Holographic quantum errorcorrecting codes: Toy models for the bulk/boundary correspondence
Abstract
We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum errorcorrecting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum errorcorrecting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the RyuTakayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum errorcorrecting features of AdS/CFT recently proposed in Almheiri14 .
1 Introduction
The AdS/CFT correspondence, an exact duality between quantum gravity on a dimensional asymptoticallyAdS space and a dimensional CFT defined on its boundary, has significantly advanced our understanding of quantum gravity, as well as provided a powerful framework for studying stronglycoupled quantum field theories. One aspect of this duality is a remarkable relationship between geometry and entanglement. This notion first appeared in the proposal Maldacena03 that two entangled CFT’s have a bulk dual connecting them through a wormhole, and was later quantified by Ryu and Takayanagi via their proposal that entanglement entropy in the CFT is computed by the area of a certain minimal surface in the bulk geometry Ryu06 ; Ryu06b . This latter proposal, known as the RyuTakayanagi (RT) formula, has led to much further work on sharpening the connection between geometry and entanglement Hubeny07 ; Headrick07 ; Raamsdonk09 ; Raamsdonk10 ; Hayden13b ; Lewkowycz13 ; Maldacena13 ; Lashkari14
In the condensed matter physics community, improved understanding of quantum entanglement has led to significant progress in the numerical simulation of emergent phenomena in stronglyinteracting systems. A key ingredient of such algorithms is the use of tensor networks to efficiently represent quantum manybody states Vidal03b ; Verstraete04b ; Verstraete08 . Vidal combined this idea with entanglement renormalization to formulate the Multiscale Entanglement Renormalization Ansatz (MERA) Vidal07 ; Vidal08 , a family of tensor networks that efficiently approximate wave functions with longrange entanglement of the type exhibited by ground states of local scaleinvariant Hamiltonians Evenbly09 ; Evenbly09b ; Evenbly10 . The key idea is to represent entanglement at different length scales using tensors in a hierarchical array.
In the AdS/CFT correspondence, the emergent radial direction can be regarded as a renormalization scale Susskind1998 , and spatial slices have a hyperbolic geometry resembling the exponentially growing tensor networks of MERA. This similarity between AdS/CFT and MERA was pointed out by Swingle, who argued that some physics of the AdS/CFT correspondence can be modeled by a MERAlike tensor network where quantum entanglement in the boundary theory is regarded as a building block for the emergent bulk geometry Swingle12 ; Swingle12b .
Recently it has been argued in Almheiri14 that the emergence of bulk locality in AdS/CFT can be usefully characterized in the language of quantum errorcorrecting codes. Certain paradoxical features of the correspondence arise naturally by interpreting bulk local operators as logical operators on certain subspaces of states in the CFT, whose entanglement structure protects these operators from boundary erasures. Moreover, inspired by Swingle12 ; Swingle12b , it was suggested that there should be tensor network models that concretely implement these ideas.
In this paper, we propose such a family of exactly solvable toy models of the bulk/boundary correspondence based on a novel tensornetwork construction of quantum errorcorrecting codes. Other authors have recently used holographic ideas Yoshida2013 ; Latorre2015 and related tensor network constructions Ferris14 ; Bacon14 to build quantum codes with interesting properties or toy models of the bulk/boundary correspondence Qi13 , but our approach differs from previous work by combining the following properties, all of which are desirable for a model of AdS/CFT:

Exactly solvable: Many of the properties of our models can be shown explicitly. In particular, an exact prescription for mapping bulk operators to boundary operators can be obtained, and we can give examples where the RyuTakayanagi formula holds exactly for all connected boundary regions.

QECC: Our models are quantum errorcorrecting codes, where the bulk/boundary legs of the tensor network correspond to input/outputs of an encoding quantum circuit. In this sense they realize explicitly the proposal of Almheiri14 .

Bulk uniformity: The tensor network is supported on a uniform tiling of a hyperbolic space, known as a hyperbolic tessellation. If the tiling is extended to an infinite system, the tensor network has no inherent directionality and all the locations in the bulk can be treated on an equal footing (see Fig. (b)b).
The rest of this paper is organized as follows: In section 2, we introduce a class of tensors called perfect tensors, which are associated with pure quantum states of many spins such that the entanglement is maximal across any partition of the spins into two sets of equal size. In section 3, we construct holographic states and codes by building networks of perfect tensors. These codes have properties reminiscent of the AdS/CFT correspondence, elucidated in the rest of the paper, where the code’s logical/physical degrees of freedom are interpreted as the bulk/boundary degrees of freedom of a CFT with a gravitational dual.
In section 4, we study the entanglement structure of holographic states, showing that the RyuTakayanagi formula is exactly satisfied for any connected boundary region, developing a graphical representation of multipartite entanglement, and confirming the negativity of tripartite information Hayden13b . In section 5, we investigate the dictionary relating bulk and boundary observables, define a lattice version of the causal wedge, and explain how bulk local operators in the causal wedge can be reconstructed on the boundary; we also define a lattice version of the entanglement wedge, and offer evidence supporting the entanglement wedge hypothesis proposed in Headrick2014 ; Wall2012 ; Czech2012 , see also Jafferis2014 . We briefly discuss how to describe black holes using holographic codes in section 6. Section 7 contains our conclusions, and many details appear in the appendices.
2 Isometries and perfect tensors
In this section we review some tools which will be used in our constructions of holographic states and codes. We begin with a standard definition:
Definition 1.
Say and are two Hilbert spaces, not necessarily of the same dimensionality. An isometry from to is a linear map with the property that it preserves the inner product.
If and have finite dimensionality, as we will assume throughout this paper, then it immediately follows that such a can exist only if their dimensionalities and obey . In the special case where , is just a unitary transformation. Clearly the composition of two isometries is also an isometry.
If is an isometry, then is the identity on and is a projector mapping to the range of . We may represent the map as a twoindex tensor acting as
(1) 
where denotes a complete orthonormal basis for and for . Then is an isometry if and only if
(2) 
We represent this graphically in figure 1, following the convention that operators are ordered from left to right, so that in the figure is applied after . We will call a tensor obeying (2) an isometric tensor.
Isometric tensors have the property that any operator acting on its “incoming” leg, can be replaced by an equal norm operator acting on its “outgoing” leg, because
(3) 
we illustrate this property in figure 2.
This operation is essential for what follows, and we will often describe it as “pushing an operator through a tensor”. It is also easy to check a useful converse of operator pushing: If the twoindex tensor has the property that any unitary transformation contracted with its incoming index can be replaced by a corresponding unitary transformation contracted with its outgoing index (i.e., ), then obeys (2) up to a scalar factor, and therefore must be proportional to an isometric tensor.
Another important property of isometric tensors is that if the input Hilbert space factorizes, we may reinterpret an input factor as an output factor while preserving (2), up to an overall rescaling. That is, if is an isometric map, acting on a basis according to
(4) 
then acting as
(5) 
obeys . We illustrate this property in figure 3.
In this paper we will be interested in a special class of isometric tensors, which we will call perfect tensors. To formulate the concept of a perfect tensor, first note that we may divide the indices of a tensor into a set and a complementary set . We use to denote the cardinality of the set ; hence . Then may be regarded as a linear map from the span of the indices in to the span of the indices in . We will usually assume that each index ranges over values, and we will use to denote both the set of indices and the corresponding vector space with dimension ; thus we say maps to .
Definition 2.
A index tensor is a perfect tensor if, for any bipartition of its indices into a set and complementary set with , is proportional to an isometric tensor from to .
It is not obvious that nontrivial perfect tensors exist, but they do! Note that for to be perfect it suffices for to be a unitary transformation when ; in that case the property illustrated in figure 3 ensures that is proportional to an isometric tensor for . In Appendix A we describe perfect tensors explicitly for the case , and for the case , ; other cases with larger and are also discussed there. To keep our discussion concrete, we will focus especially on the sixindex tensor for qubits ()^{1}^{1}1This can be obtained from the encoding map of the 5qubit code., but much of what we say applies to arbitrary index perfect tensors.
