The generative quantum eigensolver (GQE) and its application for ground state search

TL;DR

The paper introduces the generative quantum eigensolver (GQE), a non-VQA framework that uses a classical generative model to design quantum circuits for ground-state search. It implements a transformer-based GPT-QE to generate circuit sequences and optimizes them via a sampling scheme with adaptive inverse temperature, using replay buffers and two loss families (logit matching and GRPO), plus options for conditional inputs and pre-training. Empirical results on electronic-structure Hamiltonians for H$_2$, LiH, BeH$_2$, and N$_2$ show competitive or superior energy estimates relative to CCSD in key regimes and successful hardware demonstrations with error mitigation, alongside clear gains from pre-training. The approach offers a scalable, data-efficient alternative to VQAs, with potential for conditional generalization across geometries and molecules, and points to future integration with VQE-like hybrids and larger systems.

Abstract

We introduce the generative quantum eigensolver (GQE), a new quantum computational framework that operates outside the variational quantum algorithm paradigm by applying classical generative models to quantum simulation. The GQE algorithm optimizes a classical generative model to produce quantum circuits with desired properties. Here, we develop a transformer-based implementation, which we name the generative pre-trained transformer-based (GPT) quantum eigensolver (GPT-QE). We show a proof-of-concept of training and pretraining of GPT-QE applied to electronic structure Hamiltonians, and demonstrate its ability illustrated by surpassing coupled cluster singles and doubles (CCSD) for the strong bond dissociation of the nitrogen molecule and approaching chemical accuracy. We also demonstrate the method on real quantum hardware.

Figures (8)

• Figure 1: Comparison between GQE and VQE.
• Figure 2: Depiction of quantum circuit generation in GPT-QE (GQE, which employs a transformer). We also show the analogy between document generation in Large Language Models (LLMs) and quantum circuit generation in GPT-QE. The details of quantum circuit generation are described in Section \ref{['section:gptqe']}.
• Figure 3: Overview of the training and pre-training scheme in GPT-QE.
• Figure 4: Training results for each molecule showing energy vs. bond length plots with GQE results (green points), HF energy (gray dotted line), CCSD energy (blue dashed line), and FCI energy (black line). The number of tokens (circuit length) is set to 10 for $\texttt{H}_2$, 40 for $\texttt{LiH}$, 60 for $\texttt{BeH}_2$, and 100 for $\texttt{N}_2$. Note that there is a jump in the FCI calculation for $\texttt{N}_2$ around 0.93-0.94 Å, which is due to a change in the active space selection in the CAS calculation. This does not affect the objectives or conclusions of this experiment since it affect all approaches equally.
• Figure 5: Absolute error from FCI on a logarithmic scale for each molecule, with chemical accuracy threshold (blue dashed line) and HF/CCSD reference lines. The circuit complexity increases with molecular size, requiring longer sequences for more complex molecules. Note that for H$_2$ and LiH, CCSD is extremely close to FCI, so it is not plotted here. Additionally, the kink in N$_2$ around 1.75 Å is due to a sign change in the error.