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arXiv:2505.08612 · Physical Review Applied

Logical quantum phase estimation for x-ray absorption spectra

Hirofumi Nishi, Taichi Kosugi, Satoshi Hirose, Tatsuya Okayama, Yu‐ichiro Matsushita

Published Feb 6, 2026• 1 citation
arXivHTMLPDF

Abstract

In this study, we employed Fourier-based quantum phase estimation (QPE) to calculate X-ray absorption spectroscopy (XAS) spectra. The primary focus of this study is the calculation of the XAS spectra of transition metal $L_{2,3}$-edges, which are dominated by strong correlation effects. First, the Fe $L_{2,3}$-edge X-ray absorption near-edge structure of FePO$_4$ is calculated using a noiseless simulator. The present computation involves a comparison of three types of input states: a uniform superposition state, optimal entangled input state, and Slater function state. Subsequently, we investigated the resolution error of the QPE and statistical error attributed to the measurements. It was revealed that post-processing to introduce Lorentzian broadening reduces the statistical error, which becomes a significant problem for a large number of qubits. Subsequently, we implemented QPE on a trapped-ion quantum computer, encompassing three orbitals within the active space. To this end, we implemented QPE using dynamic circuits to reduce ancilla qubits and [[k+2, k, 2]] quantum error detection code to mitigate the quantum noise inherent in current quantum computers. As a result, it was demonstrated that hardware noise was reduced, and spectra close to the noiseless ones were obtained.