Perfect tensors are related to other notable ideas in quantum information theory. In general, a tensor with indices, each ranging over values, describes a pure quantum state of dimensional spins, where, up to a normalization factor,
(6) 
A perfect tensor describes a pure state of spins with a special property — any set of spins is maximally entangled with the complementary set of spins. Such states have been called absolutely maximally entangled (AME) states Helwig2012 ; Helwig2013 . Conversely any state defines a perfect tensor. Regarded as a linear map from one spin to spins, a perfect tensor is the isometric encoding map of a quantum errorcorrecting code which encodes a single logical spin in a block of physical spins, where the logical spin is protected against the erasure of any physical spins. Because is more than half of all the physical spins, this is the best possible protection against erasure errors compatible with the nocloning principle. In coding terminology this code has distance and is denoted , where is the number of physical spins in the code block, is the number of protected logical spins, and is the code distance. This code is also the basis for a quantumsecretsharing scheme called a threshold scheme Cleve1999 ; code states have the property that a party holding any spins has no information about the logical spin, while a party holding any spins has complete information about the logical spin (because erasure of the remaining spins is correctable).
3 Construction of holographic quantum states and codes
We have seen how tensors can be interpreted as quantum states or quantum codes. In this section we construct tensor networks in which the fundamental building blocks are perfect tensors. Our tensor networks describe states which we call holographic states, and codes which we call holographic codes.
We shall focus on examples based on tilings of twodimensional hyperbolic space, which are specific realizations of uniform hyperbolic tilings known as hyperbolic tessellations. These tilings have desirable symmetries for constructing a toy model of the AdS/CFT correspondence. In particular they are discretely scaleinvariant, and there exist graph isomorphisms that bring any point in the graph to the center while preserving the local structure of the tiling.^{2}^{2}2Such transformations can be directly visualized using Kaleidotile software Kaleidotile , which is freely available and has been of great aid in developing geometric intuition and producing figures of uniform hyperbolic tilings in this paper. The machinery we develop may also be straightforwardly applied to nonuniform and higherdimensional graphs.
Let’s first consider a uniform tiling of a twodimensional hyperbolic space by hexagons, with four hexagons adjacent at each vertex, as depicted in Fig (a)a. A perfect tensor with six legs is placed at each hexagon, and legs of perfect tensors are contracted with neighboring tensors at shared edges of the hexagons. We associate physical spins with the uncontracted open tensor legs on the boundary of the hyperbolic tiling; the tensor network corresponds to a pure state of these boundary spins, which we call a holographic state. Note that perfect tensors are not necessarily symmetric under all the possible permutations of tensor legs, and thus we specify some particular ordering of tensor legs in the construction.
We may similarly attach a state interpretation to more general networks constructed by contracting perfect tensors:
Definition 3.
Consider a tensor network composed of perfect tensors which cover some geometric manifold with boundary, where all the interior tensor legs are contracted. A holographic state is a state interpretation of such a tensor network, where physical degrees of freedom are associated with all uncontracted legs at the boundary of the manifold.
We now provide an example of a holographic quantum code. As in a holographic state, we consider a uniform tiling of the hyperbolic disc, this time by pentagons, with four pentagons adjacent at each vertex. A perfect tensor with six legs is placed at each pentagon, so that each tensor has one additional uncontracted open leg. This additional tensor leg is interpreted as a bulk index or logical input for the tensor network (see Fig. (b)b). The entire system can be viewed as a big tensor with logical legs in the bulk and physical legs on the boundary. We then have the following theorem:
Theorem 1.
The pentagontiling tensor network is an isometric tensor from the bulk to the boundary. We call it the holographic pentagon code.
We can prove this theorem by noting that if we order the tensors into layers labeled by increasing graph distance from the center, each tensor has at most two legs contracted with the tensors at the previous layer (this property is a consequence of the “negative curvature” of the graph). Therefore, even if we regard the pentagon’s bulk logical index as an input leg, the total number of input legs is at most three, and we may therefore regard each tensor as an isometry from input legs to output legs. Applying the perfect tensors layer by layer, and recalling that the product of isometries is an isometry, we obtain an isometry mapping all the logical indices in the bulk to the physical indices on the boundary.
We can view this isometry as the encoding transformation of a quantum errorcorrecting code, which we call a holographic code. The number of logical dimensional spins is the number of pentagons in the tiling, and the number of physical dimensional spins in the code block is the number of uncontracted boundary indices in the tensor network. We show in Appendix C that the rate of the code, meaning the ratio of the number of logical spins to the number of physical spins, approaches
(7) 
in the limit of a large number of layers.
This pentagon code was constructed by successively adding layers of tensors starting from the center and stopping after repeating this procedure a certain number of times (two layers in figure (b)b). Alternatively, we may fill the bulk using a nonuniform cutoff, so that the graph distance between the “center” and the boundary varies from one portion of the boundary to another (as occurs in figure (a)a). By exerting this freedom, we may change the corresponding value 7 for the rate of the code and even slightly increase it. By varying the choice of perfect tensor and the shape of the cutoff, a large family of holographic codes can be constructed:
Definition 4.
Consider a tensor network composed of perfect tensors which cover some geometric manifold with boundaries. The tensor network is called a holographic code if it gives rise to an isometric map from uncontracted bulk legs to uncontracted boundary legs.
Tensor networks with open legs in the bulk were first proposed by Vidal Vidal08 . More recently, Qi Qi13 constructed a tensortree model with an exact unitary mapping between the bulk and the boundary. The most important difference between their models and ours is that their states are not protected against erasure of physical spins because the code rate is asymptotically unity. In addition our models are more symmetric; since perfect tensors can be interpreted as isometries along any direction, our models have no preferred direction in the bulk and all bulk sites are treated equally. In particular, the pentagon code has the nice feature that, because the 6leg perfect tensor we construct in appendix A is symmetric under cyclic permutations of five of the legs, which we take to be the contracted legs, the symmetry of the network is just the full symmetry of the graph.
4 Entanglement structure of holographic states
In this section we explore to what extent holographic states reproduce key properties of the AdS/CFT correspondence, such as the RyuTakayanagi formula for entropy of a boundary region Ryu06 and the negativity of tripartite information Hayden13b .
4.1 RyuTakayanagi formula
The RyuTakayanagi (RT) formula says that for a CFT whose gravitational dual is wellapproximated by Einstein gravity at low energies, in any static state with a geometric bulk description the entropy of a boundary subregion at fixed time obeys
(8) 
here is Newton’s constant and is the minimalarea codimensiontwo bulk surface whose boundary matches the boundary of . In our examples the bulk theory is dimensional, so will be a spacelike bulk geodesic whose “area” is just defined as its length.
In our discrete setting, we will define as a certain cut through the tensor network which partitions it into two disjoint sets of perfect tensors. Associated with a cut is a decomposition of the tensor network as a contraction of two tensors and , where the contracted legs lie along the cut; the number of contracted legs is called the length of , denoted . If is a set of boundary legs and is the complementary set of boundary legs, then we say that the boundary of the cut matches the boundary of if the uncontracted legs of are the legs of , and the uncontracted legs of are the legs of . The minimal bulk geodesic bounded by , , is then defined as the cut of shortest length whose boundary matches the boundary of . We use to denote, not just the tensor associated with one side of the cut, but also the set of bulk lattice sites corresponding to the perfect tensors which are contracted to construct ; likewise for . We note that or might have more than one connected component, and so might when regarded as a path in the dual graph.
A standard argument for tensor network representations of quantum states shows that provides an upper bound on . If and are the tensors associated with a cut whose boundary matches the boundary of , then the holographic state may be expressed (up to normalization) as
(9) 
Here and run over complete bases for and respectively, and runs over all possible values of the indices contracted along ; the vectors in and the vectors in are not necessarily orthogonal or normalized. (See figure 5.)
Tracing out we obtain (up to normalization) the density operator on :
(10) 
Evidently the rank of is at most the number of terms in the sum over , namely . The density operator of a given rank with maximal Von Neumann entropy is proportional to the identity on its support, and has entropy equal to log of the rank. We obtain the best bound by choosing the cut with the shortest length:
(11) 
In most of what follows, we will define entropy by taking logs with base , and so suppress the factor.
If the tensors and are actually isometries from to and respectively, then and are sets of orthonormal vectors; in that case (11) is saturated and a discrete analogue of the RT formula holds exactly. Under what conditions will and be isometries? We can prove the following theorem:
Theorem 2.
Suppose that we have a holographic state associated to a simplyconnected planar tensor network of perfect tensors, whose graph has “nonpositive curvature”.^{3}^{3}3The scalar curvature of a graph is somewhat tricky to define in general; the condition we really need here is that the distance functional from one point on the dual network to another does not have interior local maxima. Then for any connected region on the boundary, we have ; in other words, the lattice RT formula holds.
