Majorana dimers and holographic quantum error-correcting codes

Majorana dimers and holographic quantum error-correcting codes

A. Jahn, M. Gluza, F. Pastawski, J. Eisert Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
Abstract

Holographic quantum error-correcting codes have been proposed as toy models describing important aspects of the AdS/CFT correspondence. In this work, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC) and demonstrate that it allows us to efficiently compute boundary state properties. We show that the dimers characterizing boundary states of the HyPeC follow discrete bulk geodesics. From this geometric picture, properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow, offering a fresh perspective on holography. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which shares many properties of the recent bit thread proposal. Our work thus elucidates the connection between bulk geometry, entanglement, and quantum error correction in AdS/CFT, and lays the foundation for new models of holography.

July 2, 2019

I Introduction

The holographic principle – the idea that certain theories of gravity are dual to lower-dimensional quantum field theory – has had wide-ranging applications within theoretical physics. In particular, the AdS/CFT correspondence has changed our understanding of theories of both (quantum) gravity and quantum field theory, by giving a specific relationship between gravity on -dimensional negatively curved Anti-de Sitter spacetime (AdS) and -dimensional conformal field theory (CFT) Maldacena (1998); Witten (1998). A number of simple models capturing key aspects of holography have been constructed Almheiri et al. (2015); Lee and Qi (2016); Pastawski et al. (2015); Hayden et al. (2016); Mintun et al. (2015); Swingle (2012), largely relying on tensor network descriptions of bulk AdS geometry and boundary states. Tensor networks have long been understood as describing a state in terms of its entanglement structure Verstraete and Cirac (2006), thus serving as an ideal tool to study holography in terms of notions of quantum information theory Fannes et al. (1992); Verstraete et al. (2008); Schollwöck (2011); Orús (2014); Eisert et al. (2010). The basis of this work is the tensor network construction of the hyperbolic pentagon code (HyPeC), a class of holographic models often named HaPPY codes after the authors’ initials Pastawski et al. (2015). These codes explicitly realize holographic quantum error correction Almheiri et al. (2015) by providing an error-correctable mapping from bulk to boundary degrees of freedom, reproducing many of the features of AdS/CFT. However, the boundary states of the HyPeC differ from other tensor network models specifically designed to produce physical CFTs, such as the MERA Vidal (2008). For computational basis bulk inputs, where the tensor network becomes Gaussian and efficiently contractible, earlier studies revealed a pair-wise correlation structure in terms of boundary Majorana modes Jahn et al. (2017). As we show in this work, HyPeC states are in fact a special case of a Majorana dimer model, and can be described by entangled fermionic pairs. Majorana dimers have previously been used to describe superconducting phases on lattices Ware et al. (2016); Tarantino and Fidkowski (2016), as instances of tensor networks that have a fermionic component Barthel et al. (2009); Kraus et al. (2010); Corboz et al. (2010); Wille et al. (2017); Bultinck et al. (2017). We show that the contraction of dimer-based tensor networks is equivalent to combining entangled Majorana pairs, replacing the computational difficulties of contraction by simple rules on dimer diagrams. This graphical language directly visualizes parities, physical correlations, and the entanglement structure of quantum states spanning the entire fermionic Hilbert space. By deriving the holographic properties of the HyPeC merely from emergent entangled pairs, we connect to recent proposals of AdS/CFT models based on bit-threads Freedman and Headrick (2017a); Cui et al. (2018). Thus, our work is also an important step towards integrating discrete tensor network models of AdS/CFT into a unified setting.

Ii A simple model of holography

Consider the boundary and bulk Hilbert spaces denoted by and respectively. A holographic quantum error-correcting code is formed by an encoding isometry from the logical states in to boundary states in . Indeed, is the projector onto the code of the boundary Hilbert space . Any bulk operator acting on the states in can be represented by at least one operator acting on with the property while preserving the code subspace (). The specific form of such a mapping from bulk to boundary is the holographic dictionary obtained in continuum AdS/CFT by equating bulk and boundary partition functions Witten (1998), which is equivalent to considering boundary CFT operators as limits of fields on the gravitational AdS background Harlow and Stanford (2011). As we visualize in Fig. 1 (left), oftentimes acts non-trivially only on a subregion of the total boundary. Given a subregion on the boundary one can perform the so-called AdS/Rindler reconstruction Susskind and Witten (1998); Banks et al. (1998); Balasubramanian et al. (1999); Harlow and Stanford (2011); Dong et al. (2016); Hamilton et al. (2006) to associate to any boundary operator a corresponding bulk operator acting within the wedge which is a subset of the bulk.

