Abstract
The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand these bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1 + 1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multiscale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error–correcting codes and tensor networks. This work is expected to stimulate a more comprehensive study of tensor network models capturing bulk-boundary correspondences.
INTRODUCTION
The notion of holography in the context of bulk-boundary dualities, most famously expressed through the anti–de Sitter space/conformal field theory (AdS/CFT) correspondence (1), has had an enormously stimulating effect on recent developments in theoretical physics. A key feature of these dualities is the relationship between bulk geometry and boundary entanglement entropies (2–4), prominently elucidated by the Ryu-Takayanagi formula (5). Because of the importance of entanglement in the context of AdS/CFT (6), it was quickly realized that tensor networks are ideally suited for constructing holographic toy models, most notably the multiscale entanglement renormalization ansatz (MERA) (7–9). The realization that quantum error correction could be realized by a holographic duality (10) further connected to ideas from quantum information theory. Despite the successful construction of several tensor network models that reproduce various aspects of AdS/CFT [see, e.g., (11–13)], a general understanding of the features and limits of tensor network holography is still lacking. Particular obstacles are the potentially large parameter spaces of tensor networks and the considerable computational cost of contraction.
In this work, we overcome some of these challenges by applying highly efficient contraction techniques developed for matchgate tensors (14, 15), which replace tensor contraction by a Grassmann-variate integration scheme. These techniques allow us to comprehensively study the interplay of geometry and correlations in Gaussian fermionic tensor networks in a versatile fashion, incorporating toy models for quantum error correction and tensor network approaches for CFT, such as the MERA, into a single framework, highlighting the connections between them. Furthermore, this framework includes highly symmetrical tensor networks based on regular tilings (see Fig. 1, A and B). We are thus in a position to efficiently probe the full space of Gaussian bulk-boundary correspondences from a small set of parameters, including the bulk curvature. We show that matchgate tensor networks with a variety of bulk geometries contain the Ising CFT in their parameter space to remarkably good approximation as a special case, with properties similar to the wavelet MERA model (16, 17). While regular hyperbolic tilings have recently been considered as a MERA alternative (18), we show that flat tilings can lead to very similar boundary states. In our studies, we restrict ourselves to tensor networks that are nonunitary and real, resembling a Euclidean evolution from bulk to boundary. In particular, we do not require the causal constraints of the MERA for efficient contraction, thus providing new approaches in the context of tensor network renormalization (19, 20). While we provide substantial evidence that tensor networks are capable of describing bulk-boundary correspondences beyond known models and introduce a framework for their study, our work is by no means exhaustive. We do hope to provide a starting point for more systematic studies of holography in tensor networks.
Discretizations of flat (A) and hyperbolic space (B and C) with a triangular tiling (blue edges), into which a tensor network is embedded (black lattice). In the matchgate formalism, joint edges between triangles correspond to an integration over a pair of Grassmann numbers, analogous to tensor network contraction over indices. While (A) and (B) show regular tilings, (C) presents a nonregular MERA-like tiling we call the matchgate MERA (mMERA).
MATERIALS AND METHODS
We constructed two-dimensional planar tensor networks with fermionic bulk and boundary degrees of freedom. The bulk degrees of freedom are associated with a set V of vertices of a tensor network. At each vertex, v ∈ V, a local tensor Tv with kv indices is placed, which can be interpreted as a local fermionic state on kv sites. After contraction over all connected bulk indices, the L remaining open indices are interpreted as boundary sites with the boundary state specified by the full contracted tensor. Because of the planarity of the network, the boundary sites form a loop. The bulk geometry can be flat or negatively curved (a positively curved network closes in on itself after finite distance). We visualized our tensor networks by representing each tensor Tv as a kv-gon whose edges correspond to indices. Thus, the tensor network is represented by a polygon tiling, which determines the bulk geometry. Adjacent edges between two polygons correspond to contracted indices and boundary edges to open ones. See Fig. 1 for examples.
Concretely, each bulk degree of freedom v ∈ V is associated with a local tensor Tv : {0,1}×r → ℂ of tensor rank r (equal to the number of edges of the corresponding tile), all of which are contracted to form tensors of higher rank. We denote the tensor component at indices j ∈ {0,1}×r as Tv(j) and the standard computational basis for r boundary spins as
For a broader introduction to tensor networks and their contractions, see (21–24).
