By taking inspiration from the backflow transformation for correlated systems, we introduce a tensor network Ansatz which extends the well-established matrix product state representation of a quantum many-body wave function. This structure provides enough resources to ensure that states in dimensions larger than or equal to one obey an area law for entanglement. It can be efficiently manipulated to address the ground-state search problem by means of an optimization scheme which mixes tensor-network and variational Monte Carlo algorithms. We benchmark the Ansatz against spin models both in one and two dimensions, demonstrating high accuracy and precision. We finally employ our approach to study the challenging $S=1/2$ two-dimensional (2D) $J_1\text{−}{J}_2$ model, demonstrating that it is competitive with the state-of-the-art methods in 2D.

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• Matrix product states
• Quantum Monte Carlo
• Spin lattice models
• Tensor network methods
• Variational wave functional methods