Introduction

Technological advances heavily rely on the design of innovative functional materials, a task chiefly driven by understanding and optimizing inherent relationships between processing, structure, and property. While AI-driven materials exploration is on the verge of gaining popularity and practical utility, numerical simulations at various length-scales are meanwhile routinely employed to support these efforts, with methods ranging from continuum and phase field modeling at the macro- and meso-scale down to ab initio approaches at the atomic level1,2,3. For the latter, density functional theory (DFT) has emerged as the most popular technique due to its convenient accuracy at moderate computational cost4, despite the limitations arising from the use of approximated exchange-correlation functionals. For strongly correlated materials, however, semi-local DFT fails to correctly capture the underlying physics, thus calling for methods reaching beyond the mean field treatment of the electron interactions5.

An increasingly prevalent approach is to tackle such strongly correlated systems by extending DFT with an embedding scheme where a small subspace of correlated orbitals is treated with a many-body method6. Dynamical mean field theory (DMFT) is a common choice7, which is based on describing correlated orbitals within a lattice as impurities embedded in a self-consistent, time-dependent mean field, and on the assumption that the lattice self-energy is local. Constructing DMFT orbitals from DFT for the subspace of partially occupied d and f shells in transition metal and rare earth elements, respectively, significantly improves the description of the electronic structures of strongly correlated condensed matter systems8,9. Such DFT + DMFT calculations have been meanwhile applied to describe the physics of a wide range of materials, including superconducting cuprates10,11, nickelates10,11,[12](https://www.nature.com/articles/s41524-025-01772-6#ref-CR12 “Chen, H., Hampel, A., Karp, J., Lechermann, F. & Millis, A. J. Dynamical mean field studies of infinite layer nickelates: physics results and methodological implications. Front. Phys. 10, https://doi.org/10.3389/fphy.2022.835942

(2022).“), and other Perovskite-type materials13,14,15,16,17,18.

The computational complexity of DMFT itself depends on the method employed to solve the underlying Anderson impurity model (AIM). Exact diagonalization methods are only applicable to small systems9,19, since the Hamiltonian matrix scales exponentially with the number of orbitals. Alternatively, an exact solution, within statistical errors, can be computed using quantum Monte Carlo (QMC) methods by expressing the impurity problem in a Lagrangian formulation in imaginary time20. The limitations of its most popular flavor, the continuous time hybridization-expansion QMC (CTHYB)21,22,23, arise from the potential, likely mild Monte-Carlo sign problem (although not relevant for a one-band model) and the possibly slow convergence, particularly in the limit of low temperatures24. Further, to obtain any physically meaningful quantities, the measured Green’s function (GF) has to be analytically continued to the real frequency axis, a task that can be tedious and is, in general, ill conditioned25. Density matrix26 or numerical renormalization group27 have been also successfully employed and are well suited to accurately treat the AIM on the real frequency axis. In addition, approximate impurity solvers have been developed, such as Hubbard I28,29, which cannot provide arbitrary precision and often trade better scaling for accuracy.

Quantum computers (QC) offer the potential to significantly improve upon the above mentioned classical impurity solvers30,31,[32](#ref-CR32 “Rungger, I. et al. Dynamical mean field theory algorithm and experiment on quantum computers. Preprint at: https://doi.org/10.48550/arXiv.1910.04735

(2020).“),33,34,[35](#ref-CR35 “Francois, J. et al. Krylov variational quantum algorithm for first principles materials simulations. Preprint at https://doi.org/10.48550/arXiv.2105.13298

(2021).“),36,37,38 in terms of efficiency and accuracy. For instance, a QC solver could be particularly effective where both exact diagonalization (ED) and QMC algorithms struggle, namely in the domain of large system sizes and at low temperatures. Several avenues to obtain the AIM GF have been proposed in the literature, the majority of which can be categorized as based on either the Lehmann representation[32](https://www.nature.com/articles/s41524-025-01772-6#ref-CR32 “Rungger, I. et al. Dynamical mean field theory algorithm and experiment on quantum computers. Preprint at: https://doi.org/10.48550/arXiv.1910.04735

(2020).“),[39](https://www.nature.com/articles/s41524-025-01772-6#ref-CR39 “Jacopo, R. et al. One-particle Green’s functions from the quantum equation of motion algorithm. Phys. Rev. Res. 4, https://doi.org/10.1103/physrevresearch.4.043011