The strategy of the proof is to show that and can in fact be interpreted as unitary transformations, from the cut together with some subregion of or to the rest of or respectively. We can then use the identity depicted in figure 3 to reinterpret these transformations as isometries from the cut to and from the cut to respectively; the RT formula follows. The key to the argument, explained in appendix B, is using a strengthened version of the maxflow mincut theorem (which is standard in graph theory Papadimitriou1998 ) to establish that the tensor network representations of and can be interpreted as unitary quantum circuits.
4.2 Bipartite entanglement of disconnected regions
Unfortunately the proof of Theorem 2 does not directly generalize to a disconnected region , nor even to connected regions for states, such as our holographic code states, where not all perfect tensor indices are contracted in the bulk. We do not consider this to be a serious problem for our models. However, we still find it worthwhile to introduce some machinery that allows us to quantify this presumption somewhat.
The first technique we will introduce is an algorithmic procedure for constructing, given a boundary region , a bulk curve bounded by such that the corresponding tensor is guaranteed to be an isometry. For a holographic state the isometry maps to , and for a holographic code maps and all incoming bulk indices of to . Furthermore, is a local minimum of the length, in the sense that no single tensor can be added to or removed from which reduces the length of the cut.
The algorithm makes essential use of the properties of perfect tensors and is quite simple. We consider a sequence of cuts each bounded by , and a corresponding sequence of isometries , such that each cut in the sequence is obtained from the previous one by a local move on the bulk lattice. The sequence begins with the trivial cut, itself; in each step we identify one perfect tensor which has at least half of its legs contracted with and construct by adding this perfect tensor to . Thus is obtained by composing with an isometry defined by a perfect tensor, and therefore is an isometry if is. The procedure halts when the cut reaches and no further local moves are possible. Though many different sequences of local moves are allowed, is well defined; tensors eligible for inclusion in remain so as other tensors are included, so the output of the algorithm does not depend on the order of inclusion. Following standard computer science terminology, we call this procedure the greedy algorithm and call the greedy geodesic. A step of the greedy algorithm is illustrated in figure 6.
When the assumptions of Theorem 2 are satisfied, the argument in appendix B ensures that the greedy algorithm will find a true minimal geodesic . If there is more than one minimal geodesic, as is sometimes the case, then the greedy algorithm might continue past a minimal geodesic and proceed through minimal geodesics of equal length. In that case, the tensors in between the successive geodesics define a unitary transformation from one cut to the other. If has more than one connected component, if there is positive curvature, or if there are uncontracted bulk indices as for a holographic code, the greedy algorithm does not necessarily succeed in finding matching minimal geodesics, as we illustrate in figure 7.
In cases where the greedy algorithm fails to find a minimal geodesic, we can still use it to prove an interesting lower bound on the entropy . Suppose that and are two greedy geodesics, produced by applying the greedy algorithm to and its complement respectively, where and are the corresponding tensors. Furthermore, suppose that is nonempty, in the sense that some links are cut by both geodesics. We can represent that state as^{4}^{4}4For holographic codes with dangling bulk legs, we assume for now that a product state is fed into all bulk legs. If the input bulk state were entangled instead, there would be additional contributions to the boundary entanglement which we are not including. This same proviso also applies to the discussion in the following subsection.
(12) 
Here denotes the index shared between and , is the index unique to , is the index unique to , and denotes the tensor that sits “in between” and . We call the set of lattice sites in the bipartite residual region (where the modifier “bipartite” draws a distinction with the multipartite residual region to be discussed in section 4.3.) Because and are isometries, both and are sets of orthonormal vectors. Therefore, the marginal density operator for is
(13) 
This density operator has support on the subspace of spanned by , which has dimension , and this subspace has a decomposition into subsystems such that the basis element may be expressed as , where and are orthonormal bases for and respectively. We may then write
(14) 
and from the additivity of the entropy, using , we obtain the following theorem.
Theorem 3.
For a holographic state or code, if is a (not necessarily connected) boundary region and is its complement, then the entropy of satisfies
(15) 
where is the greedy geodesic obtained by applying the greedy algorithm to and is the greedy geodesic obtained by applying the greedy algorithm to .
We see from Theorem 3 that violations of the RyuTakayanagi formula are closely related to the size of the bipartite residual region. In particular, if there is no bipartite residual region then ; the upper bound (11) and the lower bound (15) together imply that is in fact a minimal geodesic, and RT holds. We will argue in section 4.3 that the bipartite residual region has size when the regions and on the boundary have connected components. In this sense, the corrections to the RT formula are typically small.
4.3 A map of multipartite entanglement
So far we have emphasized the bipartite entanglement between a boundary region and its complement in a holographic state or code. But we may also divide the boundary into three or more regions and investigate the structure of the entanglement among these regions. The entanglement structure can be elucidated via an entanglement “distillation” procedure which we will now describe.
To explain this procedure we begin by revisiting the case of bipartite entanglement. We have seen that if the conditions of Theorem 2 are satisfied, then a holographic state can be expressed in the form (9), where a subsystem of of dimension is maximally entangled with a corresponding subsystem of . This entanglement shared between two systems is generally diluted, since each party may contain many more than spins. The entanglement would be more useful in a more concentrated form.
The procedure for transforming dilute entanglement into concentrated entanglement, called entanglement distillation, is particularly simple for a bipartite pure state like in (9). We choose specified spins in (the subsystem of ) and we choose spins in (the subsystem of ). Then we apply a unitary transformation acting on that transforms the basis states to the standard basis states of , and a unitary transformation acting on that transforms the basis states to the standard basis states of , thus obtaining the state
(16) 
in which the entanglement of with now resides entirely in the system . Here denotes the complement of in , denotes the complement of in , and
(17) 
is a maximally entangled EPR pair of two spins.
There is a method for constructing the unitary transformations and explicitly, which has a pleasing geometrical interpretation. The method uses the greedy algorithm for constructing , but where now each local move, in which the cut through the tensor network advances into the bulk by moving past one additional tensor, is accompanied by a local unitary transformation that decouples spins from the network. This local unitary transformation is depicted in figure 8, where entanglement distillation is performed on a pair of contracted sixleg tensors.
Since each local move of the greedy algorithm moves the cut past a tensor which initially has at least three legs crossed by the cut, the legs above the cut are always maximally entangled with the legs below, and the corresponding local unitary transformation exists. For purposes of visualization, we may imagine that the spins which remain contracted across the cut advance further into the bulk in each step, remaining adjacent to the cut, while the spins which decouple are left behind. When the greedy algorithm applied to terminates, then, all the decoupled spins of are distributed throughout the bulk region in between the greedy geodesic and the boundary, while spins of , lined up along the greedy geodesic, are contracted with tensors on the other side of the greedy geodesic. If we also apply the greedy algorithm to , then under the conditions of Theorem 2, the algorithm terminates at the same greedy geodesic. Acting together, then, the unitary transformations associated with the two greedy algorithms have decoupled all the boundary spins, except for EPR pairs, one for each of the legs crossed by the greedy geodesic, thus executing the entanglement distillation protocol.
Run backwards, the sequence of local unitary transformations associated with the greedy algorithm constitutes a holographic quantum circuit, which prepares the boundary state. The input to this circuit is EPR pairs, plus a suitable number of additional spins in a product state, distributed throughout the bulk. The circuit builds the state step by step, gradually incorporating the bulk spins as the cut advances outward from the greedy geodesic toward the boundary. The input state, envisioned as a set of EPR pairs lined up along , provides a map of entanglement, a picture characterizing the structure of the entanglement between and . (See figure 9.) The initial EPR pairs along the greedy geodesic which are deep inside the bulk encode longrange entanglement between and , while the EPR pairs closer to the boundary encode shorterrange entanglement.
We can likewise use the greedy algorithm to create a map of multipartite entanglement, whether or not the conditions of Theorem 2 are satisfied. Suppose, for example, that we divide the boundary into four regions , each of which is connected, as in figure 10. We may apply the greedy algorithm separately to each of the four regions, obtaining greedy geodesics . The bulk region in between and its greedy geodesic is called the causal wedge of , denoted . (The significance of the causal wedge in holographic codes will be discussed at length in section 5.) As figure 10 indicates, the union of the four causal wedges need not cover the entire bulk lattice — there may be a multipartite residual region in the bulk, which the greedy algorithm fails to reach when applied to the boundary regions one at a time. As we explain below, the size of the multipartite residual region is expected to be , independent of the total system size.
