Dynamical mean field theory for real materials on a quantum computer

Abstract

Quantum computers (QC) could harbor the potential to significantly advance materials simulations, particularly at the atomistic scale involving strongly correlated fermionic systems, where an accurate description of quantum many-body effects scales unfavorably with size. While a full-scale treatment of condensed matter systems with currently available noisy quantum computers remains elusive, quantum embedding schemes like dynamical mean-field theory (DMFT) allow the mapping of an effective, reduced subspace Hamiltonian to available devices to improve the accuracy of ab initio calculations such as density functional theory (DFT). Here, we report on the development of a hybrid quantum-classical DFT + DMFT simulation framework which relies on a quantum impurity solver based on the Lehmann representation of the impurity Green’s function. Hardware experiments with up to 14 qubits on the IBM Quantum system are conducted, using advanced error mitigation methods and a novel calibration scheme for an improved zero-noise extrapolation to effectively reduce adverse effects from inherent noise on current quantum devices. We showcase the utility of our quantum DFT + DMFT workflow by assessing the correlation effects on the electronic structure of a real material, Ca₂CuO₂Cl₂, which is mapped to an effective single-band Hubbard Hamiltonian and the subsequently derived Anderson impurity model solved with up to 6 bath sites on available quantum hardware. Further, we carefully benchmark our quantum results with respect to exact reference solutions and experimental spectroscopy measurements. 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.

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 level^1,2,3. 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 systems^8,9. Such DFT + DMFT calculations have been meanwhile applied to describe the physics of a wide range of materials, including superconducting cuprates^10,11, nickelates^10,11,12, and other Perovskite-type materials^{13,14,15,16,17,18}.

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 systems^9,19, 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 I^28,29, 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 solvers^{30,31,32,33,34,35,36,37,38} 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^32,39,40, a continued fraction approximation⁴¹, or time-evolution^{31,33,42,43,44}, 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 site^33,45. 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 approaches^46,47,48. On the other hand, excited states can be computed either with penalty function methods^49,50, 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 superconductor^56,57 exhibiting physical properties that are believed to arise from a single correlated d band in the low-energy spectrum^58,59. 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^39,52 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 CCOC^53,54,55, 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 O^2- 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 simulator^46,47,48. To this end, we use a hardware-efficient ansatz (HEA) with linear entanglement strategy^62,63,64, 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 optimization^64,66,67. 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 iω_n → 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 Z_CTHYB = 0.256, Z_ED = 0.271, and Z_QC = 0.265 for the converged DMFT results using the CTHYB, ED, and QC impurity solver, respectively. While the value of Z_ED is slightly larger than Z_CTHYB, 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 Z_ED and Z_QC 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 literature^72,73. 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 m_ARPES and the bare electron dispersion m_TB, 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 Z_ARPES = 0.274, in excellent agreement with our values for Z_CTHYB = 0.256, and the respective quantities from ED and QC. The residual discrepancies between the computed QPW and the ARPES values may be attributed to the inherent approximations involved in DMFT, such as the insufficient description of short-range correlations.

Discussion

In this work, we present a (charge) self-consistent DFT + DMFT simulation workflow that incorporates a QC impurity solver, and demonstrate its utility by investigating the electronic structure of a prototypical HTS material on noisy quantum hardware using up to 14 qubits. The quantum algorithm to solve the underlying AIM relies on the impurity GF in the Lehmann representation. To obtain the excited states required for its computation, we implement the qEOM algorithm and show that a truncation after the second excitation order suffices to accurately reproduce the impurity DOS from ED results. This approximation limits the computational complexity of our algorithm to ${\mathcal{O}}({N}^{5})$, a scaling behavior that would allow applications for up to a few dozen degrees of freedom, i.e., around 10–20 bath sites for a single site AIM. Note, however, that the quantum computation can be trivially parallelized over the qEOM matrix elements, a feature that we can readily exploit on next-generation quantum devices to further reduce the time to solution and hence would allow to further increase the degrees of freedom of the model.

Our experiments on IBM devices with up to 14 qubits implement the largest qEOM simulation on real QC hardware to the best of our knowledge. This achievement is only possible by addressing the inherent noise on current QC, which poses a significant challenge, as recognized in the literature⁴⁰. Even the deployment of conventional error mitigation schemes like M3 and ZNE is insufficient to obtain any meaningful results. To address this issue, we develop a novel mitigation strategy, ZNEC, which relies on training a calibration function on noisy expectation values and the corresponding gate-error-corrected counterparts to reduce systematic expectation value variances. In this work, we train an analytic calibration function that depends merely on a single fitting parameter, but more complex functional forms or an additional dependence on the circuit structure and properties of the measured observables are possible to improve the performance of the ZNEC error mitigation scheme (see also Sec. J of the SI).

Although existing classical impurity solvers currently still outperform (our) quantum approaches on both the problem presented here and on more complex ones, our results represent a significant step toward realizing utility-scale quantum computation in materials science. A key step to improve our workflow and enable the solution of more challenging state-of-the-art problems involves devising efficient and more scalable quantum algorithms for GS calculations. Variational VQE-type algorithms^46,47,48 suffer from slow convergence due to the potential presence of local minima, barren plateaus, and an overall ill-conditioned optimization behavior^48,75,76, and are thus only suited for small system sizes^46,62. While incremental progress is constantly proposed to improve the efficiency of variational approaches⁶⁴, more effective quantum algorithms like quantum phase estimation⁷⁷ and quantum imaginary time evolution (QITE)^78,79 remain impractical on noisy quantum devices due to their associated deep quantum circuits. Variational QITE⁸⁰ or Krylov subspace expansion⁸¹ might offer an affordable compromise between circuit complexity and attainable accuracy on near-term noisy quantum hardware.

