Quantum algorithm for simulating resonant inelastic X-ray scattering in battery materials

Abstract

Resonant inelastic X-ray scattering (RIXS) is the workhorse experimental technique for probing the structural degradation of higher-capacity cathode materials. However, the interpretation of experimental spectra is challenging due to the lack of accurate simulations. In this work, we propose a quantum algorithm for simulating the RIXS spectrum of molecular clusters hypothesized to form in Li-excess cathodes. The algorithm uses quantum phase estimation to sample the spectrum from a state encoding the scattering transition amplitudes of the cluster valence excitations. We prepare this state in the quantum computer using a block-encoding of the dipole operator and quantum signal processing to implement the Green's function propagator over intermediate core-excited states. To showcase the algorithm, we use a model cluster proposed in recent experimental works consisting of an oxygen dimer bonded to a manganese atom. Using the PennyLane software platform, we report resource estimation for simulating RIXS spectra for chemically motivated active spaces of increasing sizes. For a classically challenging active space with 20 orbitals, the algorithm requires $2.0 \times 10^{10}$ Toffoli gates and $414$ logical qubits.

Figures (12)

• Figure 1: A high-level description of the application pipeline for quantum simulation of RIXS spectra for structural analysis of lithium-excess cathode materials. First, experimental RIXS maps are built by measuring the intensity of scattered X-ray photons while scanning over the energies $\omega_I$ of the incoming radiation. For selected values of $\omega_I$ corresponding to resonances in the X-ray absorption spectrum (XAS), high-resolution RIXS spectra reveal the low-lying electronic excited states of the molecular clusters in the charged cathode. To interpret the spectra, we accurately simulate the RIXS cross-section of hypothesized structures -- such as oxygen dimers and clusters with transition metals -- using a quantum computer. The quantum algorithm prepares the initial state $\ket{\text{RIXS}(\omega_I)}$ and uses quantum phase estimation to sample the valence excited states of the cluster with probability given by the RIXS transition amplitude. The experimental spectra are compared with theoretical simulations of hypothesized clusters to fingerprint their RIXS features in the experimental spectra. This analysis provides a pathway to understand the causes of structural degradation in Li-excess cathodes. This is critical for informing new synthesis strategies to stabilize the structure of these materials to enable new battery cells with higher energy density.
• Figure 2: Sketch illustrating the two-step RIXS process in a molecular cluster. First, a photon with energy $\omega_I$ and polarization $\bm{\varepsilon}_I$ is annihilated, and a core electron is excited to a valence orbital ($\hat{c}_{v_1}^\dagger \hat{c}_c$). This interaction drives a resonant transition between the (many-body) ground $\ket{E_0}$ and intermediate states $\ket{E_n}$ of the electronic system. Then, a valence electron below the Fermi level (red dashed line) transitions to fill the core hole ($\hat{c}_c^\dagger\hat{c}_{v_2}$) leaving the cluster in a valence excited state $\ket{E_f}$, and the scattered photon with energy $\omega_s$ and polarization $\bm{\varepsilon}_S$ is created.
• Figure 3: Circuit for simulating the RIXS spectrum using walk-based QPE with a qubitized walk operator $\hat{\mathcal{W}}$qubitizationqromloaiza_mtd. The blue block indicates the circuit for preparing the initial RIXS state shown in \ref{['fig:aa']}. The $\mathcal{L}_\delta$ gate prepares a Kaiser lineshape vs_qsvt, which minimizes the error coming from the finite precision in QPE gqpeoptimum_qpe.
• Figure 4: Circuit for the block-encoding $\hat{\mathcal{U}}_R$ [Eq.\ref{['eq:U_R']}] to prepare the RIXS state [Eq.\ref{['eq:rixs_state']}] flagged by $\ket{1}_{succ}$. The unitary $\hat{U}_{\bm\varepsilon_I}$ prepares the dipole-perturbed state $\ket{D_{\bm\varepsilon_I}}$ [Eq.\ref{['eq:dipole_state']}]. $\textrm{GQSP}(\hat{\mathcal{W}},-K_G,K_G)$ implements a $K_G$th degree approximation to the Green's function [Eq.\ref{['eq:cheby_green']}] via the walk operator $\hat{\mathcal{W}}$ using generalized quantum signal processing, followed by a block-encoding of the dipole operator $\hat{D}^\dagger_{\bm\varepsilon_S}$. The $\ket{\cdot}_W$ register denotes ancillas used by the walk operator, while $\ket{\cdot}_D$ are the dipole block-encoding qubits, $\ket{\cdot}_{sys}$ are the qubits encoding the system's wavefunction, and $\ket{\cdot}_{\rm GQSP}$ is the additional qubit for GQSP rotations.
• Figure 5: Construction of Grover iterate $\hat{\mathcal{Q}}_R$ for amplitude estimation and amplification on block-encoding $\hat{\mathcal{U}}_R$ (\ref{['fig:state']}). The register $\ket{\cdot}_R$ collects all system and ancilla qubits (walk, GQSP, and dipole) excluding the success register, with the associated $X$ gate acting on all its qubits.