(2022).“),40, a continued fraction approximation[41](https://www.nature.com/articles/s41524-025-01772-6#ref-CR41 “Jamet, F. Agarwal, A. & Rungger, I. Quantum subspace expansion algorithm for Green’s functions. https://doi.org/10.48550/arXiv.2205.00094

(2022).“), or time-evolution31,33,42,43,44, all with their respective advantages and disadvantages. For instance, time-evolution in fermionic systems leads to deep circuits with large numbers of two-qubit operations, a challenge for current noisy QC which, up to now, could only yield accurate impurity GFs for two-site models with a single impurity site and one bath site33,45. The hybrid quantum-classical DMFT approach introduced by Sakurai et al.43 relies on a variational quantum simulation (VQS) to compute the imaginary-time GF with polynomial scaling. In their work, the authors successfully applied their method to compute the GF of a two-site and a four-site (one impurity with three bath sites) model on a QC emulator. Although the promising results presented by Sakurai et al. potentially offer better asymptotic scaling behavior compared to our near-term focused approach (see “Results” and “Methods” sections), it remains unclear how the VQS algorithm performs on real noisy quantum hardware since stability assessments were performed only in the presence of shot noise, assuming a fault-tolerant quantum device. On the other hand, the Lehmann representation or continued fraction approximation requires, in addition to the knowledge of the ground state (GS), the calculation of particle and hole excited eigenstates, a task that demands an appropriate algorithm to retain favorable scaling for any applications beyond simple systems40. Thus, an appropriate GS algorithm is a prerequisite, where most methods executable on noisy QC involve variational approaches46,47,48. On the other hand, excited states can be computed either with penalty function methods49,50, subspace search51, or the quantum equation of motion (qEOM)52.

In this work, we present our results on solving the DMFT for a real materials system on a quantum hardware using an automated hybrid quantum-classical atomistic modeling workflow. Based on the DFT + DMFT framework, we investigate the correlation effects in Ca2CuO2Cl2 (CCOC)53,54,55, a cuprate superconductor56,57 exhibiting physical properties that are believed to arise from a single correlated d band in the low-energy spectrum58,59. We map the electronic structure of CCOC to an effective single orbital Hubbard Hamiltonian, and extract the GF of a single-band single-site AIM with up to 6 bath sites based on the Lehmann representation. To this end, we employ the qEOM approach[39](https://www.nature.com/articles/s41524-025-01772-6#ref-CR39 “Jacopo, R. et al. One-particle Green’s functions from the quantum equation of motion algorithm. Phys. Rev. Res. 4, https://doi.org/10.1103/physrevresearch.4.043011

(2022).“),52 with truncated excitations up to second order to reduce computational cost, resulting in excellent scaling, and a hierarchy of mitigation schemes to alleviate the inherent errors of the quantum device. In particular, we introduce a novel algorithm to effectively suppress the variance arising from stochastic errors in zero-noise extrapolation schemes with equivalent or even lower computational cost, thereby reducing the average squared deviation from the state vector (SV) results by at least a factor of two. Using this scheme, we present QC results that show excellent agreement with classical ED reference calculations, and are able to correctly reproduce the renormalized single-particle spectrum from experimental angle-resolved photoemission spectroscopy (ARPES) of CCOC. While challenges remain to scale our approach to larger, multi-orbital and multi-site systems with more bath sites, the present work marks an important milestone towards achieving utility-scale quantum computation in materials simulation.

Results

We apply the DFT + DMFT formalism with our implementation of a QC impurity solver, which is described in detail in the “Methods” section, to study the electronic structure of a high-temperature superconductor (HTS). Our material system of choice is CCOC53,54,55, for which a maximal superconducting transition temperature of around 30 K has been observed in sodium-doped samples60. While this value is far below some of the other HTS cuprates, e.g., YBa2Cu3O761, CCOC represents a suitable model system that exhibits the main physics of an unconventional HTS. The characteristic structural motif of copper-oxide planes within a checkerboard lattice with squares composed of O2- and Cu2+ ions at the centers is shown in the inset of Fig. 1.

Fig. 1: The electronic structure of CCOC, where the DFT Kohn-Sham bands are shown in black (thin lines) and the interpolated Wannier band is shown in red (bold line).