Multipartite residual regions in the bulk can indicate multipartite entanglement among the four regions on the boundary. As discussed above for the case of bipartite entanglement, suppose we decouple spins in each of by performing suitable local unitary transformations associated with each step of the greedy algorithm. Where the greedy geodesics of adjacent regions meet, EPR pairs are distilled, in keeping with our observation in section 4.2 that the bipartite entanglement of two boundary regions is no less than the length of the greedy geodesic shared by the two regions. The tensors trapped inside a multipartite residual region however, do not necessarily have a decomposition into EPR pairs. Instead it describes a state with multipartite entanglement, which cannot be expressed as a product of states with only bipartite entanglement.
Just as for a partition of the boundary into connected regions and , we can reverse the order in which tensors are incorporated by the greedy algorithm to obtain a holographic quantum circuit of isometries which prepares the boundary state. When we partition the boundary into four connected regions, however, the input to the circuit includes more than just EPR pairs distributed along shared greedy geodesics and decoupled spins in the bulk; additional multipartite states associated with each connected component of the bulk multipartite residual region are also part of the input. The circuit factorizes into a product , with each of the four unitary transformations acting within its own causal wedge to build the corresponding connected component of the boundary. Again, the greedy geodesics encode a “map” of the entanglement among , now including a description of multipartite entanglement among all the regions as well as bipartite entanglement among pairs of regions. Two such maps are shown in figure 10; in these cases a single sixleg tensor is trapped in each connected component of the bulk multipartite residual region, though in general a more complex tensor network could be trapped inside as indicated in figure 7.
We may also argue that if the bulk has constant negative curvature, then for any partition of the boundary into connected components, the multipartite residual region is always in size. This statement is true for the Riemannian geometry of the hyperbolic plane, but is merely heuristic because it disregards subtleties arising from the discrete lattice structure of the bulk. For a twodimensional Riemannian manifold, the GaussBonnet theorem applied to the residual region states that
(18) 
here is the Gaussian curvature, is the geodesic curvature, and is the Euler characteristic of the residual region, which is when has the topology of a disk. If is the interior of an gon whose sides are geodesics, (18) says that the integral of over is the deviation of the sum of interior angles of the gon from the corresponding sum for an gon in flat space; the latter sum is because the gon can be covered by triangles. For the AdS space, the interior angles approach zero as the space becomes large compared to its curvature radius; therefore assuming uniform negative curvature (where is the AdS radius), we conclude that the volume of the residual region is
(19) 
In our tensor networks is of order the length of a link; therefore is in lattice units if is , which establishes our claim.
Likewise, the bipartite residual region arising from a partition of the boundary into two regions and , discussed in section 4.2, has size if and both have connected components. Indeed, the bipartite residual region is contained in the multipartite residual region found by applying the greedy algorithm separately to each connected component of and of .
4.4 Negative tripartite information
A useful characterization of multipartite entanglement is the tripartite information, defined as
(20) 
For a general (mixed) tripartite quantum state, can take any real value. It is zero, though, for any tripartite pure state of , since in that case and e.g. regions and , being complementary, have the same entropy and therefore make cancelling contributions to . Nor is there a contribution to from EPR pairs shared between a pair of the three regions (because e.g. a pair shared by yields positive contributions to and which are cancelled by negative contributions from and ) or from entanglement shared between one of the three regions and a fourth disjoint region. Thus, for a holographic state and for any partition of the boundary into four regions nonzero contributions to can arise only from the distilled multipartite states trapped in residual regions.
In the holographic setting, it has been shown that follows from the RT formula Hayden13b . For holographic states and codes, the nonpositivity of is not ensured in general, because of the potential (small) violations of the RT formula. In some special cases, though, RT holds exactly, and the nonpositivity of then follows. For example, suppose that we partition the boundary into four connected regions , and that each connected component of the multipartite residual region traps just one perfect tensor. In that case there is no bipartite residual region, so Theorem 3 implies that RT is exact and therefore . To see that there is no bipartite residual region in this case, consider the bipartite partition of the boundary into the two disconnected regions and , and consider an isolated index perfect tensor surrounded by three or all four of the greedy geodesics . This tensor must have at least legs crossing either or . Therefore, when we apply the greedy algorithm to the boundary regions and , one cut or the other will advance past this isolated tensor, excluding it from the bipartite residual region.
Under suitable conditions we can actually prove a stronger result — that is strictly negative. Let us say that a connected component of the multipartite residual region is three sided if surrounded by three of the four greedy geodesics, and four sided if surrounded by all four greedy geodesics. Threesided components make no contribution to ; if the three surrounding greedy geodesics are those of , the symmetry of implies , which vanishes for any pure state of . But an isolated index perfect tensor which crosses all four greedy geodesics makes a negative contribution to :
Theorem 4.
Suppose the indices of a perfect tensor state are partitioned into four disjoint nonempty sets such that . Then the tripartite information is strictly negative: .
Proof.
First we notice that for a fourpart pure state , the tripartite information is actually completely symmetric under permutations of the four subsystems, which we can see by using the property that complementary regions have the same entropy in a pure state:
(21)  
(22) 
We may therefore assume without loss of generality that which implies and . Now we use the defining property of index perfect tensors, that a set of or fewer indices is maximally entangled with its complement, which implies , with entropy expressed in units of . Therefore , and furthermore and . Now we distinguish two cases. If , then and we have
(23) 
If on the other hand , then and we have
(24) 
where to obtain the second equality we use . This completes the proof. ∎
For a holographic state with boundary partitioned into sets , the conditions of Theorem 4 are satisfied by an isolated perfect tensor trapped inside a foursided component of the multipartite residual region; fewer than of the tensor’s legs cross any greedy geodesic, because otherwise the greedy algorithm would have moved the cut forward past this perfect tensor, which therefore would not be in the multipartite residual region. Furthermore, since entropy is additive for a product state, is also strictly negative for any product of perfect tensor states shared by , provided that at least one factor has support on all four sets. Since only the foursided regions contribute to , we conclude that is strictly negative if the multipartite residual region contains at least one foursided connected component, and if each foursided connected component contains only one perfect tensor.
5 Quantum error correction in holographic codes
In this section we study the error correction properties of our holographic codes in more detail. The idea that a CFT with a gravity dual must have error correcting properties was recently proposed in Almheiri14 , and in this section we will see that our holographic codes illustrate many aspects of the proposal of Almheiri14 quite explicitly.
5.1 AdSRindler reconstruction as error correction
We begin by briefly recalling the main point emphasized in Almheiri14 , which is that in AdS/CFT a bulk local observable can be realized by many different operators in the CFT. In fact, if is any point in the bulk, and is any point on the boundary, the AdS/CFT dictionary can be chosen so that it maps the bulk local field to a CFT operator which has no support in an open set containing , and therefore commutes with any local field of the CFT supported near . Since is an arbitrary boundary point, if the CFT operator corresponding to were actually unique, we would conclude that commutes with all local fields in the CFT, and therefore is a multiple of the identity because the local field algebra is irreducible. This paradox is evaded once we recognize that the correspondence is not unique. If are two distinct boundary points, the CFT operator corresponding to can be chosen to be either , which commutes with CFT local fields supported near , or , which commutes with CFT local fields supported near , where and are inequivalent CFT operators even though they can be used interchangeably for describing bulk physics.
This novel feature of AdS/CFT, that a bulk local observable can be represented by boundary CFT operators in multiple ways, is illustrated in figure 11. The idea is that any fixedtime CFT subregion defines a subregion in the bulk, the causal wedge . For any point , bulk quantum field theory ensures that any bulk local operator can be represented in the CFT as some nonlocal operator on . This representation is called the AdSRindler reconstruction of the operator Hamilton2006 ; Morrison2014 . Because a given bulk point can lie within distinct causal wedges associated with different boundary regions, the bulk operator can have distinct representations in the CFT with different spatial support.
