Optimizing two-dimensional isometric tensor networks with quantum computers

TL;DR

This work proposes a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits using an isometric tensor network ansatz, which maps naturally to quantum circuits.

Abstract

We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional quantum systems using an isometric tensor network ansatz, which maps naturally to quantum circuits. Inspired by the density matrix renormalization group, we optimize tensors sequentially by diagonalizing a series of effective Hamiltonians. These are constructed using a tomography-inspired method on a qubit subset whose size depends only on the bond dimension. Our approach leverages quantum computers to enable accurate solutions without relying on approximate contractions, circumventing the exponential complexity faced by classical techniques. We demonstrate our method on the two-dimensional (2D) transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead -- significantly less than standard solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both noisy and fault-tolerant regimes.

Figures (5)

• Figure 1: Illustration of a $5\times5$ isoTNS. Black nodes denote unitaries with incoming (outgoing) arrows for inputs (outputs). The isometric center $\ket{\lambda}$ is drawn as a red node, and a transparent red overlay marks the corresponding central column $\ket{\Lambda}$. The white nodes denote indices fixed to some known reference value (corresponding to the $\ket{0}$ state in the quantum circuit). Uncontracted output legs are associated with physical qubits. Hence, this isoTNS encodes the state of a $7\times 5$ array of spins.
• Figure 2: Result of optimizing a $5\times 5$, $g=3.5$ TFIM using the tomography method. We compare the optimization profile obtained using different numbers of shots per circuit to the result from exact tensor contraction. The number of samples is reported per tomography circuit executed during the effective Hamiltonian estimation phase. One sweep is composed of 15 optimization steps. The relative error is calculated with respect to the exact diagonalization result.
• Figure 3: Ground-state energy optimization profile obtained for the $g=3.5$ two-dimensional transverse field Ising model. The quoted lattice sizes represent the physical lattice size (the isoTNS rows are 2 sites shorter). Results obtained with the Lanczos method for different lattice sizes are reported as a function of total sample cost. The profile obtained for the $5\times 5$ full tomography approach is also shown for comparison.
• Figure 4: Representation of the circuit diagram associated to a node in the isoTNS. The red L-shaped block in the upper right corner represents a unitary transformation that embeds the isometry of the isoTNS associated with a site $(i, j)$ in the bulk of the physical lattice. Other unitary operations associated with nodes of the lattice are also marked with matching colors in the isoTNS diagram (bottom left) and the corresponding position in the physical lattice (bottom right). The blue tensor represents an isometry located along the central column. The yellow tensor represents an isometry located at the bottom edge. Note that the physical lattice is $7\times 5$, while the isoTNS is $5 \times 5$. This is because the tensors at the edge are associated with two physical indices (as discussed in the main text).
• Figure 5: Illustration of the circuits used in the tomography method for a $3\times 3$ isoTNS. The output is the state $\ket{\tilde{\Psi}}$ defined in the main text. The auxiliary space on which tomography is performed is marked with red qubit output lines. The remaining outputs of the initial Bell pair production step are the input of the isoTNS isometry $\mathcal{U}$ and subsequently mapped to a state in physical space (black output lines). The physical leg of the isometric center itself is decoupled from the other legs, so it is omitted here.