Instead of explicit tensor contraction along pairs of indices, we used the formalism from (15) using Grassmann integration. Any tensor T can be represented by a Grassmann-variate characteristic function
Consider a rank r tensor T(x) with inputs x ∈ {0,1}×r. One calls T(x) a matchgate if there exists an antisymmetric matrix A ∈ ℂr × r and a z ∈ {0,1}×r so that one can write
A generic even matchgate has a simple Gaussian characteristic function of the form
Using Pauli matrices σα with α ∈ {x, y, z}, one can define Majorana operators γi via the Jordan-Wigner transformation
The computational basis is then equivalent to an occupational basis. In this context, we proved that any fermionic Gaussian state vector in the form of Eq. 1 has coefficients T(j) constituting a matchgate tensor. For details on this proof, refer to the Supplementary Materials. The converse statement is also true, providing a further perspective on the connection to free fermions (28).
RESULTS
The holographic pentagon code
We will now apply our framework to the highly symmetric class of regular bulk tilings, first implementing the holographic error correcting code (HaPPY code) proposed in (12) and then exploiting the versatility of our framework to extend it toward more physical setups. The HaPPY code furnishes a mapping between additional (uncontracted) bulk degrees of freedom on each tensor and the boundary state, realized by a bulk tiling of pentagons. Each pentagon tile encodes one fault-tolerant logical qubit via the encoding isometry of the five-qubit code. This [[5,1,3]] quantum error–correcting code (29) saturates both the quantum Hamming bound (30, 31) and the singleton bound (31) and can be expressed as a stabilizer code (32).
We observe that fixing the bulk degrees of freedom to computational basis states gives rise to a matchgate tensor network, as the logical computational basis states of the holographic pentagon code can be viewed as ground states of a quadratic fermionic Hamiltonian. This can be seen directly by applying Eqs. 7 and 8 onto the stabilizers Sk of the underlying [[5,1,3]] code, thus expressing it in terms of Majorana operators γi and a total parity operator
As the corresponding stabilizer Hamiltonian is given by
The holographic pentagon code of the HaPPY model for fixed computational bulk input (left) is equal to a matchgate tensor network on a hyperbolic pentagon tiling (right).
We find that the correlation structure of this model is best captured in the Majorana picture. Explicitly, consider the pentagon tiling of (12) with all bulk inputs set to the positive-parity eigenvector
(A to C) Majorana covariance matrix Γ with color-coded entries for a boundary state of a hyperbolic {5,4} tiling of the HaPPY code with fixed
(A and B) Boundary state properties of the HaPPY code at 2605 boundary sites. (A) shows average correlations at boundary distance d, computed as the relative frequency n of Majorana pairs. Dashed gray line shows an n(d)~1/d numerical fit. (B) shows the scaling of average entanglement entropy
The peculiar pairwise correlation of boundary Majorana modes, suggesting a connection to Majorana dimer models (35), is more deeply explored in a separate publication (36). However, as the correlation structure breaks the translation and scale invariance expected of CFT ground states, we now consider regular tilings with generic matchgate input.
Regular triangulations
As the boundary states of triangular tilings are necessarily Gaussian (15), we can study their properties comprehensively using matchgate tensors. The simplest such tilings are regular and isotropic, i.e., with each local tensor specified by the same antisymmetric 3 × 3–generating matrix A. Isotropy constrains its components to one parameter a = A1,2 = A1,3 = A2,3. The bulk topology follows from our choice of tiling. For triangular tilings (p = 3), setting q = 6 produces a flat tiling, whereas q > 6 leads to a hyperbolic one (see Fig. 1, A and B). Triangular tilings with q < 6 produce closed polyhedra that are positively curved and lack the notion of an asymptotic boundary. As a convention, we choose the local orientation of the triangles so that the generating matrix for the contracted boundary state satisfies
We now consider the boundary states of {3, k} bulk tilings. The falloff of correlations along the boundary generally depends on k, i.e., the bulk curvature, as shown in Fig. 5 (A and B) for the a = 0.25 case. While correlations between the boundary Majorana fermions of a flat bulk fall off exponentially, a hyperbolic bulk produces a polynomial decay (up to finite-size effects at large distances and rounding errors at very small correlations). In the hyperbolic case, geodesics between boundary points scale logarithmically in boundary distance, so the falloff is still exponential in bulk distance, as we would expect in AdS/CFT (37).