Future efforts will be aimed at extending the present formalism to more complex Hamiltonians, in particular to describe multi-orbital systems with more bath sites as well as materials with several, symmetrically inequivalent lattice sites. For example, it will be interesting to study if single and double excitations also suffice for multi-orbital systems as well and analyze in which case higher order terms are required to reach some fixed accuracy for observables of interest. Further, the implementation of the cluster extension to DMFT⁷⁴ would improve the description of short-range correlations, a feature required to better model the intricate phase diagrams of, e.g., HTS cuprates or nickelates. However, such an assessment of the phase diagram additionally requires a quantum impurity solver at finite temperatures, e.g., based on the efficient mapping of thermal states using the variational quantum thermalizer⁶⁷.

In conclusion, the present work based on applying a scalable hybrid quantum-classical workflow to challenges beyond a simple two-site model marks an important milestone en route to quantum utility for materials simulations.

Methods

We study CCOC given the structural parameters as determined experimentally in ref. ⁵⁵, with Cu–Cu distances of 3.868 Å within the CuO-planes. The overall computational workflow to map this system to a problem that is solved within a hybrid quantum-classical DFT + DMFT cycle using a quantum impurity solver is shown in Fig. 4. In the following subsections, we describe in detail all components of the flowchart, thereby illustrating the steps involved in the relevant self-consistency cycles.

Fig. 4: Flowcharts illustrating the hybrid quantum-classical DFT + DMFT workflow.

Subfigure (a) shows the overall flowchart, which starts out with solving the DFT self-consistency cycles as shown in green. The red blocks indicate the steps required to parametrize the subspace Hamiltonian by constructing localized Wannier orbitals and computing the interaction onsite and exchange parameters U and J, respectively, as well as the hopping parameters $t,t^{\prime}$, which serves as an input to the DMFT self-consistency cycle shown in blue. The impurity solver is denoted by the purple box, while the steps required for a charge-self-consistent DFT + DMFT or any post-processing steps are included in the orange block. The detailed flowchart of the QC AIM solver is shown in subfigure (b).

DFT

To construct the effective model Hamiltonian, we start out by computing the single-particle Bloch energy bands using DFT as implemented in the Quantum ESPRESSO package⁸², which expands the wave function in a plane-wave basis. We employ the Perdew-Burke-Ernzerhof approximation to the exchange-correlation functional⁸³ and norm-conserving pseudopotentials⁸⁴. The Kohn–Sham (KS) equations are solved self-consistently to obtain converged KS energies ϵ_k and orbitals φ_k, as shown in the DFT (green) block in Fig. 4a. For CCOC, a plane-wave cutoff energy of 100 Ry together with a 12 × 12 × 12k-points mesh with a spacing smaller than 0.15/Å result in total energies that are converged to within 0.5 meV/atom (see details in Sec. A of the SI).

Wannierization and cRPA

The converged KS eigenstates are fed into a Wannierization framework to obtain localized orbitals, as shown in the red block in Fig. 4a. We construct a low-energy model for the subsequent DMFT calculations by Wannierizing the single Cu ${d}_{{x}^{2}-{y}^{2}}$ state crossing the Fermi level using the Wannier90 package⁸⁵, with a large disentangling window of 18 eV and the lower and upper bounds at −8 eV and 10 eV w.r.t. the Fermi energy, respectively, with a frozen range between −0.4 eV and 2.3 eV (see also Sec. C of the SI). The spatial extent of the resulting Wannier orbitals affects the hopping parameters t_i.

To obtain the effective, screened Coulomb interaction parameter U, we perform cRPA calculations as implemented in RESPACK⁸⁶. The exchange term J can be computed in a similar fashion, a parameter that is, however, not relevant for the one-band-one-site model that we investigate in this work. For this purpose, the Wannier orbital is converted with wan2respack⁸⁷ to a suitable format. For the cRPA calculation, we include 168 virtual orbitals in addition to the 32 occupied states, resulting in converged interaction parameters that we obtain from the static limit of the real part of the screened direct Coulomb integrals (see details in Sec. D of the SI).

DMFT

The resulting, parametrized Hubbard model from the preceding step is then treated within the self-consistent DMFT GF formalism with a single impurity site, as illustrated by the blue block in Fig. 4a. Note that this procedure thus leads to a single-band model that we treat with the DMFT approach. The DMFT calculations are performed with the software infrastructure provided by the Toolbox for Research on Interacting Quantum Systems (TRIQS)⁸⁸. A detailed derivation and explanation of the steps of DMFT, as well as a discussion of advantages and disadvantages of different impurity solvers, can be found in, e.g., refs. ^74,89,90.

In general terms, the DMFT self-consistency cycle starts out by constructing the local GF G_loc by integrating the lattice GF G_latt over k, where

$${G}_{{\rm{latt}}}(\omega ,k)=\frac{1}{\omega +\mu -\Sigma (\omega )+\epsilon (k)}$$

(2)

with the chemical potential μ and the noninteracting single-particle DFT dispersion ϵ(k). We use the Dyson equation (3) to define a noninteracting Weiss field G₀,

$$\Sigma (\omega )={G}_{0}^{-1}(\omega )-{G}_{{\rm{imp}}}^{-1}(\omega ).$$

(3)

An impurity solver is required to self-consistently compute the impurity GF G_imp, a task that can be either performed by a classical solver or, as discussed later under “Quantum Algorithms” in the “Methods” section, with a quantum algorithm. After computing the self-energy Σ in Eq. (3), the DMFT cycle repeats until convergence in G_imp is reached. The computationally expensive part of solving the impurity problem is shown in purple in Fig. 4a, while the details of the QC-based solver is shown in Fig. 4b.

