Alternating and Gaussian fermionic Isometric Tensor Network States

TL;DR

This work identifies a fundamental limitation of uniform isoTNS in 2D by showing entanglement is mediated along isometric directions and can be inefficient for directions perpendicular to those arrows. It introduces alternating isoTNS (alt-isoTNS) that symmetrically mediate entanglement and maps isoTNS to sequential quantum circuits with depth $O(L_x\cdot L_y)$, enhancing representability while preserving contraction efficiency. A second major advance is the isoGfTNS framework, which imposes isometric constraints on Gaussian fermionic tensor networks and enables gradient-based ground-state optimization for free-fermion Hamiltonians; combined with alt-isoTNS, it yields improved bond-dimension scaling and variational energies across several models, including the Fermi surface, band insulators, and the $p_x+ip_y$ superconductor, as well as interacting 2D transverse-field Ising models. The results suggest alt-isoTNS (and isoGfTNS) offer a more expressive yet efficiently optimizable class of 2D tensor networks with potential implications for quantum circuit design and quantum simulation on near-term devices.

Abstract

Isometric tensor networks in two dimensions enable efficient and accurate study of quantum many-body states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward, hence the name alternating isometric tensor network states. Second, we introduce a numerical tool -- isometric Gaussian fermionic TNS (isoGfTNS) -- that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNS represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNS. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNS mediate entanglement more efficiently than conventional isoTNS. Second, alternating isoTNS correspond to a deeper, thus more representative, sequential circuit construction of depth $O(L_x \cdot L_y)$ compared to the original isoTNS of depth $O(L_x + L_y)$. Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidences of better bond-dimension scaling and variational energy of alternating isoGfTNS for ground states of various free fermionic models, including the Fermi surface, the band insulator, and the $p_x + ip_y$ mean-field superconductor. Finally, we find improved performance of alternating isoTNS as compared to the original isoTNS for the ground state energy of the (interacting) transverse field Ising model.

Figures (15)

• Figure 1: A $5\times 5$ uniform (a) and alternating (b) isoTNS with open boundary condition. The orthogonality center is the single, upper right tensor with all incoming legs. The orthogonality hypersurface is the set of red tensors. Note for the alt-isoTNS that the isometry arrow direction of the columns alternates, while for uni-isoTNS all the column arrows point up. (c) Contour plot of the $k$-space occupation number of the uni- (left) and alt-isoTNS (right) at bond dimension $\chi=16$, representing the Fermi surface ground state of Eq. \ref{['eq:fermi_surface']}.
• Figure 2: Causal structure of uni- (a) and alt-isoTNS (b). For the single site tensor colored in black, its future (past) light cone is colored on orange (blue). The uncolored tensors belong to the space-like region outside the light cone.
• Figure 3: Uni (a,c) and alt-isoTNS (b,d) representions of decoupled chains along the (1,1)-diagonals (a,b) and along the (1,-1)-diagonals (c,d). Two chains, one in blue and one in orange, are shown. The physical indices are suppressed for visual clarity. Crucially, the uni-isoTNS representation of $\ket{\hbox{$\diagdown$}}$ requires a MERA structure, while the alt-isoTNS representation mediates all entanglement locally.
• Figure 4: The sequential circuit representation of an isometric MPS. $B_i'$ is a unitary matrix obtained from adding orthogonal columns to the isometric matrix $B_i$.
• Figure 5: (a) Uni- and (b) alt-isoTNS when viewed as sequential quantum circuit to prepare a 9 qubit state. All bond dimensions are $\chi=2$, so each line represents a qubit. Different qubits are drawn in different colors. The label of the tensors corresponds to the time-step when the gate can be applied. Dotted lines indicate input qubits in the state $\ket{0}$. Due to isometry conditions on each tensor, the vertical bond dimension in the left-most column is $\chi=1$. (c) [(d)] Quantum circuit diagrams for uni-[alt-]isoTNS, where isometric tensors have been expanded into unitary gates acting on up to 3 qubits by adding orthogonal columns. Gates act on the qubits that continue through the rectangle. Gates with the same number act on disjoint sets of qubits and thus can be applied simultaneously.