Due to the computational difficulties in studying continuum AdS/CFT, discrete toy models often provide an easier approach to understanding its properties. These models usually consider a space-like slice of the full AdS spacetime, discretized by a tiling whose open boundary edges correspond to the AdS boundary. Subsets of these open edges are then identified with subregions of the boundary CFT (see Fig. 1, right).

What properties should the discretized boundary states in fulfill? As a bulk operator can be represented equivalently on different parts of the boundary, e.g. two regions and , we are led to the condition

(1)

where and are boundary representations on and of an operator inserted somewhere in the bulk. For this condition to hold for any and any suitable and , the states in must necessarily possess multi-partite and nonlocal entanglement to allow for operators that act equivalently on distant parts of the boundary.

Figure 1: Continuous (left) and discretized (right) reconstruction of an AdS bulk operator along two (causal) wedges and Almheiri et al. (2015), leading to two boundary operators and with support on boundary regions and . The AdS time slice is projected onto the Poincaré disk, with the AdS boundary corresponding to the black outer circle. The discretization is a tiling.

In this work we show that the holographic pentagon code implements these properties through an underlying fermionic structure. To motivate the use of fermions in the context of holographic quantum error correction, consider a simple toy model of entangled fermionic modes. Throughout, we denote fermionic canonically anti-commuting operators by satisfying and distinguish the vacuum state vector satisfying for any . The counterpart of a Bell pair for fermions is the so-called BCS state which has the form

(2)

By a simple calculation, we find that

(3)

which implies that if are boundary indices, we found a mapping between operators on bulk sites resembling (1). For holographic quantum error-correction, however, this mapping is insufficient: After acting with the operator, the result (3) is an unentangled Fock state vector , which is no longer in the desired code-space of entangled states. Furthermore, does not exhibit any multi-partite entanglement necessary for holography Walter et al. (2016). Fortunately, both problems can be resolved by fermionic mode fractionalization by means of Majorana dimers. Consider the action of Majorana operators, defined as

(4)

and fulfilling , on the BCS state vector (2) as

(5)
(6)

This shows that a mapping equivalent to (3) can be performed with Majorana operators without destroying entanglement. To achieve multi-partite entanglement, BCS-type states are insufficient. However, a suitable model is provided by the hyperbolic pentagon code (HyPeC). Let us briefly review its construction: The HyPeC is an isometry between bulk and boundary degrees of freedom. An AdS time slice is discretized by a finite tiling of pentagons, the Poincaré disk projection of which is shown in Fig. 1. Each pentagon is associated with one logical qubit, i.e. one bulk degree of freedom, encoded in five spins (the pentagon edges) via the quantum error-correcting code. This code can be expressed by a six-leg tensor, with one “bulk” leg corresponding to the logical qubit and the remaining five to the physical spins. The tiling is connected by tracing out spins on the edges of two adjacent pentagon tiles, i.e. by contracting the corresponding tensor indices. This contraction can be understood as a projection of the spins on the two connected edges onto a Bell pair. In this paper, we will usually consider this setup with each bulk input fixed to a certain state. Before contraction, the bulk is then effectively composed of a product state of local quantum states on five spins each. Contraction locally entangles the spins with each other, thus leading to a larger entangled state on the remaining spins at the boundary of the pentagon tiling. If we consider instead an arbitrary bulk input on each pentagon111For the purposes of this paper, bulk inputs between different pentagons are assumed to be unentangled., contraction combines the local 5-spin Hilbert spaces into a larger -spin Hilbert space that defines our code space .

