IEEE International Conference on Quantum Computing and Engineering (QCE)
Quantum Ans#X00e4;Tze for Green's Functions of Single-Impurity Anderson Models at Half-Filling
E. A. Coello Pérez, Dmytro Bykov, Markus Eisenbach, Swarnava Ghosh, Murali Gopalakrishnan Meena, Seongmin Kim, Mariia Karabin, David O. Williams Rogers, Amir Shehata, Tanvir Sohail, In‐Saeng Suh, Hanna Terletska
We introduce a set of quantum ansätze tailored for computing Green's functions of single-impurity models using the variational quantum eigensolver. These ansätze conserve particle number and spin, allowing for the efficient preparation of the impurity model's ground state and the Krylov bases required to express the Green's function as a continued fraction. Notably, the parameters preparing the particle and hole excitations used as initial Krylov vectors are fully determined by those optimized for the ground state, eliminating the need for secondary optimizations or alternative excitation preparation methods. We demonstrate the utility of these ansätze by computing Green's functions for single-impurity Anderson models with one bath site within a dynamical mean-field theory loop, where the impuritybath hybridization is updated self-consistently. The parameters defining the states in the Krylov bases are straightforwardly determined by orthonormality in the single-bath case, allowing for the direct computation of the relevant Hamiltonian matrix elements from these states. Future work will explore the robustness, scalability, and practical integration of these ansätze into hybrid quantum-classical simulations of strongly correlated materials.
We introduce a set of quantum ansätze tailored for computing Green's functions of single-impurity models using the variational quantum eigensolver. These ansätze conserve particle number and spin, allowing for the efficient preparation of the impurity model's ground state and the Krylov bases required to express the Green's function as a continued fraction. Notably, the parameters preparing the particle and hole excitations used as initial Krylov vectors are fully determined by those optimized for the ground state, eliminating the need for secondary optimizations or alternative excitation preparation methods. We demonstrate the utility of these ansätze by computing Green's functions for single-impurity Anderson models with one bath site within a dynamical mean-field theory loop, where the impuritybath hybridization is updated self-consistently. The parameters defining the states in the Krylov bases are straightforwardly determined by orthonormality in the single-bath case, allowing for the direct computation of the relevant Hamiltonian matrix elements from these states. Future work will explore the robustness, scalability, and practical integration of these ansätze into hybrid quantum-classical simulations of strongly correlated materials.
Quantum embedding methods, such as dynamical mean-field theory (DMFT), provide a powerful framework for investigating strongly correlated materials. A central computational bottleneck in DMFT is in solving the Anderson impurity model (AIM), whose exact solution is classically intractable for large bath sizes. In this work, we develop and benchmark a quantum-classical hybrid solver tailored for DMFT applications, using the variational quantum eigensolver (VQE) to prepare the ground state of the AIM with shallow quantum circuits. The solver uses a unified ansatz framework to prepare the particle and hole excitations of the ground-state from parameter-shifted circuits, enabling the reconstruction of the impurity Green's function through a continued-fraction expansion. We evaluate the performance of this approach across a few bath sizes and interaction strengths under noisy, shot-limited conditions. We compare three optimization routines (COBYLA, Adam, and L-BFGS-B) in terms of convergence and fidelity, assess the benefits of estimating a quantum-computed moment (QCM) correction to the variational energies, and benchmark the approach by comparing the reconstructed density of states (DOS) against that obtained using a classical pipeline. Our results demonstrate the feasibility of Green's function reconstruction on near-term devices and establish practical benchmarks for quantum impurity solvers embedded within self-consistent DMFT loops.
We generalize the sample-based Krylov quantum diagonalization (SKQD) method from ground-state calculations to the evaluation of single-particle Green's functions. By constructing and sampling unitary Krylov subspaces in the N +/- 1 particle-number sectors and evaluating all sector-connecting overlaps classically, the approach reconstructs the Green's function via a Lanczos continued fraction while retaining the shallow-circuit, ancilla-free character of SKQD. The quantum device is required only to prepare and sample short-time evolutions. Applied to the particle-hole-symmetric single-impurity Anderson model in chain geometry, with the discrete bath representation used in dynamical mean-field theory (DMFT), the method recovers the spectral function using a relatively small fraction of the full Hilbert space. Across a range of interaction strengths that spans the metal-insulator transition, the main spectral features are reproduced. These results suggest that SKQD-based Green's-function calculations may allow DMFT impurity solvers with a larger number of bath sites on near-term quantum hardware than is currently practical.