(A and B) Mean value of Majorana covariance
Restricting ourselves to the 0 < a < 1 region, we explore how quickly correlations decay in both settings. At a = 0 and a = 1, the boundary Majorana fermions only have neighboring pair correlations, either pairing within each edge (a = 0) or across the corners (a = 1). Thus, correlation decay becomes infinite in the limits a → 0 and a → 1, independent of bulk geometry. We use numerical fits to study the remaining region 0 < a < 1 (see Fig. 5, C and D). For a hyperbolic bulk geometry, the power law is generic with the slowest decay at a ≈ 0.61, where we see a ∝d−1 falloff over distance d. The exponential decay ∝e−d/λ generally produced by a flat bulk geometry, however, slows down to a power law (with correlation length λ diverging) around a ≈ 0.58, where correlations again decay as ∝d−1. At their critical values, the boundary states of both bulk geometries have the same average properties.
Up to finite-size effects, this critical boundary theory turns out to be the Ising CFT, as we confirm by computing a range of critical properties from the covariance matrix, shown in Table 1. The entanglement entropy scaling, shown in Fig. 4C, again matches the expected form (11) irrespective of the choice of tiling. The Ising CFT state that we observe at the critical value of a is the ground state of the Hamiltonian
Listed are the ground-state energy density ϵ0, central charge c, scaling dimensions Δϕ of the fields
Translation invariance and MERA
The regular bulk tilings considered so far have a set of discrete symmetries. When choosing identical tensors on each polygon, the boundary states necessarily inherit these symmetries, breaking translation invariance. To recover it, we consider a tiling with the same geometry as the MERA network. As we restrict ourselves to real generating matrices for the three- and four-leg matchgate tensors in this geometry, our model is not a unitary circuit but a model of Euclidean entanglement renormalization resembling imaginary time evolution, extending ideas from (19, 20). This may provide a more realistic representation of the causal structure of an AdS time slice than the standard MERA. Accordingly, the tensors of our matchgate MERA (mMERA) do not correspond to the usual (norm-preserving) isometries and disentanglers. We can still produce almost perfectly translation-invariant boundary states (Fig. 3F) while optimizing over only three parameters and recover the expected CFT properties (Table 1). In particular, at bond dimension χ = 2, the ground-state energy has a relative error of only 0.02% compared to the exact solution. Note that the optimization process only takes a few minutes on a desktop computer for a network with hundreds of tensors. We also find that the χ = 2 mMERA has a symmetry that allows us to write its four-leg tensors as contractions of simpler three-leg tensors (see Fig. 1C), yielding a nonregular triangular tiling. An interesting question to pursue is whether alternating or quasiperiodic tilings with a larger parameter space than regular tilings can also produce translation-invariant states.
DISCUSSION
In this work, we have studied bulk-boundary correspondences in fermionic Gaussian tensor networks, introducing a versatile framework and a highly efficient contraction method based on matchgate tensors (14, 15) for a wide class of flat and hyperbolic bulk tilings. We showed that our framework includes the holographic pentagon code built from five-qubit stabilizer states for fixed bulk inputs. Its boundary states correspond to a nonlocal bulk pairing of Majorana fermions, opening an avenue to studying the state properties of this holographic model at large sizes. We explicitly computed two-point correlators and entanglement entropies, which were found to exhibit critical scaling. Beyond known models, we showed that critical and gapped Gaussian boundary states can be realized by various bulk tilings. In particular, the average scaling properties of the
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/8/eaaw0092/DC1
Section S1. Tensor contractions in the Grassmann formalism
Section S2. Matchgates and fermionic Gaussian states
Section S3. Conversion of generating matrices to covariance matrices
Section S4. Contraction rules for generating matrices
Section S5. Explicit generating matrices and numerical results
Fig. S1. Combining tiles of matchgates.
Fig. S2. Tile orientations under contraction.
Fig. S3. Constructing the mMERA.
Fig. S4. Energy convergence of the mMERA.
Fig. S5. Determining scaling dimensions of flat tilings.
Fig. S6. Determining scaling dimensions of hyperbolic tilings.
Fig. S7. Determining scaling dimensions of mMERA.
Fig. S8. Determining structure constants.
Fig. S9. Correlations and entanglement with IR cutoff.
Fig. S10. Construction of triangle states with bond dimension χ = 2, 4, 8.
Table S1. Values of the critical generating matrix parameter a for different {3, k} triangular tilings and ultraviolet cutoffs.
Table S2. Exact conformal scaling dimension of various (quasi-)primary fields ϕ of the Ising CFT.
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REFERENCES AND NOTES
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