By merit of the code, the five spins on the edges of each pentagon are absolutely maximally entangled. A pure state of qubits is absolutely maximally entangled if all of its reductions to subsystems are maximally mixed Gisin and Bechmann-Pasquinucci (1998); Higuchi and Sudbery (2000); Helwig et al. (2012) and hence the states are maximally entangled over all such cuts. The isometric properties of the code follow from this construction.

A useful approach to understanding these states is to represent this spin picture of the HyPeC in terms of Majorana fermions Jahn et al. (2017). This is achieved by a Jordan-Wigner transformation between spins and Majorana modes:

(7)

where we have used the -site Pauli operators defined as

(8)

in terms of the Pauli matrices . It will be useful to define the total parity operator

(9)

In the HyPeC, we take spins for each pentagon. The logical eigenvectors and of the code have eigenvalues and , respectively, corresponding to even and odd fermionic parity. For fixed bulk input (and thus parity), the stabilizers are quadratic in Majorana operators. Thus, and are ground state vectors of a Hamiltonian describing free Majorana modes, given by

(10)

where is the eigenvalue of and indices follow periodic boundary conditions. If we replace , we recover the original stabilizer Hamiltonian with its two-fold degenerate ground state. Before considering contractions of these fermionic code states, we now develop a comprehensive framework for Majorana dimers that allows us to study the fermionic HyPeC in detail.

Iii Majorana dimers

iii.1 Definition

Majorana dimers are effectively a reordering of the vacuum state in terms of Majorana modes. The -fermion vacuum state vector is defined by being annihilated by all of the fermionic annihilation operators for as

(11)

Thus, the vacuum state effectively relates pairs of Majorana modes in an operator equation. By permuting Majorana indices, we can generalize this state to any pairing of modes. Such a Majorana dimer state is determined via conditions on distinct pairs (choosing as convention) of Majorana operators

(12)

The dimer parities give each pair an “orientation” with respect to the index ordering. We refer to as “even” and as “odd”. To recapitulate, a Majorana dimer state is defined to be a (normalized) state vector of fermionic modes which is annihilated by independent conditions of the form (12). Note that we have fixed a vacuum state which under the Jordan-Wigner transformation corresponds to a product state in spins, but non-trivial Majorana dimer states can be highly entangled, as we shall see.

Equivalently, we may characterize Majorana dimer state vectors as ground states of specific quadratic Hamiltonians: Multiplying (12) with its Hermitian conjugate from the left yields

(13)

which implies that the Hermitian operator has expectation value . We can thus construct the (parent) Hamiltonian

(14)

where we sum over all Majorana dimers . The unique ground state vector of with energy is given by , which is in the eigenspace of all the summands.

These two equivalent characterizations are most intuitively visualized through a diagrammatic notation. Consider fermionic modes, ordered as a chain visualized by an -gon, with the Majorana modes shown as dots on the edge (mode). Arrows between the Majorana modes represent the pairing. For example, for , the state visualized by

(15)

is the ground state of the Hamiltonian

(16)

An arrow along the index orientation (, blue) corresponds to a dimer parity , while an arrow against it (, orange) corresponds to . Note that these diagrams only specify the state up to a scalar , as affects neither the ground state property nor the dimer conditions (12). A particularly symmetric case is the aforementioned vacuum represented by a diagram

(17)

for . Unsurprisingly, is also the ground state vector of the Hamiltonian with the local number operators . We can construct any Majorana dimer state from the vacuum by applying swap operators onto , being the total parity operator defined in the last section. For example, the state expressed by diagram (15) is given by . It should be noted that while these swap operators violate the fermionic super-selection rule in an actual fermionic systems, we are merely interested in Majorana dimers as an effective representation of spins (such as the HyPeC).

As Majorana dimer states are Gaussian, all expectation values are determined by the entries of the covariance matrix with entries

(18)

We can read off directly from the corresponding diagram: As is constructed from by acting with a product of swap operators mapping each index to an index , is simply with interchanged rows and columns

(19)
(20)

The only non-zero entries of the vacuum covariance matrix are . We can thus infer from its diagram using the rules

(21)

For example, the covariance matrix for diagram (15) is

(22)

with color-coded entries (orange, blue). Note that we have chosen the colors to match with the dimer parities when reading the entries above the main diagonal (). We assume that the state vector is normalized. Equivalently, we can think of the swap operators as acting on the Hamiltonian, yielding . Clearly, the spectrum of is simply a permutation of the spectrum of , consistent with the covariance matrix picture.

