Holography and criticality in matchgate tensor networks

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 S_k of the underlying [[5,1,3]] code, thus expressing it in terms of Majorana operators γ_i and a total parity operator Ptot=(σz)⊗5 asS1=σx⊗σz⊗σz⊗σx⊗12=iγ7γ2S2=12⊗σx⊗σz⊗σz⊗σx=iγ9γ4S3=σx⊗12⊗σx⊗σz⊗σz=iPtotγ6γ1S4=σz⊗σx⊗12⊗σx⊗σz=iPtotγ8γ3S5=σz⊗σz⊗σx⊗12⊗σx=iPtotγ10γ5(9)

As the corresponding stabilizer Hamiltonian is given by H=−∑k=15‍Sk, we find a doubly degenerate ground state whose degeneracy is lifted by the parity operator Ptot. The resulting two states with parity eigenvalues ±1 correspond to the logical eigenstates 0¯ and 1¯, which are themselves ground states of purely quadratic Hamiltonians with different parity factors. Thus, both computational basis states are pure Gaussian, leading us to the conclusion that for fixed computational input in the bulk, the holographic pentagon code yields a matchgate tensor on the boundary (see Fig. 2). The explicit construction is given in the Supplementary Materials. Using the Schläfli symbol {p, q}, where p is the number of edges per polygon and q is the number of polygons around each corner, we can specify the hyperbolic geometry of the HaPPY model as a regular {5, 4} tiling.

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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 ∣0¯〉. The entries of the Majorana covariance matrix Γj,k=i2〈ψ∣[γj,γk]∣ψ〉 resulting from successive contraction steps are shown in Fig. 3 (A to C). As we can see, both the individual pentagon state and the larger contracted states are characterized by a nonlocal pairing of Majorana fermions. The contractions effectively connect Majorana pairs from each pentagon to a larger chain, so the pairs on the boundary of the contracted network can be seen as end points of a discretized “geodesic” spanning the bulk. While this discontinuous correlation pattern of Γ_j,k makes the computation of CFT observables difficult, we can estimate the average correlation falloff by counting the relative frequency n(d) of Majorana pairs at distance d = ∣ j − k∣ over which they connect points on the boundary. According to the results shown in Fig. 4A, correlation falloff follows a power law n(d) ∝ d⁻¹, as expected of a CFT. Furthermore, we compute the entanglement entropy S_A of a subsystem A of size l averaged over all boundary positions, defined asEl(S)=∑k=1LS[k,k+l](10)The result, shown in Fig. 4B, closely follows the Calabrese-Cardy formula for periodic 1 + 1–dimensional CFTs, given by (33, 34)SA=c3log(LπεsinπlL)≃c3loglε+O((l/L)2)(11)with a numerical fit yielding c ≈ 4.2 and ε ≈ 1.1 for a cutoff at L = 2605 boundary sites.

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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 = A_1,2 = A_1,3 = A_2,3. 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 Ai,j′>0 for i > j, corresponding to antiperiodic boundary conditions: Covariance matrix entries Γ_i,j acquire a sign flip when cyclic permutions push either index i or j over the boundary, as relative ordering is reversed.

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).

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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 HamiltonianH=i(∑k=1N−1γkγk+1+γ1γN)(12)where the sign of the boundary term γ₁γ_N signifies antiperiodic boundary conditions. Triangular tilings also incorporate more generic models: By associating each edge with a bond dimension χ > 2, it is possible to produce boundary theories with central charges c larger than 1/2. In the simplest case, we choose a generating matrix that only couples between sets of fermionic modes, resulting in a boundary theory that consists of multiple copies of the Ising CFT and a corresponding central charge that is a multiple of 1/2 (note that this construction is only possible for {n, k} tilings with even k). Furthermore, by changing the tensor content in a central region of the network, a mass gap can be introduced, highlighting how radii in a hyperbolic bulk correspond to a renormalization scale on the boundary. Details are provided in the Supplementary Materials.

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.