DOI
10.1103/cwj3-xmsg

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In this study, we employed Fourier-based quantum phase estimation (QPE) to calculate X-ray absorption spectroscopy (XAS) spectra. The primary focus of this study is the calculation of the XAS spectra of transition metal L2,3L_{2,3}L2,3​-edges, which are dominated by strong correlation effects. First, the Fe L2,3L_{2,3}L2,3​-edge X-ray absorption near-edge structure of FePO4_44​ is calculated using a noiseless simulator. The present computation involves a comparison of three types of input states: a uniform superposition state, optimal entangled input state, and Slater function state. Subsequently, we investigated the resolution error of the QPE and statistical error attributed to the measurements. It was revealed that post-processing to introduce Lorentzian broadening reduces the statistical error, which becomes a significant problem for a large number of qubits. Subsequently, we implemented QPE on a trapped-ion quantum computer, encompassing three orbitals within the active space. To this end, we implemented QPE using dynamic circuits to reduce ancilla qubits and [[k+2, k, 2]] quantum error detection code to mitigate the quantum noise inherent in current quantum computers. As a result, it was demonstrated that hardware noise was reduced, and spectra close to the noiseless ones were obtained.
X-ray absorption spectroscopy is a crucial experimental technique for elucidating the mechanisms of structural degradation in battery materials. However, extracting information from the measured spectrum is challenging without high-quality simulations. In this work, we propose simulating near-edge X-ray absorption spectra as a promising application for quantum computing. It is attractive due to the ultralocal nature of X-ray absorption that significantly reduces the sizes of problems to be simulated, and because of the classical hardness of simulating spectra. We describe three quantum algorithms to compute the X-ray absorption spectrum and provide their asymptotic cost. One of these is a Monte-Carlo based time-domain algorithm, which is cost-friendly to early fault-tolerant quantum computers. We then apply the framework to an industrially relevant example, a CAS(22e,18o) active space for an O-Mn cluster in a Li-excess battery cathode, showing that practically useful simulations could be obtained with much fewer qubits and gates than ground-state energy estimation of the same material.
X-ray absorption spectroscopy (XAS) is a leading technique for understanding structural changes in advanced battery materials such as lithium-excess cathodes. However, extracting critical information like oxidation states from the experimental spectra requires expensive and time-consuming simulations. Building upon a recent proposal to simulate XAS using quantum computers, this work proposes a highly-optimized implementation of the time-domain algorithm for X-ray absorption. Among a host of improvements to Hamiltonian representation, circuit implementation, and measurement strategies, three optimizations are key to the efficiency of the algorithm. The first is the use of product formulas with the compressed double factorized form of the Hamiltonian. The second is recognizing that for spectroscopy applications, it is sufficient to control the error in the eigenvalues of the (approximate) Hamiltonian being implemented by the product formula, rather than the generic error on the full time evolution operator. Using perturbation theory to estimate this eigenvalue error, we find that significantly fewer Trotter steps are needed than expected from the time evolution error bound. The third is the choice of an optimized distribution of samples that takes advantage of the exponentially decaying Lorentzian kernel. Through constant factor resource estimates, we show that a challenging model Li4_44​Mn2_22​O cluster system with 18 spatial orbitals and 22 electrons in the active space can be simulated with 100 logical qubits and less than 4×1084 \times 10^84×108 T gates per circuit. Finally, the algorithm is implemented on a simulator, and the reconstructed spectrum is verified against a classical computational reference. The low cost of our algorithm makes it attractive to use on fault-tolerant quantum devices to accelerate the development and commercialization of high-capacity battery cathodes.
Error-corrected phase estimation averaged over variable grids on a trapped-ion quantum computer: hyperacuity spectra of a CO molecule adsorbed onto χχχ-Fe5_55​C2_22​
Quantum phase estimation (QPE) is an underlying technology for extracting the excitation spectra of many-electron systems, yet its practical use on current hardware is hindered by low grid resolution and environmental noises. Here we propose QPE averaged over variable grids (QAVG), a vernier-type approach that combines low-resolution QPE with multiple origin shifts and physically motivated continuous parametrization to reconstruct the spectra accurately. We introduce this approach into an end-to-end workflow for the {\it ab initio}-based model system for a CO molecule adsorbed onto the χχχ-Fe5_55​C2_22​ surface. We perform experiments on Quantinuum H2-2 using both physical QPE circuits and logical QPE circuits encoded in the Steane code with offline bit-flip correction. We demonstrate that QAVG accurately reconstructs the spectra with deviations much smaller than the nominal QPE resolution, even when the noisy histograms are used. The cost landscapes averaged over the shifted grids substantially suppress the local minima arising from the spectral leakage, thereby stabilizing the optimization of trial parameters. These results indicate that QAVG provides a robust route to quantum simulations of correlated spectra toward the era of early-fault-tolerant quantum computers.
Quantum phase estimation (QPE) is a key quantum algorithm, which has been widely studied as a method to perform chemistry and solid-state calculations on future fault-tolerant quantum computers.Recently, several authors have proposed statistical alternatives to QPE that have benefits on early faulttolerant devices, including shorter circuits and better suitability for error-mitigation techniques.However, experimental investigations of the algorithm on real quantum processors are lacking.Here, we implement statistical phase estimation on Rigetti's superconducting processors.Specifically, we use a modification of the Lin and Tong [PRX Quantum 3, 010318 (2022)] algorithm with the improved Fourier approximation of Wan et al. [Phys.Rev. Lett.129, 030503 (2022)] and apply a variational-compilation technique to reduce the circuit depth.We then incorporate error-mitigation strategies including zero-noise extrapolation and readout-error mitigation with bit-flip averaging.We propose a new method to estimate energies from the statistical phase estimation data, which is found to improve the accuracy in the final energy estimates by 1-2 orders of magnitude with respect to prior theoretical bounds, reducing the cost of performing accurate phase-estimation calculations.We apply these methods to chemistry problems for active spaces up to four electrons in four orbitals, including the application of a quantum embedding method, and use them to correctly estimate energies within chemical precision.Our work demonstrates that statistical phase estimation has a natural resilience to noise, particularly after mitigating coherent errors, and can achieve far higher accuracy than suggested by previous analysis, demonstrating its potential as a valuable quantum algorithm for early fault-tolerant devices.
Dynamical response functions are fundamental quantities to describe the excited-state properties in quantum many-body systems. Quantum algorithms have been proposed to evaluate these quantities by means of quantum phase estimation (QPE), where the energy spectra are directly extracted from the QPE measurement outcomes in the frequency domain. Accurate estimation of excitation energies and transition probabilities with these QPE-based approaches is, however, challenging because of the problem of spectral leakage (or peak broadening) which is inherent in the QPE algorithm. To overcome this issue, in this work we consider an extension of the QPE-based approach adopting the optimal entangled input states, which is known to achieve the Heisenberg-limited scaling for the estimation precision. We show that with this method the peaks in the calculated energy spectra are more localized than those calculated by the original QPE-based approaches, suggesting the mitigation of the spectral leakage problem. By analyzing the probability distribution with the entangled phase estimation, we propose a simple scheme to better estimate both the transition energies and the corresponding transition probabilities of the peaks of interest in the spectra. The validity of our prescription is demonstrated by numerical simulations in various quantum many-body problems: the spectral function of a simple electron-plasmon model in condensed-matter physics, the dipole transitions of the H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecule in quantum chemistry, and the electromagnetic transitions of the 6Li^{6}\mathrm{Li}6Li nucleus in nuclear physics.
High Resolution Image Download MS PowerPoint Slide Quantum phase estimation (QPE) is a promising quantum algorithm for obtaining molecular ground-state energies with chemical accuracy. However, its computational cost, dominated by the Hamiltonian 1-norm λ and the cost of the block encoding, scales at least quadratically with the number of molecular orbitals, making it challenging to incorporate dynamic correlation by enlarging the active space. In this work, we investigate two strategies to mitigate this cost through the optimization of the basis set. First, we investigate whether adjusting the coefficients of Gaussian basis functions can minimize the 1-norm while preserving the accuracy of the ground state energy. Although this method leads to a reduction in the 1-norm up to 10%, this reduction is system-dependent and diminishes with increasing molecular size. Second, we demonstrate that employing a large-basis-set frozen natural orbital (FNO) strategy results in a substantial reduction in QPE resources without compromising accuracy. We study a data set of 58 small organic molecules and the dissociation curve of N 2, and demonstrate that an active space constructed from orbitals derived from larger basis sets captures correlation effects more effectively. This approach yields up to an 80% reduction in the 1-norm λ and also leads to a 55% reduction in the number of orbitals. Our results highlight that improving the quality, not just the size, of the orbital basis is a viable strategy for extending QPE to include dynamical correlation, making progress toward scalable and chemically accurate quantum simulations with tractable resource requirements.