By Eq. (10), the code states are ground states of Hamiltonians quadratic in Majorana operators, and can thus be represented as Majorana dimers. As diagrams, they are given by

(23)
(24)

As we will see in the next section, the code distance between these two states in terms of Pauli operations can be shown graphically.

iii.2 Pauli operations and total parity

As the Majorana operators are obtained from spin operators through a Jordan-Wigner transformation, local Pauli operations in the spin picture generally act non-locally on the Majorana dimers. Specifically, the reverse transformation of (7) is given by

(25)

A Majorana operator acting on a Majorana dimer state flips the parity of the dimer ending on site . We show this by noting that if a state vector is annihilated by the operator (with dimer parity and ), then both and are annihilated by :

(26)
(27)

All other dimer conditions remain unaffected. As a graphical notation, we highlight the affected edges of the state in red. Some examples of these operations on a Majorana dimer state vector are shown here,

(28)
(29)
(30)

When both ends of a dimer are acted upon with a Majorana operator, the local parity stays the same. Note that operations only affect the th edge, while and combine a local Majorana operation with a string on the first edges. Using this graphical calculus, we can now see that it requires three Pauli operations to map (23) into (24) or each into itself. These correspond to bit flip (e.g. ) and phase flip errors (e.g. ), respectively.

Now consider the total parity operator , which affects all Majorana sites at once. Clearly, acting with leaves the state invariant (up to the parity eigenvalue), which implies that all Majorana dimer states have definite total parity . In fact, this parity is given by

(31)

depending on the dimer parities of all dimers as well the number of crossings between dimers. This statement can be proven inductively: We start with the vacuum with . It corresponds to a diagram with for all dimers and no crossings. We can now construct any state vector from by applying swap operators . Since anticommutes with , each swap inverts . To see that (31) reflects this, consider how a swap affects the pairs ending in and for each possible initial configuration as

(32)

Up to mirroring, relabeling and relative shifting of indices, all possible swaps belong to one the four categories shown above. The first two swaps flip one local parity but create no additional crossings; the last two either add or remove one crossing while flipping an even number of parities. Thus (31) is always satisfied. Note that we are free to move around the dimer curve between the fixed endpoints, which means we can make two (or more) paths overlap. However, this will always change the number of crossings by an even number. For example, the logical state of the code corresponds to both of the following diagrams (each with ten crossings):

(33)

As expected, (31) tells us that this state has positive parity. For a fixed dimer configuration but variable dimer parities , only the second factor of (31) is relevant. Thus we find that acting with an or operator, which changes an odd number of dimer parities, also flips the total parity. A error, which always flips two dimer parities, leaves the total parity invariant.

iii.3 Contracting dimers

We will now show how the notion of tensor network contraction applies to Majorana dimer states. To begin with, consider a state of spins

(34)

Here the amplitudes can be viewed as a tensor which fully specifies the state vector . A tensor network is a means of specifying a tensor describing a state of a large number of spins through multiple contractions of tensors of a smaller rank. Specifically, the contraction of two tensors and of rank and between the last index of and the first index of is defined to be a new tensor of rank , with entries

(35)

We see that by contracting the respective tensors, this operation allows us to merge two state vectors and into a larger one given by

(36)

A tensor network state can thus describe a large state by the relatively few entries of its contracted tensors. This process can be generalized to fermions by identifying the spin basis with a fermionic one as

(37)

In this picture, tensors are associated with pure fermionic states. As these expressions only use creation operators, they obey a Grassmann algebra. The tensor contraction (III.3) can then be expressed by a Grassmann integration over fermionic degrees of freedom Bravyi (2009). Specifically, a contraction of two fermionic state vectors and into a state vector over the same indices as in (III.3) has the form

(38)
(39)

where we have used the Grassmann integration (for more information, see Refs. Berezin (1966); Cahill and Glauber (1999); Bravyi (2009, 2005)). Note that acts like an annihilator on a fermionic state, with a subsequent projection onto the fermionic subspace excluding the th mode. This requires a relabeling of the remaining degrees of freedom and a truncation of the Jordan-Wigner string in the corresponding spin representation.