Initially, classical reference results are obtained using the CTHYB QMC impurity solver²³ at a temperature of 386 K in a paramagnetic setting. The total number of QMC cycles for each DMFT iteration is set to 1 × 10⁸ with a cycle length of 400, resulting in an autocorrelation time of roughly 4. We perform a total of 25 DMFT iterations with the solid_dmft workflow manager^91,92, which conveniently incorporates DFT calculations with the DMFT toolbox of TRIQS. These pre-converged results subsequently serve as a starting point for self-consistent DMFT calculation with a discretized bath and an ED solver at 0 K with the same frequency mesh as in CTHYB.

In principle, the solid_dmft framework offers the possibility to feed a charge correction from the DMFT results back into the DFT cycle (e.g., in Quantum ESPRESSO) within an upfolding scheme, allowing a fully charge-self-consistent DFT + DMFT loop (shown by the orange box in Fig. 4a). In this work, however, we terminate the cycle after one full DFT + DMFT step, a procedure commonly referred to as one-shot (OS) DFT + DMFT. In many cases, OS DFT + DMFT alone already offers a good description of the many-body effects governing the physics of strongly correlated materials⁹², especially since full charge self-consistency is qualitatively merely of particular importance if DMFT leads to significant charge transfers for multiple correlated sites^11,93.

Mapping to Anderson impurity Hamiltonian

The following sections outline the methods involved to prepare and execute the quantum impurity solver within the DMFT cycle. The sequence of steps are illustrated in Fig. 4b, which will be referenced throughout.

Bath discretization

In contrast to QMC impurity solvers, methods based on ED or quantum computing work in a Hamiltonian formulation of the impurity problem and require a mapping to an AIM with a finite number of bath sites, i.e., qubits. In our algorithm, we perform this mapping (first block in Fig. 4b) by fitting a discrete, noninteracting model with N_b bath sites to the hybridization function describing the impurity problem:

$$\Delta (i{\omega }_{n})=i{\omega }_{n}-{G}_{0}^{-1}(i{\omega }_{n})-{\epsilon }_{0}.$$

(4)

Here, G₀ is the noninteracting impurity GF of our submodel and ϵ₀ is the effective atomic energy level obtained from the Wannier orbital. The functions depend on the discrete, imaginary, fermionic Matsubara frequencies iω_n. We use the discretization function from the TRIQS toolbox⁸⁸, which fits the parameters of an impurity problem with a bath where each site is directly connected to the impurity site but not to other bath sites (star topology, denoted by a bar, see Fig. 5). The hybridization of such a model is given by

$${\Delta }_{{\rm{disc}}}(i{\omega }_{n})=\mathop{\sum }\limits_{j=1}^{{N}_{b}}\frac{{\bar{{V}_{j}}}^{2}}{i{\omega }_{n}-\bar{{\epsilon }_{j}}},$$

(5)

where $\bar{{V}_{j}}$ are the hopping strengths between the impurity and bath site j, and $\bar{{\epsilon }_{j}}$ are the corresponding energy levels. We optimize the bath parameters with an additional 1/∣ω_n∣ weight to improve the fitting in the vicinity of the Fermi level, which is especially important for the DMFT convergence when N_b < 3,

$$\mathop{\min }\limits_{\{\bar{{\epsilon }_{j}}\},\{\bar{{V}_{j}}\}}\quad \sum _{{\omega }_{n}}\frac{| {\Delta }_{{\rm{disc}}}(i{\omega }_{n})-\Delta (i{\omega }_{n})| }{| {\omega }_{n}| }.$$

(6)

For our AIM derived for CCOC, we carefully examine the convergence behavior with respect to the number of bath sites, and conclude that N_b = 6 is more than sufficient to accurately reproduce the CTHYB hybridization function at the given finite temperature T = 386 K(see Sec. E in the SI).

Fig. 5: Mapping and topology of the AIM for 4 bath sites.

The star topology is shown in subfigure (a) and subfigure (b) shows the chain topology. Subfigure (c) shows the Jordan-Wigner mapping of spin-orbitals to qubits, where the order is based on qubit indexing, and subfigure (d) illustrates the same mapping with re-indexed qubits. The gray lines in c, d connect sites which are coupled in the Hamiltonian.