We present a variational quantum eigensolver (VQE) approach for solving the Anderson Impurity Model (AIM) arising in Dynamical Mean-Field Theory (DMFT). Recognizing that the minimal two-site approximation often fails to resolve essential spectral features, we investigate the efficacy of VQE for larger bath discretizations while adhering to near-term hardware constraints. We employ a symmetry-adapted ansatz enforcing conservation of particle number (N), spin projection (Sz=0), and total spin (S2=0) symmetry, benchmarking the performance against exact diagonalization across different interaction strengths using bath parameters extracted from the DMFT self-consistency loop. For a four-site model, the relative error in the ground state energy remains well below 0.01 with a compact parameter set (Np≤30). Crucially, we demonstrate that the single-particle Green's function-the central quantity for DMFT-can be accurately extracted from VQE-prepared ground states via real-time evolution in the intermediate to strong interaction regimes. However, in the weak interaction regime, the Green's function exhibits noticeable deviations from the exact benchmark, particularly in resolving low-energy spectral features, despite the ground state energy showing excellent agreement. These findings demonstrate that VQE combined with real-time evolution can effectively extend quantum-classical hybrid DMFT beyond the two-site approximation, particularly for describing insulating phases. While this approach offers a viable pathway for simulating strongly correlated materials on near-term devices, the observation that accurate ground state energy does not guarantee accurate dynamical properties highlights a key challenge for applying such approaches to correlated metals.
In this paper, we present a quantum computational method to calculate the many-body Green's function matrix in a spin orbital basis. We apply our approach to finite-sized fermionic Hubbard models and related impurity models within Dynamical Mean Field Theory, and demonstrate the calculation of Green's functions on Quantinuum's H1-1 trapped-ion quantum computer. Our approach involves a cumulant expansion of the Lanczos method, using Hamiltonian moments as measurable expectation values. This bypasses the need for a large overhead in the number of measurements due to repeated applications of the variational quantum eigensolver (VQE), and instead measures the expectation value of the moments with one set of measurement circuits. From the measured moments, the tridiagonalised Hamiltonian matrix can be computed, which in turn yields the Green's function via continued fractions. While we use a variational algorithm to prepare the ground state in this work, we note that the modularity of our implementation allows for other (non-variational) approaches to be used for the ground state.
We present a hardware-reconfigurable ansatz on N q -qubits for the variational preparation of many-body states of the Anderson impurity model (AIM) with N imp + N bath = N q / 2 sites, which conserves total charge and spin z component within each variational search subspace. The many-body ground state of the AIM is determined as the minimum over all minima of O ( N q 2 ) distinct charge-spin sectors. Hamiltonian expectation values are shown to require ω ( N q ) < N meas. ≤ O ( N imp N bath ) symmetry-preserving, parallelizable measurement circuits, each amenable to postselection. To obtain the one-particle impurity Green's function we show how initial Krylov vectors can be computed via midcircuit measurement and how Lanczos iterations can be computed using the symmetry-preserving ansatz. For a single-impurity Anderson model with a number of bath sites increasing from one to seven, we show using numerical emulation that the ease of variational ground-state preparation is suggestive of linear scaling in circuit depth and subquartic scaling in optimizer complexity. We therefore expect that, combined with time-dependent methods for Green's function computation, our ansatz provides a useful tool to account for electronic correlations on early fault-tolerant processors. Finally, with a view towards computing real materials properties of interest like magnetic susceptibilities and electron-hole propagators, we provide a straightforward method to compute many-body, time-dependent correlation functions using a combination of time evolution, midcircuit measurement-conditioned operations, and the Hadamard test.
The accurate determination of the electronic structure of strongly correlated materials using first principle methods is of paramount importance in condensed matter physics, computational chemistry, and material science. However, due to the exponential scaling of computational resources, incorporating such materials into classical computation frameworks becomes prohibitively expensive. In 2016, Bauer et al. proposed a hybrid quantum-classical approach to correlated materials [B. Bauer et al., Hybrid quantum-classical approach to correlated materials, Phys. Rev. X 6, 031045 (2016).PRXHAE2160-330810.1103/PhysRevX.6.031045] that can efficiently tackle the electronic structure of complex correlated materials. Here, we experimentally demonstrate that approach to tackle the computational challenges associated with strongly correlated materials. By seamlessly integrating quantum computation into classical computers, we address the most computationally demanding aspect of the calculation, namely the computation of the Green's function, using a spin quantum processor. Furthermore, we realize a self-consistent determination of the single impurity Anderson model through a feedback loop between quantum and classical computations. A quantum phase transition in the Hubbard model from the metallic phase to the Mott insulator is observed as the strength of electron correlation increases. As the number of qubits with high control fidelity continues to grow, our experimental findings pave the way for solving even more complex models, such as strongly correlated crystalline materials or intricate molecules.