We can now apply this machinery to Majorana dimer states. In our graphical language, tensor contraction is equivalent to connecting two polygon edges and integrating out the four Majorana modes on them. What happens to the dimers of the original states? It is easy to see that dimers of a state vector not connected to the contracted edges remain dimers, i.e., if vanishes, we also find

(40)

as and commute with the integration. We now claim that the dimers connected to the contracted edge become new dimers of the contracted state . This leads to the following statement.

Theorem 1 (Contractions of Majorana dimer states).

The contraction of two Majorana dimer state vectors and yields either a new Majorana dimer state vector or zero.

An example for the contraction of two pentagon state vectors and is given by

(41)

We have visualized the contraction by a pair of dashed lines. In this example, dimers not connected to the contracted edges are omitted. The upper diagram corresponds to the conditions

(42)
(43)

We now prove that (42) implies for the contracted state vector , i.e., that the two original dimers fuse into a larger one:

(44)

where we have used the identities and . A similar proof using (43) leads to .

The full proof for all possible dimer contractions is given in Appendix B. The resulting rules are:

  • Contracting neighbouring edges and removes the Majorana modes . The dimers ending on and as well as and are fused into larger dimers.

  • The dimer parity of a fused dimer is the product of parities of the original dimers. In addition, every crossing of the path of a contracted dimer with itself reverses .

  • Every contraction that creates closed loops leads to a vanishing contracted state if at least one loop has an odd dimer parity.

The last case refers to diagrams such as the following:

(45)

Loops with even total dimer parity only change the contracted state by a non-zero constant.

iii.4 Computing entanglement

The entanglement entropy of a subsystem and its corresponding reduced density matrix can be evaluated diagrammatically. Given the Majorana covariance matrix of the subspace belonging to (i.e., the rows and columns of the full covariance matrix whose Majorana modes are contained in ), we can perform a special orthogonal transformation to the form

(46)

where are the eigenvalues of , some of which may be zero. From this form, the entanglement entropy is computed as

(47)

As we have found in Section III, the covariance matrix entries of Majorana dimer states can only be or zero. Consider the th row (or column) of the sub-matrix : The dimer connected to Majorana mode ends on another mode (with .) If , the and th row will jointly contribute to a of , i.e., zero entanglement entropy. However, if , the th row of will be zero. As the number of such “dimer leaks” must be even, each pair contributes to a vanishing . Thus each dimer connecting with its complement contributes an entanglement entropy of , i.e., half of an EPR pair. Graphically, the entanglement entropy reduces to counting such dimers:

(48)

Consider the following example.

(49)

The subsystem comprises four edges with the Majorana modes to . As four dimers connect with , the entanglement entropy is given by . Effectively, counts the number of dimers across the cut separating from (shown as a dashed line). For contracted states, , where is the length of the shortest cut through the contracted network. Thus, we recover the tensor network interpretation of the Ryu-Takayanagi surface , which appears in continuum AdS/CFT in the holographic entanglement entropy formula Ryu and Takayanagi (2006)

(50)

which expresses in terms of the area of the minimal surface , denoted , and Newton’s constant . In our two-dimensional bulk space, is simply a geodesic and its length. We will see later how the discrete analog of (50), where , is saturated in the HyPeC.

Note that this method of calculating entanglement entropies, as well as the more advanced methods presented in Appendix C, rely on the assumption of the subsystem being simply connected. We can relate a disjoint subsystem to a connected subsystem by transposing indices. Fermionic transpositions can only affect dimer parities, and thus leave invariant. However, for many cases – including the HyPeC model – Majorana dimers are only an effective spin representation, so to compute we need to transpose spin indices and describe the result in terms of Majorana dimers. Unfortunately, a spin transposition usually does not preserve the state’s Majorana dimer structure, as it leads to fermionic states that are not ground states of Hamiltonians quadratic in Majorana operators. As a result, the entanglement entropy of a disjoint subsystem can differ substantially between a system of fundamental spins and a fermionic system (where (48) still holds).