In order to obtain a model with a suitable topology for current noisy quantum devices, we perform a Lanczos tridiagonalization procedure, which produces an impurity model in a chain bath topology (see Fig. 5)⁹⁴. This model, with the impurity denoted with site index j = 0, is described by the Hamiltonian

$$\begin{array}{ll}\hat{H}=\,{\epsilon }_{0}\left({\hat{n}}_{0\uparrow }+{\hat{n}}_{0\downarrow }\right)+U{\hat{n}}_{0\uparrow }{\hat{n}}_{0\downarrow }\\ \quad\,+\mathop{\sum }\limits_{j=1}^{{N}_{b}}\mathop{\sum}\limits _{\sigma }\left[{\epsilon }_{j}{\hat{n}}_{j\sigma }+{V}_{j}\left({\hat{c}}_{j\sigma }^{\dagger }{\hat{c}}_{j-1\sigma }+h.c.\right)\right],\end{array}$$

(7)

and can be efficiently mapped to a linear chain of 2 × (N_b + 1) qubits by enumerating the orbitals in the following order: N_b↑, ⋯, 1↑, 0↑, 0↓, 1↓, ⋯, N_b↓. In this notation ${\hat{c}}_{j\sigma }^{\dagger }$, ${\hat{c}}_{j\sigma }$, and ${\hat{n}}_{j\sigma }$ are the creation, annihilation, and number operators on site j with spin σ, respectively, henceforth referred to as computational basis operators. Using the Jordan–Wigner transformation⁹⁵, the energy EV can be obtained by measuring the quantum circuit three times (independent of N_b), as all terms only involve a coupling of nearest neighbors. Therefore, all Pauli-strings include exclusively either Pauli X, Y, or Z operators. The sampling in the Pauli bases XX ⋯ XX, YY ⋯ YY and ZZ ⋯ ZZ thus suffices to construct the EV of all terms in the Hamiltonian and subsequently obtain the energy.

Exact diagonalization

Up to a small number of bath sites (N_b ≈ 7) one can obtain with reasonable numerical effort the zero-temperature GF by ED using the Lehmann representation:

$${G}_{{\rm{imp}}}(z)=\sum _{k > 0}\frac{| \left\langle k\right\vert {\hat{c}}_{0}^{\dagger }\left\vert 0\right\rangle {| }^{2}}{z+({E}_{0}-{E}_{k})}+\sum _{k > 0}\frac{| \left\langle k\right\vert {\hat{c}}_{0}\left\vert 0\right\rangle {| }^{2}}{z-({E}_{0}-{E}_{k})},$$

(8)

where the sum runs over all energy eigenstates $\left\vert k\right\rangle$ with their associated energies E_k, and k = 0 denotes the GS. Note that the ${\hat{c}}_{0}$ and ${\hat{c}}_{0}^{\dagger }$ are the annihilation and creation operators, respectively, at the impurity site. Depending on the purpose, we can choose the complex argument z of the impurity GF in different ways: (i) we select the Matsubara frequencies z = iω_n to continue DMFT iterations, or (ii) a real frequency with a small imaginary part z = ω + i0⁺ to obtain the final quantities of interest like the spectral function or the DOS. In contrast to the QMC methods operating on the imaginary time axis where the measured GF has to be analytically continued to the real frequency axis²⁵, this process is not required for the real frequency GFs obtained by the Lehmann representation, offering a significant advantage.

However, a challenge remains when the impurity GF G_imp from a discrete spectrum is used to calculate the electronic self-energy in the Dyson equation on the real axis in Eq. (3). The peaks in the inverse Weiss field ${G}_{0}^{-1}(\omega )$ need to be canceled by peaks in ${G}_{{\rm{imp}}}^{-1}(\omega )$, otherwise the imaginary part of the self-energy will be positive at some frequencies. This would lead to a negative, unphysical spectral function, which is given by −1/π times the imaginary part of the lattice GF of Eq. (2). Such spurious artifacts may arise not only due to the noise of current quantum devices, but already appear from the limited machine precision of classical computers.

To avoid this issue, we adapt the ideas of Lu et al. in ref. ⁹⁴ where the GF is stored as N poles (E_k0, λ_k), while E_k0 ≔ E_k − E₀ is the energy difference in the denominator of Eq. (8) and λ_k is the corresponding overlap in the numerator. The inverse of the GF can again be stored as a pole GF by diagonalizing an N × N matrix. In order to reduce the size of this matrix, one can eliminate some of the poles by distributing the overlap of small poles onto their neighbors. Ref. ⁹⁴ describes a method to perform this procedure that locally preserves the 0th and 1st moments, a method we also use in the evaluation of the Dyson equation in Eq. (3) to redistribute the peaks of ${G}_{0}^{-1}(\omega )$ with positive weights until only poles with negative weights remain. In our workflow, we use this pole-reduction purely as a post-processing step to compute, e.g., the spectral function or the QPW (see orange box in Fig. 4a).

Quantum algorithms

The following sections describe in detail the quantum algorithms within our workflow, as shown in Fig. 4b.

Ground state

After constructing the AIM Hamiltonian as described under “Bath Discretization” in the “Methods” section, preparing an accurate GS is a necessary prerequisite for the qEOM method under “Quantum Equation of Motion” in the “Methods” section to compute the impurity GF in the Lehmann representation (second block in Fig. 4b). Approximating the GS of a many-body Hamiltonian $\hat{H}$ is a central challenge in electronic structure calculations, but it is not the main focus of our work. We therefore employ the conventional VQE for this task^46,47,48. It is important to highlight, however, that our computational strategy does not depend strongly on this choice: indeed, thanks to its versatile and modular design, it can easily allow for the application of alternative and potentially more effective approaches to construct the GS.

VQE is a hybrid quantum-classical algorithm designed to find an approximate solution to the Schrödinger equation for a given Hamiltonian $\hat{H}$. It utilizes a parameterized quantum circuit to prepare an ansatz wavefunction $\left\vert \psi ({\boldsymbol{\theta }})\right\rangle$, where θ is a set of variational parameters. These parameters are iteratively adjusted using classical optimization techniques to minimize the energy EV of the Hamiltonian $\left\langle \psi ({\boldsymbol{\theta }})\right\vert \hat{H}\left\vert \psi ({\boldsymbol{\theta }})\right\rangle$ estimated on a quantum device. The iterative refinement continues until convergence, at which point the ansatz with the final converged parameters provides an approximation to the GS of the system.