In Almheiri14 the nonuniqueness of the CFT operator corresponding to the bulk operator was interpreted as indicating that is a logical operator preserving a code subspace of the Hilbert space of the CFT. This code subspace is protected against “errors” in which parts of the boundary are “erased.” If the boundary operator corresponding to acts on a subsystem of the CFT which is protected against erasure of the boundary region , then this operator can be represented in the CFT as an operator supported on , the complement of the erased region. Thus we may interpret the AdSRindler reconstruction of on boundary region as correcting for the erasure of ; choosing the erased portion of the boundary in different ways leads to different reconstructions of . Moreover, operators near the center of the bulk are “well protected” in the sense that a large region needs to be erased to prevent their reconstruction, while operators near the boundary can be erased more easily by removing a smaller part of the boundary Almheiri14 .
We may think of this code subspace as the lowenergy sector of the CFT corresponding to a relatively smooth dual classical geometry. All CFT operators are physical, and thus have some bulk interpretation, but the “logical” operators are special ones which map lowenergy states to other lowenergy states. The same logical action can be realized by distinct CFT operators, as these distinct operators act on highenergy CFT states (those outside the code subspace) differently even though they act on lowenergy states in the same way.
5.2 The physical interpretation of holographic codes
The errorcorrecting properties of the AdS/CFT correspondence were motivated in Almheiri14 by bulk calculations, together with plausibility arguments regarding the CFT. Our central observation in this paper is that analogous statements are provably true in holographic codes.
We emphasize that in holographic codes the uncontracted bulk legs hanging from each tensor should not be thought of as tensor factors in addition to the boundary legs. Rather the entire physical Hilbert space is spanned by states of the boundary legs only. The bulk legs just provide a way of conveniently describing states in a certain code subspace of this boundary Hilbert space, obtained by feeding states of the bulk legs through the isometry defined by the entire tensor network; this code subspace can be regarded as a simplified model of the lowenergy states in a CFT.
Likewise, operators acting on the dangling bulk indices correspond to nonlocal operators in the boundary theory whose algebra and action on the code subspace resembles what we would expect for the CFT description of how bulk local operators act on lowenergy CFT states. When we speak of a “bulk local operator” we really mean the nonlocal boundary operator obtained by pushing an operator acting on a dangling bulk index out to the boundary using the isometry defined by the network.
5.3 Bulk reconstruction from tensor pushing
We now explain how holographic codes realize the AdSRindler reconstruction of figure 11. The basic idea is that, instead of using the full isometry of the entire network to push a local bulk operator to the boundary, we can instead successively push it through individual perfect tensors in a manner of our choosing by using the operation of figure 2. We illustrate the reconstruction for two different bulk points of the pentagon code in figure 12. Here we use the defining property of perfect tensors — that the tensor provides a unitary transformation which maps any three legs of the tensor to the complementary set of three legs, and therefore also an isometry mapping any set of three or fewer “incoming” legs to any disjoint set of three “outgoing” legs. In figure 12, each bulk vertex with arrows showing incoming and outgoing directions indicates such an isometry, and the complete set of blue legs is a product of such isometries, and hence also an isometry. The blue operator on the boundary is obtained by conjugating the blue bulk operator by the blue isometry, and the same applies to the green bulk and boundary operators. In the construction of the isometry, we regard the dangling bulk index on each tensor as an incoming index, and therefore require that no more than two contracted indices are incoming for each blue (or green) tensor. The same blue isometry, then, can be used to push not just the central blue bulk index to the boundary, but also any of the other incoming bulk indices (which are not shown in the figure) on blue tensors.
The boundary operation corresponding to a given bulk local operator manifestly has the nonuniqueness we described in our discussion of the AdSRindler reconstruction. For example, we could move one of the three blue arrows directed outward from the central blue vertex to a different edge, thus reconstructing the central bulk operator on a different boundary region, or we could have sent the green arrows in the opposite direction and reconstructed the green bulk operator on a considerably larger boundary region on the opposite side. No matter which reconstruction we use, the boundary operator is obtained from the isometric embedding of the bulk indices into the code subspace of the boundary Hilbert space, and therefore each reconstructed operator corresponding to a given bulk operator acts on the code subspace in the same way.
In the theory of quantum errorcorrecting codes, we say that an error is an erasure (or equivalently a located error) if the set of spins damaged by the error is known, so this information can be used in recovering from the error. Holographic codes also provide protection against errors which act at unknown locations on the boundary, but for the purpose of developing the analogy with the AdS/CFT correspondence we will focus on protecting against erasure. A logical system can be protected against erasure of a set of spins in the code block if the full algebra of logical operations has a realization supported on the complementary set of unerased spins. In AdS/CFT we might only require reconstruction of a subalgebra of the full logical algebra; for example, the pentagon code provides better protection for the degrees of freedom deep within the bulk than for those closer to the boundary. The framework in which a quantum code protects only a subalgebra of the code’s full logical algebra has been called operator algebra quantum error correction Kribs2005 ; Kribs2006 ; Beny2007 ; Beny2007a .
5.4 Connected reconstruction and the causal wedge
Given a subregion of the boundary, which bulk local operators can be reconstructed on ? This is not an easy question to answer in general, but at least we can give a simple description of a large logical subsystem reconstructable on , namely those logical operators acting on bulk sites which are reachable using the greedy algorithm explained in section 4.
Recall that the greedy algorithm associates with any boundary region a greedy geodesic whose boundary matches the boundary of , such that and enclose a tensor which defines an isometry mapping free bulk legs in together with the legs cut by to . Using this isometry applied to any operator acting on a bulk leg in (tensored with the identity acting on all the rest of the isometry’s input indices), we may push that logical operator through the isometry to obtain its reconstruction on . Let’s call the position of a perfect tensor in the network a bulk point and say that the greedy algorithm reaches a bulk point if it moves the cut past that tensor, hence using it in the construction of
This operator reconstruction procedure can be applied to any boundary region . In the special case where is connected, it provides a precise analog of the AdSRindler reconstruction in holographic codes, which we can formalize with a definition and theorem:
Definition 5.
Suppose that is a connected boundary region. The causal wedge of , denoted , is the set of bulk points reached by applying the greedy algorithm to .
We then have:
Theorem 5.
Suppose is a connected boundary region. Then any bulk local operator in the causal wedge can be reconstructed as a boundary operator supported on .
We could have formulated a geometric notion of the causal wedge, defining it as the set of bulk points enclosed between and the actual minimal geodesic , rather than the greedy geodesic. This geometrical definition is closer in spirit to how the term “causal wedge” has been used in the context of AdS/CFT. But we prefer this greedy notion of causal wedge instead, so that Theorem 5 is correct as stated.
As figure 12 illustrates, bulk operators near the boundary can be reconstructed on smaller connected regions than bulk operators near the center, just as for the AdSRindler reconstruction in AdS/CFT. It is natural to wonder how large the connected region should be for the operator at the center of the bulk to be reconstructable on . This question is studied for the pentagon code in appendix C by investigating whether the greedy algorithm applied to reaches the central tensor in the network. We find that a connected region of boundary spins necessarily allows reconstruction of all operators acting on the center provided that covers a sufficiently large fraction of the boundary, namely
(25) 
The analogous result for the AdSRindler reconstruction is , but the discreteness of our lattice introduces some additional overhead. It turns out, though, that because the tensor network is not invariant under translations of the boundary, whether the connected region allows reconstruction of the center depends not just on the size of but also on its location. In appendix C we show that, while the condition (25) is needed to guarantee reconstruction of the central operator on an arbitrary connected region, there are some connected regions with that suffice for the reconstruction.
5.5 Disconnected reconstruction and the entanglement wedge
Now let’s consider what bulk operators can be constructed on boundary regions with more than one connected component. First we extend the definition of the causal wedge to disconnected regions:
Definition 6.
Suppose that is a boundary region, which is a union of connected components . The causal wedge of , denoted , is defined as the union of the causal wedges of the components of , .
Since we have already established that any bulk operator in is reconstructable on if is connected, it follows immediately from this definition that even for disconnected regions any bulk operator in is reconstructable on .
The causal wedge contains bulk operators which can be reconstructed when we apply the greedy algorithm to the connected components of one at a time. But the greedy algorithm might advance further into the bulk, beyond the causal wedge, when applied to instead. Specifically, there could be a index tensor just beyond the causal wedge of with or more legs crossing the union of greedy geodesics , even though fewer than legs cross or individually. Then applying the greedy algorithm to moves the cut past this tensor. This step may then render further tensors eligible for inclusion, and in fact we will see that sometimes the greedy algorithm can move far beyond the causal wedge
A concrete first example illustrating reconstruction of a bulk operator outside the causal wedge is shown in figure 13. In this example, is the union of two connected components and , and the full operator algebra of the central tensor can be pushed to either or . This implies that no nontrivial operator acting on the central tensor can be pushed to either or . For every nontrivial operator in the algebra there is another operator which does not commute with . If can be pushed to , then surely cannot be pushed to , because operators supported on complementary regions must commute. The same argument applies to . Yet the greedy algorithm applied to reaches the central tensor, showing that its full operator algebra can be pushed to the union of and .