In the latter case, however, we can easily generalize (48) to more complicated entanglement measures for disjoint subsystems, such as the mutual information

(51)

Compared to (48), each dimer in (III.4) is counted twice. In terms of the geometry of the dimer graph itself, (III.4) corresponds to a system with an exact area law. Bao et al. (2015) One of the properties of this form of the mutual information is an always vanishing tripartite information Casini and Huerta (2009)

(52)

This implies that Majorana dimer models are compatible with holographic theories, where Hayden et al. (2013). Furthermore, as we show in Appendix C, the spectrum of Rényi entropies

(53)

is flat, a property of the underlying stabilizer state structure Flammia et al. (2009). We show in Appendix C that this property also follows from the Majorana dimer picture for arbitrary local superpositions of bulk input in the HyPeC under certain constraints on the (compact) boundary region considered.

To clarify the connection between Majorana dimers and EPR pairs, we can explicitly construct Bell states from pairs of dimers. Consider the following two even-parity dimers connecting edges and (with ) without crossing:

(54)

This corresponds to two conditions on the total state vector ,

(55)

As no entanglement between edges and and the rest of the system exists, should be factorizable with regards to these degrees of freedom:

(56)

where includes terms containing creation operators with . Up to a complex phase, the parameters can be fixed using (55), which leads to and (assuming normalization). This corresponds to a Bell state vector on sites and . This analysis can be repeated for all possible dimer configurations, yielding Table 1. Conveniently, this allows us to form superpositions of dimers, for example

(57)

Each diagram in this expression corresponds to a normalized Majorana dimer state. Note that this diagram confirms our intuition that a contraction, which is the sum of projections onto and , is equivalent to connecting pairs of Majoranas via dimers. In a mild abuse of notation, we may thus write

(58)

to express a contraction (dashed lines). This also allows us to fix relative factors that appear through contraction, such as in the following projection of (57) onto a state vector:

(59)

The second term vanishes from the condition , in agreement with the rule that loops of total odd parity vanish (compare Eq. (45)). Note that the arrow orientation of the dimer for is reversed, as it is used in its adjoint form (more on Hermitian conjugates in the next section). Projections like (III.4) can be evaluated for each of the entries in Table 1, always leading to a resulting factor of . This result is heavily used in Appendix C, where we study the entanglement properties of superpositions of HyPeC code states, where norms of Majorana dimer states become relevant.

Majorana dimer Bell state Majorana dimer Bell state
Table 1: Bell states expressed as Majorana dimers.

iii.5 Orthogonality and completeness

Our diagrammatic notation can also express inner products. Consider the bra corresponding to a ket . Clearly, if then holds for the adjoint. Thus, we can visualize adjoints by inverting all arrows and corresponding parities, for example (omitting labels):

(60)

The inner product is a contraction between and over all indices, expressed as

(61)

The right-hand side, showing a “flipped” , represents a Choi-Jamiolkowski isomorphism expressing in the same Hilbert space as . This involves an inversion of the orientation which flips all dimer parities. As all diagrams are defined for normalized state vectors that satisfy . Furthermore, we can easily evaluate whether two diagrams correspond to orthogonal states as a contraction of and vanishes for any odd-parity loop (see Appendix B). In particular, and are orthogonal if they share the same correlation structure (i.e., pairing of Majorana modes) but differ in at least one dimer parity. This allows us to construct the complete Hilbert space with Majorana dimer states on edges by fixing a correlation structure and then flipping through all possible dimer parities, resulting in mutually orthogonal state vectors. Since the Hilbert space is also -dimensional, we can express any state in it by a superposition of Majorana dimer states under the given correlation structure. This is equivalent to obtaining a orthogonal stabilizer state basis by considering all sign combinations of stabilizer generators.

Iv The HyPeC with Majorana dimers

iv.1 Overview

As we saw in the previous section, the computational basis logical code states of the quantum error-correcting code can be expressed as Majorana dimers. Furthermore, we showed that identifying Majorana dimer states as tensors and contracting them yields new Majorana dimer states, and that these contractions are easy to evaluate diagrammatically. Because the HyPeC is built from tensors each representing the code, we find the following key result:

Theorem 2 (Representing the HyPeC with Majorana dimers).