The Fermi level is set to zero. The fat bands show, together with the associated colors, the band character based on the projection on the atomic Cu ({d}_{{x}^{2}-{y}{2}}) orbital. The inset shows a unit cell of CCOC along (001), with the large blue, small blue, and small red spheres representing the Ca, Cu, and O atoms, respectively (note that the Cl atoms are not depicted here). The positive and negative lobes of the maximally localized Wannier function are shown as red and green isosurfaces, respectively.

We start out by constructing a low-energy model from the converged DFT electronic structure of CCOC, and transform the relevant Bloch states to localized orbitals by Wannierizing the Cu ({d}_{{x}^{2}-{y}{2}}) state, resulting in an effective tight-binding model that reproduces well the non-interacting single-particle band crossing the Fermi level, as shown by the red interpolating line in Fig. 1. The screened interaction parameter from our constrained random phase approximation (cRPA) calculation of (U=\Re \left[U(\omega =0)\right]=3.2) eV is in good agreement with values for other, similar cuprate materials, e.g., of CaCuO2 in ref. 11. To obtain classical reference values and pre-converged impurity GFs for subsequent QC experiments, we first perform sufficient DMFT iterations with the CTHYB QMC impurity solver at finite temperatures to reach self-consistency.

The converged results from the CTHYB-DMFT cycles are then fed into our fitting procedure to construct a non-interacting model with up to 6 bath sites based on the hybridization function describing the impurity problem. We then proceed by performing a self-consistent DMFT update using the classical ED solver at 0 K with the identical frequency mesh as in CTHYB-DMFT, then iterate by again refitting the bath parameters. The chief reason why we continue the DMFT calculations at 0 K is due to the limitations of our quantum algorithm and, in general, current quantum devices that pose challenges to prepare the correct thermal state. In principle, a true DMFT calculation at vanishing temperatures would require an increasingly larger number of bath sites.

A small number of merely 10 iterations suffices to reach self-consistency. The output from the final ED iteration, namely the self-energy and bath parameters, serves as the input to our quantum algorithm for computing the impurity GF for the final iteration of the DMFT cycle, the detailed procedures of which will be discussed next. Note that in this paper, we compare the self-energy obtained using the qEOM algorithm to classical benchmarks using ED or CTHYB. We especially focus on assessing if the behavior around the Fermi level, which is the most important for DMFT convergence in imaginary frequencies, is well reproduced. However, we note that the remaining small deviations can influence the overall DMFT results slightly, as presented in ref. 40. In this work, we do not perform full self-consistency cycles of the DMFT loop on the quantum hardware due to the lack of an efficient GS algorithm, which scales at least polynomially in the number of qubits. Relevant benchmarks for performing full self-consistency cycles on the quantum hardware will be published elsewhere.

Ground-state calculations with VQE

Our converged ED-DMFT AIM parameters are used to construct an AIM Hamiltonian in the qubit representation to be used in quantum hardware experiments. The first step within our quantum impurity solver is to compute an accurate, high-quality GS, which we obtain using the variational quantum eigensolver (VQE) algorithm on the noiseless SV simulator46,47,48. To this end, we use a hardware-efficient ansatz (HEA) with linear entanglement strategy62,[63](#ref-CR63 “Polat, F., Tuncer, H., Moin, A. & Challenger, M. Model-driven engineering for quantum programming: a case study on ground state energy calculation. 2024 IEEE 48th Annual Computers, Software, and Applications Conference (COMPSAC), Osaka, Japan, 2353−2360, https://doi.org/10.1109/COMPSAC61105.2024.00378

(2024).“),64, and follow the procedure as described under “Ground State” in the “Methods” section to produce shallow quantum circuits that support the limited qubit connectivity on IBM Quantum hardware.

We quantify the quality of our GS by measuring the fidelity with respect to the exact GS from the ED results. To obtain an accurate GS, we optimize this figure of merit with respect to: ansatz architecture (number of layers and types of rotation and entangling gates), initial state circuit (zero or generalized Hartree–Fock (GHF)), initialization scheme for parameters θ (random or identity block65), and random seed (used to generate initial values of parameters θ)66, using a grid-search based global optimization64,66,67. This strategy explores diverse regions of the parameter space, improving the likelihood of finding a globally optimal solution with the highest fidelity with respect to the ED GS.