That operators beyond the causal wedge of can be reconstructed on has deep potential implications for AdS/CFT. Perturbative gravity techniques like the AdSRindler reconstruction can be used to construct bulk operators in the causal wedge but not beyond. Yet there has been speculation in the literature that reconstruction should be possible in a larger region, the entanglement wedge Headrick2014 , see also Wall2012 ; Czech2012 ; Jafferis2014 . In AdS/CFT, the entanglement wedge is defined by first finding the minimal area surface used in the RT formula, and then drawing a codimension one (i.e., twodimensional for ) spatial slice in the bulk whose only boundaries are and . The bulk domain of dependence of this slice is then defined as the entanglement wedge . The entanglement wedge contains the causal wedge, but can be much larger in some cases. Figure 14 illustrates a simple example highlighting the distinction between the causal and entanglement wedges.^{5}^{5}5In excited states where the geometry deviates from pure AdS, there are differences between the entanglement wedge and the causal wedge even for connected boundary regions. We will not try to capture this in our toy models, since without a theory of dynamics we cannot capture the full spacetime definitions of these regions. Our discussion is limited to the case where we stick with states near the vacuum, in which case needs to be disconnected for its causal wedge and entanglement wedge to differ.
We would like to investigate whether bulk operators in the entanglement wedge are reconstructable for holographic codes, but how should the entanglement wedge be defined? A definition of close to that used in AdS/CFT is:
Definition 7.
Suppose is a (not necessarily connected) boundary region. The geometric entanglement wedge of is the set of bulk points in the bulk region bounded by and , where is the minimal bulk geodesic whose boundary matches the boundary of . If there is more than one minimal bulk geodesic, is chosen to make the geometric entanglement wedge as large as possible.
The main motivation for the conjecture that operators in the entanglement wedge are reconstructable in AdS/CFT comes from the validity of the RT formula for disconnected regions. (Additional evidence was given in Almheiri14 based on a typicality argument.) But we have already seen above that the RT formula does not hold exactly in holographic codes, so we should not necessarily expect the entanglement wedge conjecture to hold in detail for the geometric entanglement wedge. Instead, as in defining the causal wedge, we prefer a definition that makes the reconstructability manifest:
Definition 8.
Suppose is a (not necessarily connected) boundary region. The greedy entanglement wedge of , denoted , is the set of bulk points reached by applying the greedy algorithm to all connected components of simultaneously.
With this definition, bulk local operators in are automatically reconstructable in , using the isometry defined by to push these operators to the boundary. The greedy algorithm also ensures that the interior boundary of is the greedy geodesic , though not necessarily the minimal geodesic .
A drawback of this definition is that includes only the bulk local operators which can be reconstructed on using the greedy algorithm; it might miss additional bulk operators which can be reconstructed by other methods. In fact we can find examples of codes such that some bulk local operators lying outside can be reconstructed on , as discussed in appendix E. These codes typically have special properties, such as symmetries, which make the reconstruction possible. If we know nothing more about the perfect tensors used to construct the code, aside from their perfection, we have no general reason to expect that bulk operators far outside the greedy entanglement wedge will be reconstructable. That said, we confess that we lack a complete understanding of when reconstruction is possible, and hope that further progress on this issue can be achieved in future work.
5.6 Erasure threshold
If the entanglement wedge conjecture is true for AdS/CFT, if holographic codes faithfully model the entanglement structure of boundary theories with classical gravitational duals, and if the greedy entanglement wedge is a reasonable standin for the entanglement wedge, then we should be able to find holographic codes and boundary regions such that the greedy entanglement wedge reaches far outside the causal wedge. In this section we provide examples which confirm this expectation. One way to formalize this is to choose to be a randomly chosen set of boundary spins, whose size is a specified fraction of the total boundary. The geometry of the hyperbolic plane suggests that, if is large enough, the causal wedge will stick close to the boundary, yet the entanglement wedge reaches the center of the bulk with high probability; we illustrate this in figure 15. We will see that not all holographic codes have this property, but we are able to provide concrete examples that do.
Another, perhaps better, way to formulate this case is to imagine a probabilistic noise model which acts independently (without any noise correlations) on each of the physical boundary spins, where each spin is either erased with probability or left untouched with probability . If is small, the set of unerased boundary spins breaks into many connected islands, where a typical island contains spins and has a causal wedge which reaches into the bulk by only a constant distance. We can show, though, that if the holographic code is properly chosen and the erasure probability is less than a threshold value , then contains the central bulk spin with a success probability deviating from one by an amount which becomes doubly exponentially small as the radius of the bulk increases.
Which codes have an erasure threshold? One necessary requirement is that the code must have a distance that increases with the system size. For the purpose of reconstructing the central tensor in the bulk, this means that there should not be any logical operator supported on a constant number of boundary spins which acts nontrivially on the central bulk index. That’s because erasure of any constant number of spins occurs with a nonzero constant probability, and recovery from the erasure error is not possible if a nontrivial logical operator has support on the erased qubits.
The pentagon code fails to fulfill this necessary condition. To illustrate the problem, it is helpful to consider first a simpler code, the “triangle code” constructed by contracting fourindex perfect tensors, where each leg is a 3level spin, a qutrit. Each triangle in the bulk has a dangling bulk index, and the code is constructed as a tensor network forming a tree, the Bethe lattice; each triangle is contracted with one triangle closer to the center and two triangles further from the center, as shown on the left side of figure (a)a. (Qi’s model Qi13 is based on a tensor network with a similar structure.) One way to describe the greedy algorithm is to say that it propagates erasures from the boundary toward the center of the bulk — the inward directed leg of a triangle is erased if either of its outward directed legs is erased, and the central triangle can be reconstructed only if at least two of that triangle’s legs are unerased.
It is easy, then, to prevent the greedy algorithm from reaching the center — only two spins need to be erased. A single erasure on the boundary propagates all the way up to the center of the network, erasing one of the central triangle’s legs. A single erasure on a different branch of the tree propagates up to another of the central triangle’s legs, blocking the reconstruction of the central tensor on the remaining unerased spins.
The greedy algorithm fails for a good reason. As described in appendix A.2, The logical algebra for the threequtrit code represented by a single triangle is generated by logical operators of the form , where is a generalized Pauli operator; in fact the code is symmetric under permutation of the three qutrits, so we can choose and to act on any two of the three qutrits without changing the operator’s action on the code space. Now choose a path through the Bethe lattice which begins on one leaf, travels to the center, exits the center on a different branch, and finally reaches another leaf on that branch. Apply the operator to each of the logical bulk indices visited by this path. Then for each leg along the path the from the triangle on one side cancels the coming from the triangle on the other side, except for one uncanceled on one leaf and one on the other. We conclude that the code admits a logical operator acting nontrivially on the central triangle which has support on only two boundary spins. That is why the central bulk spin can be damaged by erasing only two boundary spins.
For the pentagon code the situation is only slightly better. If we pick just four spins at the positions shown on the right side of figure (b)b, then the greedy algorithm applied to the complement of these four spins never absorbs any of the tensors adjacent to the dashed line. This failure is just a property of the graph defining the holographic code, but once again we can understand the failure by noting that there is a logical operator acting on the central pentagon supported on these four boundary spins, so erasing these four spins prevents central bulk operators from being reconstructed on their complement. Now we may consider a product of bulk logical operators acting on the pentagons just above and just below the dashed line. We use the logical operator of the fivequbit code described in appendix A.1, where and are Pauli operators (which square to one), and the operator’s action is unchanged by cyclic permutations of the five qubits. Now ’s applied from either side of the cut cancel on the legs crossed by the cut, and ’s applied from either side cancel for the legs just above and below the cut, leaving only four uncanceled ’s acting on the boundary qubits.
Of course, uncorrectable damage deep inside the bulk caused by erasing just a few boundary spins is not at all what we expect in AdS/CFT, where according to the entanglement wedge conjecture we should always be able to reconstruct the center of the bulk from a sufficiently large fraction of the boundary, whatever its shape or location. To obtain a better model for AdS/CFT we should modify the holographic code, thinning out the algebra of bulk logical operators, and hence reducing the rate of the code.