The hyperbolic pentagon code (HyPeC) with logical bulk input fixed to local basis states or yields a Majorana dimer state on the boundary. Each input corresponds to a (unique) pattern of dimer parities on the boundary state.

While fermionic modes require an explicit ordering, we show in Appendix D that different contraction orderings lead to equivalent boundary states. We will now show how the geometry of the dimers in the HyPeC determines its properties, using the tools developed in the previous section.

iv.2 Dimers and entanglement structure

First, we will consider the physical properties of the HyPeC for logical inputs fixed locally to or . The code is constructed from a hyperbolic tiling222This Schläfli symbol denotes a polygon tiling with four pentagons at each vertex., with each tile now set to (23) or (24) (the full HyPeC also allows for superpositions between the two). As the model consists of asymptotically infinite tiles, we have to define a UV cutoff at which the tiling is truncated. We do this by starting with a central tile and iteratively adding tiles on all free edges. The number of iterations thus gives the graph distance between each boundary tile and the centre, determining the cutoff. Such a series of iterations for an all- bulk input is visualized in Fig. 2.

Figure 2: Iterative contraction of the HyPeC for fixed bulk inputs of state vectors, with a cutoff after 3 iterations. Each step involves contracting a further layer of tiles, starting from the centre at . The asymptotic boundary of the Poincaré disk is drawn as a black circle.

The contracted dimers are drawn as geodesics in the Poincaré disk. This is not an arbitrary choice, as the dimers follow discrete geodesics (i.e., shortest paths) in the tiling. Fig. 3 shows the limit both for an example of two dimers and the whole contraction. Because of the particular property of the tiling that the pentagon edges connect to continuous geodesics, the asymptotic endpoints of a contracted dimer are also endpoints of such a geodesic.333In the dual tiling, each four-sided tile contains an intersection of two such geodesics meeting at right angles. Tracing this geodesic back into the bulk, we see that it passes along all tiles that contained the uncontracted dimer pieces. Furthermore, as Fig. 3 also shows, there are always two dimers with the same pair of asymptotic boundary points, resulting in a bulk geodesic that is dual to a boundary Bell state. While Fig. 3 only shows a uniform bulk input, the dimer parities generally differ with the input. The dimer pairs then correspond to different types of Bell states, as in Table 1. Note that the Majorana modes composing the effective fermions of these Bell states are located on neighbouring boundary edges at any finite cutoff. This elucidates the code’s error correction properties: Any product of pairs of Majorana operators acting on dimer endpoints can only change the state up to a total sign, and is thus a representation of a logical parity operation in the bulk. While single Majorana operators are nonlocal in terms of spins, pairs of Majorana operators on neighboring sites can be expressed by a local pair of Pauli operators (compare (25)). For each pair of HyPeC dimers, there then exists a boundary operator of weight , i.e., consisting of four Pauli operators acting on the boundary, which represents a logical bulk operation. Thus, even for an infinitely large number of HyPeC tiles, the code distance never exceeds , as such an represents an error on the code space.

Figure 3: Left: A Majorana dimer pair in an infinitely large contraction of HyPeC tiles. The endpoints of both dimers meet at the asymptotic boundary, thus the dimer pair can be drawn as a double-geodesic. Right: Full contraction for a input on all tiles, with all dimers pairing up.

Given this picture of pair entanglement on the boundary through bulk geodesics, the dependence of the entanglement entropy on the boundary subsystem size can be explicitly computed. Clearly, the position of this subsystem affects the value of , as the distribution of entangled pairs in Fig. (3) does not respect translation invariance on the boundary. Thus, we consider the average expectation value of the entanglement entropy instead. The results in Fig. 4 show an approximate logarithmic growth , as expected of a critical theory. Fitting against the expected logarithmic scaling expected for a CFT Calabrese and Cardy (2004),

(62)

we find a central charge (dashed line in Fig. 4). For a finite system of boundary size , reaches its maximum at , in agreement with the full form of (