudits, then maps to a Pauli under . The operator is not unique, and from Lemma 3.4 we know that its action on qudits can be chosen to be any element of . So we can choose that acts as on the first qudits, for . ∎

3.3 Existence of (pseudo-)perfect stabilizer tensors

3.3.1 Classical coding theory

A classical linear code, , encodes -dimensional dits of information in dits. It can be described by a generator matrix , where information is encoded as for . Equivalently, admits a description as the kernel of a parity check matrix . Consistency of the two descriptions implies , , and hence the rows of are orthogonal to the rows of .

The minimum distance of a classical code is defined as the minimum Hamming distance between any two code words. It is bounded by the classical Singleton bound, . Codes which saturate the classical Singleton bound are referred to as classical MDS codes.

Reed-Solomon codes are a class of classical MDS codes [34].999Reed Solomon codes can be defined over any finite field, but we only require the definition of Reed Solomon codes over for our construction.

Definition 4.

Let be a prime, and let be integers such that . For a set , the Reed-Solomon code over is defined as:

 CRS[n,k]={(P(α1),P(α2),...,P(αn))∈Znp∣P(X)∈Zp[X],deg(P)≤k−1} (26)

where is the polynomial ring in over .101010The polynomial ring in over , , is the set of polynomials where .

To encode a message in the Reed-Solomon code define the polynomial:

 Pa(X)=a0+a1X+a2X2+...+ak−1Xk−1 (27)

and construct the codeword .

Reed-Solomon codes are linear codes, with generating matrix:

 (28)

The generator matrix can be put into standard form (where is a matrix) using Gauss-Jordan elimination over the field . The parity check matrix is then given by .

3.3.2 Constructing AME stabilizer states

An AME stabilizer state can be constructed from a classical MDS code with . The state is given by [22]:

 |Φ⟩=1dl2∑x∈Zlp|Gx⟩ (29)

and has stabilizers for all , and where for all . The full set of stabilizers is given by the generator matrix [22]:

 M=(G00H) (30)

where for .

Reed-Solomon codes can be constructed for any satisfying [34], so by increasing this construction can provide AME stabilizer states for arbitrarily large . By Theorem 3.2 the tensor which describes the AME stabilizer states will be a stabilizer (pseudo-)perfect tensor.

This construction is not optimised to minimise for a given . It is possible to construct generalised Reed-Solomon codes which exist for [37], which if used in this construction will give (pseudo-)perfect stabilizer tensors acting on lower dimensional qudits for certain values of . There are also methods for constructing stabilizer perfect tensors for which using cyclic and constacyclic classical MDS codes [17], but this method is significantly more involved than the one presented here, and does not work for pseudo-perfect tensors. In our construction there is no benefit to minimising , so we have selected the simplest, most universal method for constructing (pseudo-)perfect stabilizer tensors.

3.4 Hyperbolic Coxeter groups

3.4.1 Coxeter systems

The holographic error correcting codes presented in this paper are tensor networks embedded in tessellations of . We will use Coxeter systems to analyse these tessellations.111111An overview of hyperbolic Coxeter groups can be found at [11].

Definition 5 (Coxeter system [40]).

Let , be a finite set. Let be a matrix such that:

• ,

• ,

• ,

is called the Coxeter matrix. The associated Coxeter group, , is defined by the presentation:121212A group presentation , where is a set of generators and is a set of relations between the generators, defines a group which is (informally) the largest group which is generated by and in which all the relations in hold.

 W=⟨S∣(sisj)mij=1∀i,j∈I⟩ (31)

The pair is called a Coxeter system.

To understand the connection between Coxeter systems and tesselations of hyperbolic space we need to introduce the notion of a Coxeter polytope.

Definition 6.

A convex polytope in or is a convex intersection of a finite number of half spaces. A Coxeter polytope is a polytope with all dihedral angles integer submultiples of .

A Coxeter system can be associated to every Coxeter polytope. Let be the facets of , and if set , where is the dihedral angle between facets and . Set , and if set . Let be the reflection in . The Coxeter group with Coxeter matrix is a discrete subgroup of , generated by reflections in the facets of , and tiles [10].

Coxeter systems can be represented by Coxeter diagrams. In a Coxeter diagram a vertex is associated to every (or equivalently to every facet in the corresponding Coxeter polytope). Vertices are connected by edges in the following manner:

• If (i.e. facets and in the Coxeter polytope are orthogonal) there is no edge between the vertices representing and

• If (i.e. the dihedral angle between and is ) there is an unlabelled edge between vertices representing and

• If (i.e. the dihedral angle between and is ) there is an edge labelled with between vertices representing and

• If (i.e. facets and in the Coxeter polytope diverge) there is a dashed edge between the vertices representing and

A Coxeter group is irreducible if its Coxeter diagram is connected.

Faces of correspond to subsets of