Challenges in applying VQE for GS preparation include the selection of a valid ansatz and the initialization of the circuit parameters for the classical optimizer, the overcoming of barren plateaus and local minima in the energy landscape, and the mitigation of inherent noise on near-term quantum devices. The choice of ansatz and the number of variational parameters significantly impact the accuracy of the GS approximation in VQE. Balancing ansatz expressiveness with computational resource requirements is essential for a successful application of the VQE.

Our implementation utilizes a HEA with the linear entanglement structure^62,63,64 illustrated in Fig. S8 of the SI. This ansatz choice shows a good circuit expressibility for our Hamiltonians while featuring shallow circuit depths and supporting the limited qubit connectivity on IBM Quantum hardware. One can also use a symmetry-preserving ansatz⁹⁶, but they will generally have a larger depth than a HEA. Since the VQE preparation is not the main focus of the paper, we implement a VQE state on the IBM hardware that has a small amount of transpiled gates, while also having a high overlap with the exact GS. The ansatz circuit is complemented by an initial state circuit for preparing an initial state for the classical optimization. The initial state circuit can be either an empty circuit for preparing a zero initial state⁶⁴, or a Slater determinant (SD) circuit⁹⁷ for preparing the mean-field (MF) state of the AIM Hamiltonian. The MF state is obtained by solving the AIM Hamiltonian in a GHF approximation using the PySCF software package⁹⁸. The molecular orbital (MO) coefficients from the GHF solution are then used to parameterize the initial state circuit that prepares a SD following the method of Jiang et al.⁹⁷. The GHF optimization is performed using the number of particles determined by ED of the AIM Hamiltonian. We observe that for GSs with an odd number of particles, the GHF state has higher fidelity compared to the unrestricted Hartree–Fock (UHF) state, suggesting that GHF better captures the multi-reference character of the GS than UHF.

The VQE optimization is performed using the limited-memory Broyden-Fletcher-Goldfarb-Shanno bound optimizer (L_BFGS_B)^99,100, which performs well for our optimization problems in the absence of noise and with access to gradients^47,64,66,101, which we compute through finite differences, employing Qiskit’s default convergence criteria (2.22 × 10⁻¹⁵ as a relative tolerance for termination and 15,000 as a maximum number of iterations). For variational parameter initialization, we adopt random initialization as well as several different implementations of the identity block method⁶⁵. When combined with the GHF initial state, the identity block approach ensures that the optimization starts closer to the optimal point in the parameter space compared to random initialization. This, in turn, reduces the probability of encountering barren plateaus during optimization. Since the focus of this work is mainly on ultimately computing the impurity GF on a quantum computer, we do not perform the VQE calculations on the IBM Quantum hardware, but rather on a noiseless SV simulator.

Quantum equation of motion

We employ the qEOM method to compute the excited states required for the Lehmann representation of the impurity GF. The qEOM was first introduced in ref. ⁵² and was initially applied to compute molecular excitation energies. The method was subsequently used to calculate the GF of a 2-site Hubbard model in ref. ³⁹ and it has seen recent extensions and applications to industrial use-cases by Asthana et al.¹⁰². Below we describe the qEOM method in more detail.

In general, any excited state ${\left\vert \psi \right\rangle }_{k}$ may be expressed as an operator ${\hat{O}}_{k}$ applied to the GS $\left\vert 0\right\rangle$ with N particles

$${\left\vert \psi \right\rangle }_{k}={\hat{O}}_{k}\left\vert 0\right\rangle .$$

(9)

In the qEOM method, one imposes the extra so-called annihilation condition on the operator ${\hat{O}}_{k}$

$${\hat{O}}_{k}^{\dagger }\left\vert 0\right\rangle =0.$$

(10)

The time-independent Schrödinger equation for ${\left\vert \psi \right\rangle }_{k}$ can be re-expressed as

$${[\hat{H},{\hat{O}}_{k}]}_{-}\left\vert 0\right\rangle ={E}_{k0}{\hat{O}}_{k}\left\vert 0\right\rangle ,\quad {[\hat{A},\hat{B}]}_{\pm }:= \hat{A}\hat{B}\pm \hat{B}\hat{A},$$

(11)

where E_k0 = E_k − E₀ corresponds to the excitation energy. Operating on both sides of Eq. (11) with the state $\left\langle 0\right\vert {\hat{O}}_{k}^{\dagger }$ and using the annihilation condition (10) gives

$$\left\langle 0\right\vert {[{\hat{O}}^{\dagger },{[\hat{H},{\hat{O}}_{k}]}_{-}]}_{+}\left\vert 0\right\rangle ={E}_{k0}\left\langle 0\right\vert {[{\hat{O}}_{k}^{\dagger },{\hat{O}}_{k}]}_{+}\left\vert 0\right\rangle .$$

(12)

Finally, adding to the left and right hand side of Eq. (12) their corresponding hermitian conjugate leads to the following equation (note that Eq. (13) also follows from applying the annihilation condition to Eq. (12) once again):

$$\left\langle 0\right\vert {[{\hat{O}}_{k}^{\dagger },\hat{H},{\hat{O}}_{k}]}_{+}\left\vert 0\right\rangle ={E}_{k0}\left\langle 0\right\vert {[{\hat{O}}_{k}^{\dagger },{\hat{O}}_{k}]}_{+}\left\vert 0\right\rangle ,$$

(13)

where we used the definition of the double-commutator:

$${[\hat{A},\hat{B},\hat{C}]}_{+}:= \frac{1}{2}\left({[\hat{A},{[\hat{B},\hat{C}]}_{-}]}_{+}+{[{[\hat{A},\hat{B}]}_{-},\hat{C}]}_{+}\right).$$

(14)

The anti-commutator in Eq. (13) typically leads to cancellations in the operators (see also Sec. H in the SI), such that the amount of operator terms is reduced compared to another similar method to compute eigenstates, called quantum subspace expansion¹⁰³.