A code that works better can be obtained by a simple modification of the pentagon code — the modified tensor network is constructed by starting with a pentagon at the center and adding alternating layers of hexagons (with no dangling bulk indices) and pentagons (each with one bulk index) as the network grows radially outward. The associated network is depicted in figure 17. This change suffices to remove all the constantweight logical operators acting nontrivally on the center and in fact we can prove that this pentagon/hexagon code has an erasure threshold. Numerical studies show that erasure can be corrected by the greedy algorithm with high success probability for ; the erasure threshold achieved by the optimal recovery method might be higher than if the tensors have further special properties aside from just being perfect.
Since our main interest is in the reconstruction of the center of the bulk, in appendix D we study a code for which the only logical index resides at the center, also shown in figure 17. This code is almost the same as the holographic state obtained by contracting sixleg perfect tensors (hexagons), except that the tensor network contains one pentagon at the center; we therefore call it the singlequbit hexagon code. We prove the existence of an erasure threshold for this code, and also derive an analytic lower bound on the threshold erasure rate . Numerical evidence indicates that the threshold is actually quite close to .
The lower bound on the threshold is derived using a simplified and less powerful version of the greedy algorithm, the hierarchical recovery method, which begins at the boundary and proceeds inward toward the center of the bulk. A tensor at level of this hierarchy is connected to at least four tensors at level , and the level tensor is erased if two or more of its level neighbors are erased. The proof proceeds by recursively deriving an upper bound on the erasure probability at level , finding
(26) 
where and . Thus the erasure probability for the central tensor drops doubly exponentially with the radius of the bulk if , which means that the central tensor can be reconstructed on the set of unerased boundary qubits with very high probability.
A tricky aspect of the proof is that, because a single level tensor couples to two level tensors, there are noise correlations which propagate from level to level. Fortunately, the hyperbolic geometry controls the spread of correlations, making the analysis manageable. In fact, correlations beyond nearest neighbors never arise. This is one advantage of using the hierarchical recovery method rather than the greedy algorithm. A similar proof strategy may also be applied to other holographic codes.
5.7 Holographic stabilizer codes
Stabilizer codes have been extensively studied in quantum coding theory, and are often used in applications to faulttolerant quantum computing Gottesman2009 . Here we describe how to construct a family of holographic codes which are also stabilizer codes. We introduce the stabilizer formalism to pave the way for section 5.8, where we study some geometrical properties of holographic stabilizer codes.
Stabilizer codes can be defined for higherdimensional spins as well, but here we will assume the spins are qubits for simplicity. A Pauli operator acting on qubits is a tensor product of Pauli matrices, that is, one of the operators contained in the set
(27) 
where is the identity matrix and are the Pauli matrices (often denoted ). We use to denote a quantum code with logical qubits embedded in a block of physical qubits. We say that an code is a stabilizer code (also called an additive quantum code), if the code space can be completely characterized as the simultaneous eigenspace of commuting Pauli operators. These commuting Pauli operators are called the code’s stabilizer generators because they generate an abelian group called the code’s stabilizer group. The special case of a stabilizer code is called a stabilizer state. We say that an index tensor is a stabilizer tensor if the corresponding qubit state is a stabilizer state.
For example, the sixindex perfect tensor is a perfect stabilizer tensor, and holographic codes defined by tiling a hyperbolic geometry with pentagons are stabilizer codes. More generally, we may formulate the following theorem:
Theorem 6.
Consider a holographic code defined by a contracted network of perfect stabilizer tensors, and suppose that the greedy algorithm starting at the boundary reaches the entire network. Then the code is a stabilizer code.
To understand why Theorem 6 is true we need to see how to construct the code’s stabilizer generators. To be concrete, consider holographic codes constructed from tilings by hexagons and pentagons. The sixindex perfect tensor defines a [[6,0]] stabilizer code, whose stabilizers are enumerated in appendix A.1. As we have already noted, it also defines isometries from any set of 1, 2, or 3 indices to the complementary set of indices; these isometries may be regarded as the encoding maps for , , and stabilizer codes respectively.
To be specific, consider the code, and let denote its isometric encoding map taking a onequbit input to the corresponding encoded state in the code block of five qubits. We can characterize by specifying how it acts on Pauli operators, which (together with the identity) span the space of operators acting on a single qubit. Since the Pauli group is generated by and it suffices to specify
(28) 
where and are the code’s logical Pauli operators, given explicitly in appendix A. Similarly, the action on Pauli operators defines isometric encoders for the and stabilizer codes, except that for e.g. the code we specify the action on the four independent Pauli operators , where the subscript labels the code’s two logical qubits. For stabilizer codes the encoding isometry is always a Clifford isometry, meaning its action by conjugation maps qubit Pauli operators to qubit Pauli operators.
We already explained in section 3 that when the condition of Theorem 6 is satisfied then the encoding isometry for the holographic code can be obtained by composing the isometries associated with each perfect tensor in the network. A given tensor may have 0, 1, 2, or 3 incoming legs, including the dangling bulk leg (if the tensor is a pentagon) and all the incoming contracted legs, which are output legs from previously applied isometries. To prove Theorem 6 then, it is enough to know that composing the encoding isometries of two stabilizer codes yields the encoding isometry of a stabilizer code.
To see how this works, it is helpful to think about the simple special case of a concatenated quantum code, for which the tensor network is a tree. Consider in particular a code with just one logical qubit — the central pentagon has one incoming logical leg and five outgoing legs, while every other tensor is a hexagon with one incoming leg and five outgoing legs. If the code is concatenated just once, the tensor network has five hexagons and describes a stabilizer code. To obtain this code’s isometric map, we first apply the encoding isometry of the to the logical qubit, and then apply again to each one of the five outgoing qubits. If denotes the stabilizer group of the code, then the stabilizer of the code will include acting on each one of the five subblocks corresponding to the five hexagons in the tensor network. But it also includes elements which act collectively on four of the five hexagons. For example, as described in appendix A.1, one of the stabilizer generators for the code is the Pauli operator . The isometries associated with the five hexagons map this operator to , where now are the logical Pauli operators acting on the five outgoing qubits emanating from a single hexagon.
The same idea applies to more general compositions of code isometries. Suppose that are the stabilizer group and encoding isometry for an stabilizer code and that are the stabilizer group and encoding isometry for an stabilizer code. We may apply to of the output qubits from along with additional input qubits (where and ), thus obtaining an code. In fact this code is a stabilizer code, whose stabilizer group is generated by and ; here we use a streamlined notation, in which it is understood that operators and maps are extended by identity operators where necessary, and we note that the elements of are Pauli operators because is a Clifford isometry. Thus we have proven Theorem 6. It is also worthwhile to note that the stabilizer group and encoding isometry for the holographic code can be efficiently computed by composing the isometries arising from the perfect tensors in the network.
5.8 Are local gauge constraints enough?
It has recently been argued that in AdS/CFT gauge constraints in the boundary CFT may pick out a small enough subspace of states to explain the error correcting properties of AdS/CFT Mintun2015 . The idea is that any gaugeinvariant state already possess some nonlocal entanglement via the imposition of the gauge constraints, and that this might be enough to resolve the various paradoxes of Almheiri14 .^{6}^{6}6The word “gauge” is sometimes used in quantum information theory in a way that is nonstandard from the point of view of quantum field theorists. In quantum field theory, states that are not gaugeinvariant have no physical interpretation, and are not really part of the Hilbert space of the theory; they appear only as a mathematical convenience. This is what the authors of Mintun2015 meant by gauge constraints, and it is what we mean here.
We can try to test this idea for the holographic stabilizer codes discussed in section 5.7. Since gauge constraints are spatially local, the argument of Ref. Mintun2015 suggests that the code’s stabilizer group should be locally generated, in the sense that it has a complete set of generators, each with support on a constant number of neighboring boundary qubits. In fact, though, holographic stabilizer codes do not have this property in cases where the greedy entanglement wedge reaches outside the causal wedge. This property poses no problem for the proposal of Almheiri14 however, as those authors argued that energetic constraints should also be included in defining the code subspace.