In total, we sample 12 candidate ansatz architectures, encompassing all possible combinations of the following parameters:

Number of layers = 4, 6, 8,

Rotation gates = RY, RY + RZ,

Entangling gates = CZ, CX.

For each architecture, we explore various combinations of initial state and initial parameters θ:

Zero initial state and random initial parameters θ,

GHF initial state and random initial parameters θ,

GHF initial state and four variants of the identity block method65 for initializing parameters θ:

Initialization close to 068 (adding a small random noise with an amplitude of 0.01),

Initialization close to π69 (adding a small random noise with an amplitude of 0.01),

The onion-initialization scheme64,

The inverse-initialization scheme, where the ansatz circuit with random initial parameters θ is inverted and appended to the original circuit, creating an identity block and doubling the number of layers.

Each set of initial variational parameters undergoes replication using 64 different random seeds. This results in 4608 combinations of parameter sets and initializations. The parameters of all these candidate circuits are variationally optimized on the SV simulator. Among these optimized circuits, we select an ideal GS that balances low circuit depth and high fidelity F67. We reach F = 0.959 with 1026 single-qubit and 164 two-qubit gates for the 5-bath system, and F = 0.989 with 252 single-qubit and 104 two-qubit gates for the 6-bath system (see Fig. S8 in the SI). These GS serve as the basis for the subsequent qEOM algorithm to compute the impurity GF on a quantum computer. While the VQE optimization itself is not carried out directly on the IBM Quantum hardware, we note that the complexity of the circuit structure is sufficiently low to be mapped onto a noisy quantum device. The quantum circuit, combined with the classically optimized parameters, is used to prepare the GS state and measure all qEOM operators, as described in the next sections. Performing the full VQE optimization on noisy quantum hardware to high precision would naturally represent a significant achievement, which however, lies beyond the scope of the present work.

Converged selection of the excitation order

To compute the impurity GF, we employ the qEOM method as described under “Quantum Equation of Motion” in the “Methods” section. Since it is challenging to a priori determine the excitation order required to achieve a desired precision for the GF of our AIM, we first carefully compare the results for singles and doubles Bogoliubov excitation operators (BEO) to determine the convergence behavior, using noiseless SV simulation for all EV. Figure 2a, b compares the impurity density of states (DOS) with 5 and 6 bath sites (12 and 14 qubits), respectively, taking into account single and double excitations using SV with the ED results. Here, and in all subsequent sections, we use the VQE state as described under “Ground-State Calculations with VQE” in the “Results” section to approximate the true GS. Note that the exact results from ED shows a rugged peak structure that stems from the bath discretization. With increasing number of bath sites, the GF would converge to a continuous function with a quasi particle peak in the vicinity of the Fermi level and two emerging lower and upper Hubbard peaks to its left and right, respectively.

Fig. 2: Impurity DOS for 5 and 6 bath sites using 12 and 14 qubits, respectively.

Subfigures (a), (b) show the results computed with the SV qEOM method using excitation operators singles, doubles, and exact diagonalization (ED) for 5 and 6 bath sites, respectively. Subfigures (c), (d) show the results computed with the qEOM method using excitation operators up to doubles with SV and on IBM hardware. The plots show the qEOM results with (red) and without (orange) ZNE calibration. The hardware experiments are conducted on “ibm_torino” with 8192 shots to evaluate the circuits, as well as with M3 (orange in (d)), M3+ZNE (orange in (c)), and M3+ZNEC (red in (c, d)) error mitigation as explained under “Error Mitigation” in the “Methods” section. In all plots, a small imaginary part of 0.1 is added in the denominators of the GF in Eq. (8).

We observe that singles and doubles together suffice to obtain a DOS in good agreement with the ED result, both for 5 and 6 bath sites. The remaining small difference between SV and ED arise from the infidelity of the VQE state and the missing higher order excitations, which would also be required to fully account for the Kondo effect. For other, smaller number of bath sites, we find the same behavior. Based on this observation, we elect to use singles and doubles for the calculation on noisy quantum hardware. The crucial finding that we can safely neglect all excitations exceeding doubles is the foundation for the favorable polynomial ({\mathcal{O}}({N}^{5})) scaling (further supporting details can be found in Sec. K of the SI).