The set $\{{\hat{O}}_{k}\}$ can be written in terms of a set of fermionic operators:

$${\hat{O}}_{k}=\sum _{j}{({X}_{k})}_{j}{\hat{R}}_{j},$$

(15)

where the operators $\{{\hat{R}}_{j}\}$ are any basis of excitation operators and themselves a sum of products of annihilation and creation operators ${\hat{c}}_{k},{\hat{c}}_{l}^{\dagger }$ of the fermions in the system (the index j does not necessarily indicate a j-body term. In fact, each ${\hat{R}}_{j}$ can in general be a mixed sum of single-, two-, and higher-body terms, such that the set $\{{\hat{R}}_{j}\}$ forms a basis of excitation operators). A generalized eigenvalue problem (GEP) for the coefficients ${({X}_{k})}_{j}$ can now be derived by solving for E_k0 and imposing the stationary condition $\frac{\partial {E}_{k0}}{\partial {({X}_{k}^{\dagger })}_{j}}=0$ in Eq. (13),

$${\bf{A}}{{\bf{X}}}_{k}={E}_{k0}{\bf{B}}{{\bf{X}}}_{k},$$

(16)

$${A}_{ij}=\left\langle 0\right\vert {[{\hat{R}}_{i}^{\dagger },\hat{H},{\hat{R}}_{j}]}_{+}\left\vert 0\right\rangle ,\quad {B}_{ij}=\left\langle 0\right\vert {[{\hat{R}}_{i}^{\dagger },{\hat{R}}_{j}]}_{+}\left\vert 0\right\rangle .$$

(17)

The matrix elements of A, B correspond to EV of the operators and can be evaluated on a QC.

For the GF, we are interested in the particle and hole states with N + 1 and N − 1 particles, respectively (first and second sum in Eq. (8)). This corresponds to the ionization potential (IP-EOM) and electron affinity equation-of-motion methods, respectively, that are well established for calculating excitation energies to ionized states with coupled-cluster ansatz using classical computation^104,105,106. After the entries of A and B are calculated on the QC, we solve the GEP (16) classically to find the coefficients ${({X}_{k})}_{j}$ and the excitation energies E_k0 needed for the Lehmann representation of the GF. The transition amplitudes in the numerators of the impurity GF in Eq. (8) are expressed in terms of EV w.r.t. the prepared GS that can be computed on the QC,

$$\frac{\left\langle k\right\vert {\hat{c}}_{0}\left\vert 0\right\rangle }{\sqrt{\left\langle k\right\vert k\left.\right\rangle }}=\frac{\left\langle 0\right\vert {\hat{O}}_{k}^{\dagger }{\hat{c}}_{0}\left\vert 0\right\rangle }{\sqrt{\left\langle 0\right\vert {\hat{O}}_{k}^{\dagger }{\hat{O}}_{k}\left\vert 0\right\rangle }}=\frac{{({X}_{k})}^{l}\left\langle 0\right\vert {\hat{R}}_{l}^{\dagger }{\hat{c}}_{0}\left\vert 0\right\rangle }{\sqrt{{({X}_{k})}^{i}{({X}_{k})}^{j}\left\langle 0\right\vert {\hat{R}}_{i}^{\dagger }{\hat{R}}_{j}\left\vert 0\right\rangle }},$$

(18)

with the Einstein summation convention used above for the repeating upper and lower indices. A graphical representation of the workflow for the impurity GF is shown in Fig. 4b. Further details on our implementation of the qEOM method can be found in Sec. H of the SI.

Excitation operators

The choice of excitation operators ${\hat{R}}_{j}$ in Eq. (15) significantly affects the solution of the GEP (16) and thereby the precision of the resulting GF, whenever one restricts the basis to a subset (of the complete basis of particle and hole operators). A particular choice is based on subdividing them into excitation orders singles (s), doubles (d), triples (t), etc., and take the following general form for charged excitations of particle states

$${\hat{R}}_{i}^{(s)}=\sum _{j}{\alpha }_{ij}{\hat{c}}_{j}^{\dagger },\quad {\hat{R}}_{i}^{(d)}=\sum _{j,k,l}{\alpha }_{ijkl}{\hat{c}}_{j}^{\dagger }{\hat{c}}_{k}^{\dagger }{\hat{c}}_{l},\ldots ,$$

(19)

with the same so-called computational basis operators $\hat{c},{\hat{c}}^{\dagger }$ that enter the AIM Hamiltonian in Eq. (7). The full set $\{{\hat{R}}_{j}\}$ therefore equals $\{\{{\hat{R}}_{{i}_{s}}^{(s)}\},\{{\hat{R}}_{{i}_{d}}^{(d)}\},\ldots \}$, where j runs over all basis elements and i_s, i_d over the singles and doubles, respectively.

