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arXiv:2511.13827

Optimizing two-dimensional isometric tensor networks with quantum computers

Sebastian Leontica, Alberto Baiardi, Julian Schuhmacher, Francesco Tacchino, Ivano Tavernelli

quant-phcond-mat.str-elSubmitted Nov 17, 2025• Announced Nov 19, 2025• 0 citations
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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.

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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.
The variational quantum eigensolver (VQE) combines classical and quantum resources in order simulate classically intractable quantum states. Amongst other variables, successful VQE depends on the choice of variational ansatz for a problem Hamiltonian. We give a detailed description of a quantum circuit version of the 2D isometric tensor network (isoTNS) ansatz which we call qisoTNS. We benchmark the performance of qisoTNS on two different 2D spin 1/2 Hamiltonians. We find that the ansatz has several advantages. It is qubit efficient with the number of qubits allowing for access to some exponentially large bond-dimension tensors at polynomial quantum cost. In addition, the ansatz is robust to the barren plateau problem due emergent layerwise training. We further explore the effect of noise on the efficacy of the ansatz. Overall, we find that qisoTNS is a suitable variational ansatz for 2D Hamiltonians with local interactions.
Simulating the dynamics of finite two-dimensional (2D) quantum many-body systems poses challenges to numerical methods due to the exponential growth of the many-body Hilbert space. In this work, the authors explore the variational power of recently introduced isometric tensor network states (isoTNS) --- a subset of general tensor-network states that allow an efficient numerical simulation. As a efficient tool to find ground states, the variational density matrix renormalization group (DMRG2{}_{2}2​) is introduced. It is then demonstrated that dynamical correlation functions for the 2D transverse field Ising model and the Kitaev Honeycomb model can be obtained.
Tensor-network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm for approximating of the ground state of a Hamiltonian as an isometric TNS-which we demonstrate for the 2D transverse field Ising model.
Variational quantum algorithms are practical approaches to prepare ground states, but their potential for quantum advantage remains unclear. Here, we use differentiable 2D tensor networks (TN) to optimize parameterized quantum circuits that prepare the ground state of the transverse field Ising model (TFIM). Our method enables the preparation of states with high energy accuracy, even for large systems beyond 1D. We show that TN pre-optimization can mitigate the barren plateau issue by giving access to enhanced gradient zones that do not shrink exponentially with system size. We evaluate the classical simulation cost evaluating energies at these warm-starts, and identify regimes where quantum hardware offers better scaling than TN simulations.
Hybrid tensor networks offer a promising route to enhance the expressivity of classical tensor network methods by incorporating quantum states prepared on a quantum computer. Existing approaches are limited by the variational optimization of the quantum component of the tensor network. In this work, we introduce an alternative strategy that combines a non-parametric quantum state prepared through quantum annealing and a classical isometric tensor network. The latter is variationally optimized while the former is used as a fixed, boundary tensor resource in the form of classical shadows. We demonstrate the feasibility of this approach through extensive numerical simulations on the transverse-field Ising model, showing that the optimization procedure remains robust under statistical and hardware noise. Moreover, our results indicate that our newly proposed setup improves the accuracy of the obtained ground state approximation compared to the original quantum simulation, without increasing the depth of the applied quantum circuits. Therefore, this setup offers a practical route to scale variational quantum algorithms towards the quantum utility scale.
The variational quantum eigensolver (VQE) is a promising algorithm for demonstrating quantum advantage in the noisy intermediate-scale quantum (NISQ) era. However, optimizing VQE from random initial starting parameters is challenging due to a variety of issues including barren plateaus, optimization in the presence of noise, and slow convergence. While simulating quantum circuits classically is generically difficult, classical computing methods have been developed extensively, and powerful tools now exist to approximately simulate quantum circuits. This opens up various strategies that limit the amount of optimization that needs to be performed on quantum hardware. Here we present and benchmark an approach where we find good starting parameters for parameterized quantum circuits by classically simulating VQE by approximating the parameterized quantum circuit (PQC) as a matrix product state (MPS) with a limited bond dimension. Calling this approach the variational tensor network eigensolver (VTNE), we apply it to the 1D and 2D Fermi-Hubbard model with system sizes that use up to 32 qubits. We find that in 1D, VTNE can find parameters for PQC whose energy error is within 0.5% relative to the ground state. In 2D, the parameters that VTNE finds have significantly lower energy than their starting configurations, and we show that starting VQE from these parameters requires non-trivially fewer operations to come down to a given energy. The higher the bond dimension we use in VTNE, the less work needs to be done in VQE. By generating classically optimized parameters as the initialization for the quantum circuit one can alleviate many of the challenges that plague VQE on quantum computers.