Hardware results with ZNE-calibration

Next, we compute the impurity GF on the real QPU “ibm_torino”, using a hierarchy of error-mitigation techniques to improve the inherently noisy hardware results. To evaluate the circuits we set the number of shots to 8192, a value that produces sufficiently low statistical variances. As discussed in under “Error Mitigation” in the “Methods” section, we then employ at the first stage the M3 method to mitigate read-out errors.

The resulting DOS of the impurity GF using up to double excitations is shown by the orange lines in Fig. 2c, d for 5 and 6 bath sites, respectively, using conventional ZNE and M3 data from the real hardware. Note that for the 14 qubit experiment we do not show the ZNE results due to limited compute resources. A comparison with the SV DOS (blue, dashed line) shows that the ZNE DOS in Fig. 2c and the M3 DOS in Fig. 2d exhibit the overall qualitative features of the SV results, and in particular reproduce relatively accurate excitation energies. However, the QC ZNE results produce inaccurate DOS values close to the Fermi level for the 12 qubit experiment, while the M3 DOS for the 14 qubit experiment lacks the precision required to capture the correct overlaps, and the peak positions and amplitudes diverge significantly the further away we move from the Fermi energy.

To mitigate this issue, we employ our new error calibration scheme, ZNEC (including M3 mitigation), which we introduce under “ZNE-Calibration” in the “Methods” section. The resulting calibrated DOS is shown as a red, solid line in Fig. 2c, d for the 12 and 14 qubit experiment, respectively (QC M3+ZNEC). The improvement of the impurity DOS using ZNEC with respect to conventional M3 + ZNE is evident from Fig. 2c for the system with 5 bath sites, especially at the Fermi level. Compared to M3, the calibrated ZNEC in Fig. 2d effectively eliminates large portions of the remaining artifacts and corrects the majority of overlaps, in particular also those in the vicinity of the Fermi level, the most relevant portion in the energy spectrum. Residual discrepancies only remain at the peaks close to ω = 3 eV, an energy regime we deem not very pertinent for the subsequent steps in our workflow. In fact, the DMFT convergence is performed in imaginary frequencies, where only the portions of the DOS close to the Fermi level have a significant influence. Additionally, we are primarily interested in determining the quasi-particle weight, a quantity which is also evaluated at the Fermi level. The significantly improved agreement with the reference SV result demonstrates that it is of utmost importance to include our newly developed ZNEC error mitigation scheme to capture the correct physics close to the Fermi energy.

While our choice of the ZNEC calibration function is well motivated under “ZNE-Calibration” in the “Methods” section, further improvements of the error mitigation could be potentially achieved by:

adding auxiliary Pauli strings to the subset measured for the calibration. If these strings are chosen to commute with the existing measurements, no additional measurement effort is required while producing more data to train the regression model f(x).

improving the functional form of the regression model f(x) under the constraint that it remains monotonic in the interval [− 1, 1] and maps the values −1 and +1 to itself. In most of our calculations, the presented function f(x) is able to correct a large portion of the systematic error, but a more flexible functional form, e.g., using Gaussian process models with an appropriate, monotonic and noise-aware kernel, might further improve the results.

applying the same calibration scheme to other gate-error mitigation methods, e.g., for probabilistic error cancellation or amplification70.

Comparison of spectral properties

To obtain any physically meaningful quantities that can be compared to experimental measurements, the lattice GF has to be constructed from the impurity self-energy and the noninteracting single-particle dispersion. The imaginary part of this GF corresponds to the renormalized spectral function A(k, ω), which depends on the wavevector k and energy ω, a quantity that is in fact experimentally accessible through ARPES. However, before diving into a comparison of our results with ARPES data, we discuss the convergence behavior of our QC spectral properties with respect to classical reference calculations.

Figure 3a shows the spectral functions along a predefined path in reciprocal space Γ → X (where X corresponds to (π, π) in the 2D Brillouin zone of the CuO-plane), using the classical CTHYB and QC impurity solver executed on IBM Quantum hardware for the 6 bath sites in the left and right panel, respectively. The heat map corresponds to the spectral intensity A(k, ω), clearly showing the quasiparticle band crossing the Fermi energy at around 0.43× (π, π). Within an energy window of ±0.2 eV around the Fermi level, there is good agreement between the CTHYB and QC results. However, stronger deviations are observed at energies further away from the Fermi level. These discrepancies can be attributed, on one hand, to the different temperatures employed for the two calculations, but predominantly to the bath discretization required for the QC solver, an artifact that would disappear with a larger number of bath sites.

