Holographic Representation of One-Dimensional Many-Body Quantum States via Isometric Tensor Networks

TL;DR

This work introduces holographic isoTNS, a (1+1)D isometric tensor network with a holographic axis that enables efficient contractions while representing states with volume-law entanglement in 1D. It shows that random holographic isoTNS realize volume-law $S^{[2]}$, and that important classes such as MPS, fermionic Gaussian states, and Clifford states are exactly representable with $\chi=2$, including certain time-evolved states. Variational arguments and circuit-mapping perspectives illustrate broad representational power for highly entangled yet low-complexity states, while TEBD-based real-time evolution demonstrates both the potential and limitations of current algorithms due to orthogonality-surface shifts. The results suggest holographic isoTNS as a promising path to explore highly entangled, low-complexity quantum dynamics and motivate future scalable update schemes and higher-dimensional generalizations.

Abstract

Isometric tensor network states (isoTNS) allow for efficient and accurate simulations of higher-dimensional quantum systems by enforcing an isometric structure. We bring this idea back to one dimension by introducing a holographic isoTNS ansatz: a (1+1)-dimensional lattice of isometric tensors where the horizontal axis encodes physical space and an auxiliary "holographic" axis boosts expressivity. Despite the enlarged geometry, contractions and local updates remain computationally efficient due to isometric constraints. We investigate this ansatz and benchmark it in comparison to matrix product states (MPS). First, we show that randomly initialized holographic isoTNS typically display volume-law entanglement even at modest bond dimension, surpassing the representational limits of MPS and related ansätze. Second, through analytic constructions and variational optimization, we demonstrate that holographic isoTNS can faithfully represent arbitrary fermionic Gaussian states, Clifford states, and certain short-time-evolved states under local evolution -- a family of states that is highly entangled but low in complexity. Third, to exploit this expressivity in broad situations, we implement a time-evolving block decimation (TEBD) algorithm on holographic isoTNS. While the method remains efficient and scalable, error accumulation over TEBD sweeps suppresses entanglement and leads to rapid deviations from exact dynamics. Overall, holographic isoTNS broaden the reach of tensor-network methods, opening new avenues to study physics in the volume-law regime.

Figures (7)

• Figure 1: Graphical notation for tensor networks. Each circle is a tensor, with blue bonds for physical legs and black bonds for virtual legs connecting the tensors. Arrows indicate isometries. (a) Generic MPS. (b) MPS in isometric form, where all tensors are isometries directed toward the orthogonality center (filled red). (c) Contraction scheme for the local two-site expectation value $\langle\Psi|\hat{O}_{i,i+1}|\Psi\rangle$. Due to the isometric form, most of the network contracts to identity, leaving a non-trivial contraction of only four tensors. (d) PEPS representing a 2D quantum state on the square lattice. (e) isoTNS for the same 2D system, where the orthogonality surface is drawn in red and the orthogonality center tensor is filled red.
• Figure 2: (a) Diagrammatic representation of the holographic isoTNS ansatz for $L=8$ with all possible positions of the orthogonality surface. The horizontal axis represents physical space, and the vertical axis represents the virtual time domain. Physical legs are shown in blue, and bulk tensors are colored black. The pink surface represents the orthogonality surface, with the orthogonality surface tensor highlighted as a filled circle. The bond dimension along the orthogonality surface is upper-bounded by $\chi$, whereas all other bonds are given their full dimension of $d$. (b) Circuit picture of the ansatz. Each bulk tensor from (a) is mapped to a two-qudit unitary (black) acting on adjacent sites. The isometric tensors are represented by pink unitaries, which have multiple legs and act on fixed ancilla qudits [see Eq. \ref{['isometrygate']}]; for clarity, we omit the ancilla qudits from this visualization, and only show one leg (corresponding to the case with $\chi=2$). The execution time of the circuit flows in the opposite direction of the isometric arrows.
• Figure 3: (a) Contraction scheme for the calculation of the half-chain second Rényi entropy when the orthogonality surface is placed at the center of the system. (b), (c) Rényi-2 entanglement entropy $S^{[2]}$ as a function of the system size $L$. The blue lines represent results for holographic isoTNS with varying the bond dimension $\chi$, and the black dashed line plots the Page value. The orthogonality surface is placed at $L/2$ for (b) and at $1$ for (c). The data is averaged over $128$ random realizations.
• Figure 4: Variational optimization of holographic isoTNS (blue) and MPS (red) while changing the bond dimension $\chi$ (system size $L=14$). The left and right panels show, respectively, the Rényi-2 entropy $S^{[2]}$ and the variational error $||\ket{\Psi}-\ket{\Psi_\text{ref}}||^{2}$ as a function of time. (a) Dynamics under the KIC Hamiltonian at $(J,g,h)=(\pi/4,\pi/4,0.5)$ from the product state $\ket{\Psi_0} = \ket{\uparrow\uparrow\dots\uparrow}$. (b) TFIM dynamics at $(J,g)=(1,1)$ from the highly entangled volume-law rainbow state defined in Eq. \ref{['twoqubit']}. For completeness, we note that our results provide only an upper bound on the true variational error, as this computationally hard constrained optimization problem yields only local minima of the cost function.
• Figure 5: The TEBD algorithm on holographic isoTNS. The time-evolution operator is approximated via the 1st-order Suzuki-Trotter decomposition into nearest-neighbor gates. Gates are applied in a left-to-right sweep at odd time steps; for even time steps , a right-to-left sweep is performed (only odd time steps is shown). At each gate application, the orthogonality surface is shifted accordingly to one of the sites where the gate is applied.