Consider for example the disconnected boundary region in the pentagon code, depicted in figure 13. We have already seen that the full logical algebra of the central pentagon can be reconstructed on the disconnected region , but that no nontrivial logical operator acting on the central pentagon is supported on either one of the connected components , . In a stabilizer code, a logical Pauli operator supported on is a tensor product of Pauli operators supported on and separately. In order to preserve the code space, this logical Pauli operator must commute with all of the code’s stabilizer generators. But if the two components and are distantly separated and the stabilizer generators are geometrically local, then no stabilizer generator has nontrivial support on both and . Any stabilizer generator with no support on trivially commutes with , and if it commutes with then it must commute with as well. Likewise, a stabilizer generator with no support on must commute with if it commutes with . Therefore and are logical operators, and at least one is nontrivial if their product is, contradicting the hypothesis that no nontrivial logical operator is supported on either connected component of . The conclusion is that the stabilizer generators cannot be geometrically local.
The above argument applies even to higherdimensional holographic stabilizer codes. In the case were the boundary is one dimensional, we may simply appeal to a known result in quantum coding theory, that a stabilizer code in one dimension with geometrically local generators has constant distance Bravyi09 ; Pastawski15 . Therefore, a onedimensional code with a local stabilizer cannot have a positive erasure threshold.
6 Black holes and holography
In holographic codes, bulk operators are reconstructed only on a subspace of the boundary Hilbert space. This may seem troubling, since the holographic correspondence is supposed to assign a bulk interpretation to all possible states on the boundary. A resolution of this confusion was proposed in Almheiri14 — a particular bulk operator might not always be reconstructable because it lies deep inside a black hole for most boundary states.^{7}^{7}7We are currently agnostic about the reconstruction of bulk operators just inside the horizon, which must be needed in some form to describe the experience of an infalling observer. This is a topic of much recent controversy Almheiri13 ; Harlow2014 , but we will not take sides here. In fact we can see this directly in our models if we incorporate black holes in a manner that we now describe.
To illustrate the idea, consider the pentagon code, but with the central tensor removed. The central tensor’s one free bulk index has been replaced by five bulk indices, those which had previously been contracted with legs of the missing pentagon; the tensor network now provides an isometry mapping these five indices, together with the bulk legs on the remaining pentagons, to the boundary. Thus the code subspace of the boundary Hilbert space is larger than for the pure pentagon code. We interpret this enlarged code space as describing configurations of the bulk with a black hole in the center, whose microstate is determined by the input to the new bulk legs. The entropy of the black hole is the logarithm of the dimension of the Hilbert space of black hole microstates, or
(29) 
since only four of the bulk spins are new and we shouldn’t count states that were part of the original pentagon code subspace. We depict this construction in figure 18a.
We can construct larger black holes by removing more central layers of the network; it is clear that their entropy scales with their horizon area, as predicted by Bekenstein and Hawking Bekenstein73 ; Hawking75 . As the black hole grows, the number of bulk legs outside the black hole decreases, so we can reconstruct fewer and fewer bulk local operators. Eventually the black hole eats up the entire network, and our isometry becomes trivial (and unitary). Thus our model really does assign a bulk interpretations to all boundary states, as demanded by AdS/CFT — most boundary states correspond to large black holes in the bulk.
It is amusing to note that we can also describe configurations corresponding to the twosided wormhole of Maldacena03 ; we just prepare two networks with central black holes of equal size, and maximally entangle the bulk legs at their horizons, as shown in figure 18b. It would be interesting to make contact with recent speculations about how the length of the wormhole relates to the complexity of the tensor network describing the state Hartman2013 ; Susskind2014 ; Roberts2014 , although for that purpose we would probably need to incorporate dynamics into our model.
7 Open problems and outlook
A remarkable convergence of quantum information science and quantum gravity has accelerated recently, propelled in particular by a vision of quantum entanglement as the foundation of emergent geometry. We expect this interface area to continue to grow in importance, as practitioners in both communities struggle to develop a common language and toolset. This paper was spurred by the connection between AdS/CFT and quantum error correction proposed in Almheiri14 . We have strived to make this connection more concrete and accessible by formulating toy models which capture the key ideas, and we hope our account will equip a broader community of scientists to contribute to further progress. Indeed, much remains to be done.
First of all, the entanglement structure of holographic codes is not yet completely understood. We would like a more precise characterization of the violations of the RyuTakayanagi formula which can occur, and of the relationship between bulk residual regions and the multipartite entanglement of the boundary state. How is the greedy entanglement wedge different from the geometric entanglement wedge, and to what extent does the greedy entanglement wedge reach beyond the causal wedge?
We have not yet discussed the correlation functions of boundary observables in holographic codes because we do not have much to say. In a stabilizer state , where and are Pauli operators, the expectation value is either zero (if anticommutes with an element of the stabilizer) or a phase (if commutes with the stabilizer); the same conclusion applies to a stabilizer code unless is a nontrivial logical operator preserving the code subspace. In contrast, twopoint correlations in a CFT decay algebraically with distance; how might we recover this behavior in holographic codes? Perhaps algebraic decay is recovered for nonstabilizer holographic states, by defining suitable coarsegrained observables, or by injecting an encoded state such that bulk correlation functions decay exponentially as in Ref. Qi13 . Or we might replace perfect tensors by tensors which are nearly perfect.
The behavior of twopoint correlators highlights one way our toy models differ from fullblown AdS/CFT, but there are other ways as well; for one, there is no obvious analog of diffeomorphism invariance in a lattice model. What features in our lattice model correspond to the corrections in the continuum theory? In AdS/CFT the AdS radius is large compared to the Planck scale when the bulk theory is weakly coupled, yet in the pentagon model for example the curvature scale is comparable to the lattice cutoff. To approximate flatter bulk geometries we should study more general tessellations, including higher dimensional ones. A particularly serious drawback of our toy models so far is that we have not introduced any bulk or boundary dynamics. Can holographic codes illuminate dynamical processes like the formation and evaporation of a black hole?
Finally, we have emphasized that holographic states and codes provide a concrete realization of some aspects of AdS/CFT, but they may also be interesting for other reasons, for example as models of topological matter. Furthermore, holographic codes generalize the concatenated quantum codes that have been extensively used in discussions of faulttolerant quantum computing Gottesman2009 , and might likewise be applied for the purpose of protecting quantum computers against noise. For this application it would be valuable to develop the theory of holographic codes in a variety of directions, such as studying tradeoffs between rate and distance, formulating efficient schemes for correcting more general errors than erasure errors, and finding ways to realize a universal set of logical operations acting on the code space.
Acknowledgment
We thank Ning Bao, Oliver Buerschaper, Glen Evenbly, Daniel Gottesman, Aram Harrow, Isaac Kim, Seth Lloyd, Nima Lashkari, Hirosi Ooguri, Grant Salton, Kristan Temme, Guifre Vidal and Xiaoliang Qi for useful comments and discussions. We also have enjoyed discussions with Ahmed Almheiri, Xi Dong, and Brian Swingle, and with Matthew Headrick, about their independent and upcoming related work. FP, BY, and JP acknowledge funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation (Grants No. PHY0803371 and PHY1125565). BY is supported by the David and Ellen Lee Postdoctoral fellowship. DH is supported by the Princeton Center for Theoretical Science.
Appendix A Perfect tensor examples
In this section we will present the 5qubit code, the 3qutrit code and discuss possibilities of constructing perfect tensors with a larger number of legs. The 5qubit code is a qubit stabilizer code of the form introduced in section 5.7 whereas the 3qutrit code can be described through a natural generalization of the stabilizer formalism to higher spin dimensions.
a.1 qubit code and qubit state
The fivequbit code is a perfect code with distance 3 encoding one logical qubit in five physical qubits. It is a stabilizer code with a stabilizer subgroup given by , where
(30) 
Note that and hence the group is manifestly invariant under cyclic permutations. As is the case in the stabilizer formalism, codes are characterized by an abelian stabilizer subgroup () and codespace is the joint eigenspace for this group, code states satisfy
(31) 
In this case, there are two orthogonal codeword states.
Logical operators are unitary operators which preserve the codeword space, but may act nontrivially on it. They are given by
(32) 
One may see that both and commute with all the stabilizer generators, so they indeed preserves the codeword space. Yet, they anticommute with each other, so they characterize one logical qubit, and and behave as logical Pauli and  operators for a logical qubit. Namely, one can denote two codeword states by and such that , , , . Applications of stabilizer generators to logical operators do not change the action on the codeword space, so representations of logical operators are not unique. Then one can introduce the following equivalence relations among logical operators