Fig. 3: A comparison of spectral properties.

Subfigure (a) shows the spectral function along Γ → X in the 2D Brillouin zone of the CuO-planes, with minimal and maximal values in light blue and yellow, respectively. The left panel is computed with the classical impurity solver CTHYB, while the right panel stems from our QC experiments with 14 qubits on “ibm_torino” with 6 bath sites. The inscribed dashed line stems from a linear fit to the quasi particle peaks at fixed energies from experimental ARPES data (ARPES), and the solid orange line (TB) shows the non-interacting single-particle band derived from the Wannierized tight-binding model. Subfigure (b) contains a comparison of the spectral functions at discrete points along the reciprocal path Γ → X. The first panel shows the experimental ARPES data from refs. 72,73 with a subtracted background using a Gaussian process regression model. The panels denoted by CTHYB, ED, and QC show the corresponding spectral functions using the QMC impurity solver CTHYB, the ED impurity solver for the discretized representation, and the QC algorithm executed on the “ibm_torino” quantum device, respectively. All panels also include the noninteracting single-particle δ-peaks from the TB model in orange.

To quantify the differences in the spectral properties, we compare the quasi particle weight (QPW)

$$Z={\left[1-\frac{\partial \Re \left[\Sigma (\omega )\right]}{\partial \omega }\right]}^{{-1}={\left[1-\frac{\partial \Im \left[\Sigma (i{\omega }_{n})\right]}{\partial {\omega }_{n}}\right]}}{-1}$$

(1)

at the Fermi level ω → 0, or in Matsubara frequencies i**ωn → 071. This quantity can be directly computed from the real or imaginary part of the self-energy Σ, which contains the information about the renormalization of the bare electronic dispersion, and is readily available within each DMFT cycle. We obtain values of ZCTHYB = 0.256, ZED = 0.271, and ZQC = 0.265 for the converged DMFT results using the CTHYB, ED, and QC impurity solver, respectively. While the value of ZED is slightly larger than ZCTHYB, the difference is within the range one would expect from the bath discretization. In fact, the self-energy obtained from the Lehmann representation contains spurious peak structures, which influence the numerical robustness when computing the derivatives in Eq. (1). We therefore eliminate and redistribute all peaks within ∣ω∣ < 0.1 as described under “Exact Diagonalization” in the “Methods” section. A justification of this procedure is presented in Sec. M of the SI. The resulting peak representation of the self-energy allows an analytic computation of the derivative in Eq. (1). To evaluate the value of Z with CTHYB, we compute the numerical derivative in the Matsubara frequencies. The relative error between ZED and ZQC is merely 2.2%, demonstrating the excellent agreement of our QC results with respect to the ED reference values.

Finally, we turn our attention to comparing the computational results to the experimentally measured ARPES spectra of CCOC available in the literature72,73. The first panel in Fig. 3b plots the single-electron ARPES spectra with subtracted background signals, showing the evolution of the quasi particle peak at various wavevectors (see also Sec. L of the SI). The second, third, and fourth panels contain the corresponding spectral lines computed from the converged DMFT results using the CTHYB, ED, and QC impurity solver with 6 bath sites, respectively, however, convolved with a Fermi-Dirac function around the Fermi level to eliminate any spurious signals from the unoccupied states. Again, the peak evolution within an energy window of −0.2 eV below the Fermi level is in good agreement between the ARPES and all computed spectra.

Since the self-energy cannot be directly extracted from the ARPES spectrum to quantify the agreement, we approximate the QPW using an alternate, approximate approach by fitting a line through the maxima along the energy as a function of the wavevector k close to the Fermi level (fit within ϵ ∈ [−0.4, 0] eV and k ∈ [0.25, 0.5] × (π, π), shown as a dash-dotted line in Fig. 3a). The ratio of its slope mARPES and the bare electron dispersion mTB, which we approximate with the TB band and is shown as a solid line in Fig. 3a, provides an estimate of the QPW, ({Z}_{{\rm{ARPES}}}=\frac{{m}_{{\rm{ARPES}}}}{{m}_{{\rm{TB}}}})74. Using this procedure, we obtain for the ARPES data a value of ZARPES = 0.274, in excellent