In this work, we use a GHF MF solution of the AIM Hamiltonian to derive our basis of excitation operators. The GHF SD from PySCF⁹⁸ with N particles that approximates the exact GS of the AIM is then expressed as

$${\left\vert 0\right\rangle }_{{\rm{GHF}}}={\hat{b}}_{1}^{\dagger }\cdots {\hat{b}}_{N}^{\dagger }\left\vert {\rm{vac}}\right\rangle ,\quad {\hat{b}}_{i}^{\dagger }=\sum _{k}{W}_{ij}{\hat{c}}_{j}^{\dagger },$$

(20)

where W_ij are the MO coefficients obtained from the GHF calculation. The operators ${\hat{b}}_{i}$ are typically called Bogoliubov operators and diagonalize the corresponding MF Hamiltonian ${H}_{{\rm{MF}}}=\sum _{i > 0}{\epsilon }_{i}{\hat{b}}_{i}^{\dagger }{\hat{b}}_{i}$, with ϵ_i ≤ ϵ_i+1 for all i.

The SDs that approximate the particle and hole states of the AIM are similarly subdivided into singles (s), doubles (d), etc. and are defined by acting on the GHF GS with the appropriate amount of creation ${\hat{b}}_{i}^{\dagger }$ and annihilation ${\hat{b}}_{i}$ Bogoliubov operators for empty (i > N) and occupied (i ≤ N) orbitals, respectively, of the GHF GS. Inspired by this, we define our set of BEO as

$${\hat{R}}_{i}^{{\rm{Bog}},(s)}={\hat{b}}_{i}^{\dagger },\quad {\hat{R}}_{ijk}^{{\rm{Bog}},(d)}={\hat{b}}_{i}^{\dagger }{\hat{b}}_{j}^{\dagger }{\hat{b}}_{k},\ldots ,$$

(21)

where for each BEO, the indices of all creation operators are either all simultaneously virtual (i > N) or all simultaneously occupied (i ≤ N) orbital indices of the GHF GS. The indices of the annihilation operators in a BEO in Eq. (21) are all occupied orbital indices when the creation operators are virtual orbital indices and vice versa. Plugging in Eq. (20) for the operators ${\hat{b}}_{i}$ appearing in Eq. (21), one finds that the BEO are a specific choice for the basis in Eq. (19).

We note that the set of BEOs in Eq. (21) forms a basis for the excitation operators ${\hat{O}}_{k}$ that results in particle states when we apply ${\hat{O}}_{k}$ on the GS, while it also forms a basis for operators ${\hat{O}}_{k}$ that give the hole states when applying their adjoint, ${\hat{O}}_{k}^{\dagger }$, to the GS. The BEOs therefore form an independent set of fermionic operators that make up a complete basis set for deriving all charged particle and hole eigenstates together in the qEOM method whenever the maximal excitation order is taken (see also Sec. H of the SI). In Sec. I of the SI, we show a comparison of the above BEO with another excitation operator basis choice, by computing the impurity DOS, when the GS is approximated by a VQE state.

Error mitigation

We employ a hierarchy of error-mitigation techniques to obtain accurate results from noisy quantum computing experiments, which constitutes the 5th step shown in the flowchart of Fig. 4b.

Readout errors

First, we employ mitigation schemes to reduce read-out errors (readout error mitigation, REM). A method proposed in ref. ¹⁰⁷ suggests the construction of a readout matrix to model the readout errors. However, constructing such a 2ⁿ × 2ⁿ matrix for an n-qubit simulation would demand substantial computational resources as the size of the simulation increases, making this approach unsuitable for practical applications.

In this work, we employ the Matrix-free Measurement Mitigation (M3) method¹⁰⁸ to mitigate noisy computational results when estimating the EV of observables. With this approach, we can leverage the qubit-wise commutation rules¹⁰⁹ which reduce the number of Pauli string measurements (see also Sec. H of the SI). The M3 method works on a subspace of the entire readout matrix defined by the unique noisy bit strings to be corrected. The matrix elements are approximately computed using the single-qubit calibration data. The number of those unique bit strings are typically smaller than the total 2ⁿ bit strings, a behavior which is particularly advantageous for mitigating readout errors of simulations for large systems.

Gate operation errors

Second, we utilize the zero-noise extrapolation (ZNE) technique^110,111 to mitigate other sources of errors, such as gate operation errors (gate error mitigation, GEM). The ZNE method estimates the noiseless EV of observables by extrapolating the measured values at different noise levels to the zero-noise limit. The noise is systematically amplified by inserting additional gates or by stretching the duration of the microwave pulses. In this work, we amplify the noise by adding additional CNOT gates, leveraging the fact that applying (2n + 1) CNOT gates to the same qubit pair produces the same outcome as a single CNOT, but with larger noise. We carefully investigate the error mitigation using the ZNE method, and find that a linear extrapolation with scale factors of n = 1.0, 1.5, and 2.0 is most suitable for our 14 qubit system, while for the 12 qubit system we use quadratic extrapolation at noise factors n = 1.0, 2.0, and 3.0. The ZNE prototype code is used for our ZNE error mitigation¹¹².

The M3 method plays an important role for small-scale problems where gate noises from the device are relatively low and the readout error is dominant, while ZNE is useful for larger simulations with a high number of gate operations. In this work, we find that combining the two methods is essential to obtain meaningful EV for our observables. However, even the combination of the two mitigation schemes (M3 and ZNE) is still not enough to provide a sufficiently accurate EV. To improve on that, in the next section, we develop an additional mitigation scheme that allows to achieve results in qualitative agreement with the reference calculations.

