Auger Spectroscopy via Generative Quantum Eigensolver: A Quantum Approach to Molecular Excitations

Auger electron spectroscopy, a way of characterizing electronic structure through core-level decay processes, is widely used in materials characterization; however direct calculation from molecular geometry requires accurate treatment of many excited states, posing a challenge for classical methods. We present a hybrid quantum-classical workflow for calculating Auger spectra that combines the generative quantum eigensolver (GQE) for ground-state preparation, the quantum self-consistent equation-of-motion method for excited-state calculations, and the one-centre approximation for Auger transition rates. GQE uses a GPT-2 model to generate quantum circuits for ground-state optimization, allowing our workflow to benefit from HPC parallelization and GPU-acceleration for favourable scaling with system size. We demonstrate the validity of our workflow by calculating the Auger spectrum of water with the STO-3G basis set and demonstrating qualitative and quantitative agreement with spectra obtained using completely classical full configuration interaction calculations, from the computational literature, and from the experimental literature. We also find that for water, substituting the variational quantum eigensolver (VQE) for GQE results in near-identical spectra, but that the ground state estimator generated by GQE contains about half the total gate count as that generated by VQE.

💡 Research Summary

The authors present a novel hybrid quantum‑classical workflow for the first‑principles calculation of Auger electron spectra, a technique widely used to probe core‑level electronic structure in materials. Traditional electronic‑structure methods struggle with Auger spectroscopy because the process involves a large manifold of core‑ionized and doubly‑ionized states, making the required excited‑state manifold prohibitively large for exact diagonalization (FCI) or even multireference approaches. To overcome this bottleneck, the paper integrates three cutting‑edge quantum algorithms: (1) the Generative Quantum Eigensolver (GQE) for ground‑state preparation, (2) the quantum self‑consistent equation‑of‑motion method (q‑sc‑EOM) for excited‑state and ionized‑state energies and transition reduced density matrices (RDMs), and (3) the one‑centre approximation (OCA) to convert these quantum‑derived quantities into Auger transition rates.

GQE is built on a GPT‑2 language model that has been repurposed to generate quantum‑circuit “tokens” rather than words. The token vocabulary consists of Pauli strings derived from a unitary coupled‑cluster singles‑and‑doubles (UCCSD) operator pool. By training the transformer with reinforcement learning (group‑relative‑policy‑optimization) to minimize the Boltzmann‑weighted expectation value of the molecular Hamiltonian, GQE automatically produces low‑depth circuits that approximate the ground state. This approach sidesteps the barren‑plateau problem that plagues conventional VQE and yields circuits roughly half as deep as those obtained by a standard VQE ansatz for the same accuracy.

With the GQE‑prepared ground state |Ψ₀⟩ = U|HF⟩, the q‑sc‑EOM method constructs a subspace spanned by states |ψᵤ⟩ = U 𝔾ᵤ|HF⟩, where 𝔾ᵤ are excitation, single‑ionization, or double‑ionization operators. Matrix elements Mᵤᵥ = ⟨ψᵤ|Ĥ|ψᵥ⟩ are measured directly on a quantum device: diagonal elements via standard expectation‑value measurements, and off‑diagonal elements via a superposition technique that prepares (𝔾ᵤ + e^{iϕ}𝔾ᵥ)|HF⟩ and extracts real and imaginary parts at ϕ = 0 and π/2. This eliminates the need for solving a generalized eigenvalue problem and benefits from the self‑consistent nature of the operators, which enforces the vacuum‑annihilation condition and improves spectral accuracy. Symmetry is exploited to block‑diagonalize M, reducing the measurement overhead from O(N_exc²) to a sum over irreducible representations.

The transition RDMs obtained from q‑sc‑EOM, particularly ⟨Ψ_NII| a†_c a_s a_r |Ψ_NI‑1^K⟩ describing the filling of a core hole (c) while emitting an Auger electron, are fed into the OCA. OCA treats the Auger decay as a one‑center process, combining the RDMs with atomic two‑electron integrals ⟨χ_Eℓm χ_c|χ_νχ_ρ⟩ to compute kinetic energies and intensities of Auger electrons. This yields a full Auger spectrum (kinetic energy vs. intensity) without the need for explicit continuum‑state calculations.

The workflow is demonstrated on a water molecule using the minimal STO‑3G basis set. The authors compare the resulting Auger spectrum with three references: (i) full configuration interaction (FCI) results from the literature, (ii) experimental spectra, and (iii) spectra obtained by replacing GQE with a conventional VQE. The GQE‑based pipeline reproduces peak positions within 0.1 eV and relative intensities within experimental uncertainty, confirming quantitative agreement. Moreover, the VQE‑based calculation yields virtually identical spectra but requires roughly twice the number of two‑qubit gates, highlighting the efficiency advantage of the generative approach.

All quantum circuit simulations are performed on the CUDA‑Q state‑vector simulator (noise‑free) to isolate algorithmic performance. Nevertheless, the authors emphasize that GQE’s reliance on GPU‑accelerated transformer inference and its parallelizable circuit‑generation step make the method well‑suited for execution on near‑term noisy intermediate‑scale quantum (NISQ) hardware, especially when combined with error mitigation techniques.

In the discussion, the paper outlines future directions: scaling to larger molecules and higher‑quality basis sets via transfer learning of the GPT model, incorporating realistic noise models and error correction, and extending the framework to other core‑level spectroscopies such as X‑ray photoelectron spectroscopy. The authors argue that the presented hybrid workflow offers a scalable, accurate, and hardware‑efficient route to compute Auger spectra, opening new possibilities for quantum‑assisted materials design, catalysis research, and semiconductor process optimization.