A rigorous hybridization of variational quantum eigensolver and classical neural network

TL;DR

This paper introduces U-VQNHE, a normalization-free hybrid quantum-classical method that improves the accuracy and robustness of variational quantum eigensolvers by addressing limitations of existing neural post-processing techniques.

Contribution

The paper develops U-VQNHE, a novel hybrid approach that guarantees variational safety and overcomes normalization issues in neural post-processing for VQE.

Findings

U-VQNHE outperforms traditional VQE and DNP methods in accuracy.

Numerical experiments show improved robustness on transverse-field Ising models.

The method reduces sampling costs compared to existing approaches.

Abstract

Neural post-processing has been proposed as a lightweight route to enhance variational quantum eigensolvers by learning how to reweight measurement outcomes. In this work, we identify three general desiderata for such data-driven neural post-processing -- (i) self-contained training without prior knowledge, (ii) polynomial resources, and (iii) variational consistency -- and show that current approaches, such as diagonal non-unitary post-processing (DNP), cannot satisfy these requirements simultaneously. The obstruction is intrinsic: with finite sampling, normalization becomes a statistical bottleneck, and support mismatch between numerator and denominator estimators can render the empirical objective ill-conditioned and even sub-variational. Moreover, to reproduce the ground state with constant-depth ansatzes or with linear-depth circuits forming unitary 2-designs, the required reweighting range (and hence the sampling cost) grows exponentially with the number of qubits. Motivated by this no-go result, we develop a normalization-free alternative, the unitary variational quantum-neural hybrid eigensolver (U-VQNHE). U-VQNHE retains the practical appeal of a learnable diagonal post-processing layer while guaranteeing variational safety, and numerical experiments on transverse-field Ising models demonstrate improved accuracy and robustness over both VQE and DNP-based variants.