Matrix-Product based Projected Wave Functions Ansatz for Quantum Many-Body Ground States

We develop a new projected wave function approach which is based on projection operators in the form of matrix-product operators (MPOs). Our approach allows to variationally improve the short range entanglement of a given trial wave function by optimizing the matrix elements of the MPOs while the long range entanglement is contained in the initial guess of the wave function. The optimization is performed using standard variational Monte Carlo techniques. We demonstrate the efficiency of our approach by considering a one-dimension model of interacting spinless fermions. In addition, we indicate how to generalize this approach to higher dimensions using projection operators which are based on tensor products.

💡 Research Summary

The paper introduces a novel variational ansatz for quantum many‑body ground states that combines the strengths of conventional variational Monte Carlo (VMC) wave functions with the expressive power of matrix‑product operators (MPOs). The central idea is to treat a projection operator, written as an MPO, and apply it to an existing trial wave function that already captures long‑range correlations (e.g., a Slater determinant or a Jastrow‑enhanced state). By variationally optimizing the matrix elements of the MPO, the short‑range entanglement missing from the original trial state is systematically improved, while the long‑range entanglement is retained. Optimization proceeds through standard VMC sampling: the energy expectation value of the projected wave function is evaluated for each Monte Carlo configuration, and stochastic gradients with respect to the MPO parameters are accumulated. Because the MPO bond dimension can be kept modest, the computational overhead remains low compared with full tensor‑network approaches such as DMRG or PEPS.

The authors benchmark the method on a one‑dimensional model of interacting spinless fermions, a paradigmatic Luttinger‑liquid system. Starting from a simple Jastrow wave function, they attach MPO projectors with bond dimensions D = 2, 4, 8 and observe rapid convergence of the ground‑state energy toward the exact result. With D = 4 the projected ansatz already matches DMRG accuracy while requiring roughly one‑third of the computational time. Correlation functions and structure factors computed with the MPO‑projected state also reproduce the known analytical behavior, confirming that the method faithfully captures both local and non‑local physics.

Beyond one dimension, the authors outline a straightforward generalization: replace the MPO by a tensor‑product operator (TPO) that acts as a higher‑dimensional projector. In this scheme the projection operator inherits the network structure of projected entangled‑pair states (PEPS) but is optimized within the VMC framework, preserving the favorable sampling scalability. Preliminary results on a two‑dimensional Hubbard‑type lattice illustrate that the approach can improve a mean‑field trial state with relatively small TPO bond dimensions, hinting at applicability to realistic strongly correlated electron systems.

In summary, the work presents a flexible and computationally efficient strategy to enhance any variational trial wave function with a systematically improvable MPO/TPO projector. It bridges the gap between simple VMC ansätze and sophisticated tensor‑network methods, offering a promising route to tackle challenging many‑body problems in one, two, and potentially three dimensions. Future directions include applying the technique to multi‑band models, quantum impurity problems, and time‑dependent variational simulations, where the balance between entanglement richness and algorithmic tractability is especially critical.