Hybrid Quantum-Classical Eigensolver with Real-Space Sampling and Symmetric Subspace Measurements

TL;DR

This paper introduces a hybrid quantum-classical eigensolver that combines real-space sampling, symmetric subspace measurements, and tensor networks to efficiently simulate strongly correlated quantum systems, achieving high accuracy with low bond dimensions.

Contribution

The method innovatively integrates real-space sampling with symmetry-aware measurements and tensor networks to enhance scalability and accuracy in quantum many-body simulations.

Findings

Achieved energy error of 10^{-5} for 64-site chain and 6x6 torus.

Successfully incorporated symmetries to improve entanglement representation.

Demonstrated scalability with low bond-dimension tensor networks.

Abstract

We propose a hybrid quantum-classical eigensolver to address the computational challenges of simulating strongly correlated quantum many-body systems, where the exponential growth of the Hilbert space and extensive entanglement render classical methods intractable. Our approach combines real-space sampling of tensor-network-bridged quantum circuits with symmetric subspace measurements, effectively constraining the wavefunction within a substaintially reduced Hilbert space for efficient and scalable simulations of versatile target states. The system is partitioned into equal-sized subsystems, where quantum circuits capture local entanglement and tensor networks reconnect them to recover global correlations, thereby overcoming partition-induced limitations. Symmetric subspace measurements exploit point-group symmetries through a many-to-one mapping that aggregates equivalent real-space configurations into a single symmetric state, effectively enhancing real-space bipartition entanglement while elimilating redundant degrees of freedom. The tensor network further extends this connectivity across circuits, restoring global entanglement and correlation, while simultaneously enabling generative sampling for efficient optimization. As a proof of concept, we apply the method to the periodic $J_1\!-\!J_2$ antiferromagnetic Heisenberg model in one and two dimensions, incorporating translation, reflection, and inversion symmetries. With a small matrix product state bond dimension of up to 6, the method achieves an absolute energy error of $10^{-5}$ for a 64-site periodic chain and a $6\times6$ torus after bond-dimension extrapolation. These results validate the accuracy and efficiency of the hybrid eigensolver and demonstrate its strong potential for scalable quantum simulations of strongly correlated systems.