ZNE-calibration

In this section, we introduce a new error mitigation strategy, which sits on top of the standard ZNE (see “Error Mitigation” in the “Methods” section) and that we name ZNE-calibration (ZNEC). The main idea of this approach is to calibrate a function which maps results without gate-error mitigation (raw data or REM results) to the mitigated results (GEM) using ZNE data. Compared to standard ZNE, this procedure has the advantage of reducing the variance of the mitigated results while correcting the systematic error present in the raw as well as REM results (see Fig. 6a). In contrast to methods like probabilistic error cancellation or amplification (PEC¹¹³ or PEA⁷⁰), ZNEC has a similar or even reduced overhead compared to standard ZNE. For any quantum algorithm that requires measuring a large set of Pauli strings $\hat{{P}_{i}}$ for a quantum state, we perform the following steps:

• 1. Choose a subset of calibration Pauli strings $\hat{{C}_{j}}$ of the complete set $\hat{{P}_{i}}$. In our calculations we choose the 100 largest groups of commuting Pauli strings in order to reduce the amount of measurements while simultaneously taking the most data points for ZNEC calibration.

• 2. Measure the Pauli strings $\hat{{C}_{j}}$ using ZNE for different noise factors. This will give the raw/REM EV $\bar{{C}_{j}}$ at noise factor c = 1 and the EV $\tilde{{C}_{j}}$ at noise factor c = 0 using an extrapolation method. Additionally, measure all remaining Pauli strings of $\hat{{P}_{i}}$ without using ZNE to obtain their raw/REM EV $\bar{{P}_{i}}$.

• 3. Fit the parameter α of the function $f(x)=\frac{2}{\pi }{\tan }^{-1}\left(\alpha \left(\tan \left(\frac{\pi }{2}x\right)\right)\right)$ to map all raw/REM results to the GEM results $\tilde{{C}_{j}}\approx f(\bar{{C}_{j}})$. The idea here is to fit only one value α for all Pauli strings simultaneously (see Fig. 6b). This function corrects the damping of the EV due to noise of the hardware, which manifests in a slope smaller than 1 in Fig. 6a. We choose this particular form of the function since it has a sigmoidal shape, odd symmetry, and maps the two possible eigenvalues of each Pauli operator (−1 and 1) onto itself.

• 4. Apply the calibrated function f(x) to the raw/REM results to obtain the final EV ${P}_{i}=f(\bar{{P}_{i}})$.

Fig. 6: Demonstration of the ZNE-calibration technique for our 14 qubit experiment performed on the “ibm_torino” quantum device.

Subfigure (a) shows the different set of expectation values obtained using only the readout error mitigation M3 (orange, $\bar{{C}_{j}}$), using M3 and ZNE (blue, $\tilde{{C}_{j}}$), and using the ZNE-calibration technique (ZNEC, green, ${C}_{j}=f(\bar{{C}_{j}})$). The y value of each cross represents the expectation value obtained on “ibm_torino” for the calibration Pauli strings used in the ZNEC measurements of the qEOM algorithm, while the corresponding x value stems from the simulated SV result. The data obtained with a perfect quantum computer would be placed on the diagonal black line. Subfigure (b) illustrates the calibration procedure. The green crosses show the expectation value with ZNE versus without ZNE for all calibration Pauli strings, again from the “ibm_torino” device. The purple line shows the fitted function f(x) resulting in a optimized value of α = 1.85, used to obtain the red data in (a). We include the error domain as a shaded region that encompasses all data points within the α-range where the root mean square error (RMSE) is within 1.2 times the least squares fit residual ${{\rm{RMSE}}}_{\min }$, i.e., representing the spread of the data points that are within 20% of the residual. For our data, this range corresponds to α ∈ [1.50, 2.23].

We demonstrate the ZNEC procedure in Fig. 6 for a 14 qubit qEOM calculation on the “ibm_torino” device. We clearly see the advantage of ZNEC over ZNE or REM (labeled with M3), without additional measurements compared to ZNE. While ZNE is strongly affected by the statistical errors of the measurements at each noise factor c_i, ZNEC uses the data from 1949 operators, which is a subset of all 23,354 measured operators and can be sorted into 100 commuting groups, to calibrate a functional mapping f(x). This reduces the required quantum resources by a factor of approximately 3 compared to standard ZNE. For our 12 qubit results we calibrate using ZNE measurements for all Pauli strings as the total number of Pauli strings is significantly smaller than in the 14 qubit calculation. For this calculation, we obtain root mean squared errors (RSME) of all values with a significant EV (the SV simulated result $| \langle {\hat{P}}_{i}\rangle | > 0.1$) to be 0.127 (M3), 0.130 (M3 + ZNE), and 0.077 (ZNEC). Around the origin (SV value $| \langle {\hat{P}}_{i}\rangle | < 0.1$), the RSMEs are 0.031 (M3), 0.082 (M3 + ZNE), and 0.054 (ZNEC). In this regime, all methods deliver relatively accurate results, however, it seems that ZNE or the calibration function are not able to capture the systematic offset correctly and thus the ZNEC result is slightly worse than plain M3. The influence of the statistical error in ZNEC is much lower than in the M3 + ZNE case, and, therefore, we believe that this new approach can be used to significantly increase accuracy, or lower the shot budget for the evaluation of EV. Further details concerning the ZNEC scheme are discussed in Sec. J of the SI.