Optimizing two-dimensional isometric tensor networks with quantum computers

We propose a hybrid quantum-classical algorithm for approximating the ground state of two-dimensional (2D) quantum systems using an isometric tensor network (isoTNS) ansatz. Inspired by the density matrix renormalization group, we optimize the isoTNS sequentially by diagonalizing a series of effective Hamiltonians. The latter are constructed with a quantum-computing algorithm that first represents the isoTNS as a quantum circuit and, then, applies a tomography-inspired method. The first step can be performed efficiently because the tensor network is composed of isometries, which can be naturally embedded within unitary operations that are then mapped to quantum gates. The classical cost of the second step remains manageable because the tomography is applied on a number of qubits that depends only on the bond dimension. By leveraging quantum computers, our approach does not rely on approximate contractions that are used to reduce the computational cost of classical 2D TNS methods. Moreover, it circumvents the exponential complexity of classical techniques through quantum computations. We demonstrate our method on the 2D transverse-field Ising model, achieving ground-state optimization on up to 25 qubits with modest quantum overhead (in terms of both circuit depth and number of shots) -- significantly less than solutions based on variational quantum eigensolvers. Overall, our results offer a path towards scalable variational quantum algorithms in both utility and fault-tolerant regimes.