We propose an improved quantum algorithm to calculate the Green's function through real-time propagation, and use it to compute the retarded Green's function for the two-, three-, and four-site Hubbard models. This protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the two-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the $|N\ifmmode\pm\else\textpm\fi{}1\ensuremath{\rangle}$ states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.
We propose an improved quantum algorithm to calculate the Green's function through real-time propagation, and use it to compute the retarded Green's function for the two-, three-, and four-site Hubbard models. This protocol significantly reduces the number of controlled operations when compared to those previously suggested in literature. Such reduction is quite remarkable when considering the two-site Hubbard model, for which we show that it is possible to obtain the exact time propagation of the ∣N\ifmmode±\else\textpm\fi1\ensuremath⟩ states by exponentiating one single Pauli component of the Hamiltonian, allowing us to perform the calculations on an actual superconducting quantum processor.
We present a quantum-classical hybrid implementation of the Liouvillian recursion method to compute many-body Green's functions using a quantum computer. From an approximate ground state preparation circuit, this algorithm produces the local (r=r′) and inter-site (r=r′) Green's functions Grr′(ω) by measuring observables generated recursively. We demonstrate the approach on a superconducting quantum processor for the open-boundary four-site Hubbard model. We then use the computed Green's functions as input to the Galitskii-Migdal formula to produce better ground state energy estimation than the expectation value of the Hamiltonian for the approximate circuit. Empirical results indicate exponential convergence in the number of iterations, yielding a computational complexity polynomial in the Green's-function accuracy, as measured with the Wasserstein distance. Our results also indicate significant robustness to noise and to inaccuracies of the ground state preparation, providing evidence that Liouvillian recursion is well adapted to the constraints of near-term quantum computing.
Computation of the Green's function is crucial to study the properties of quantum many-body systems such as strongly correlated systems. Although the high-precision calculation of the Green's function is a notoriously challenging task on classical computers, the development of quantum computers may enable us to compute the Green's function with high accuracy even for classically-intractable large-scale systems. Here, we propose an efficient method to compute the real-time Green's function based on the local variational quantum compilation (LVQC) algorithm, which simulates the time evolution of a large-scale quantum system using a low-depth quantum circuit constructed through optimization on a smaller-size subsystem. Our method requires shallow quantum circuits to calculate the Green's function and can be utilized on both near-term noisy intermediate-scale and long-term fault-tolerant quantum computers depending on the computational resources we have. We perform a numerical simulation of the Green's function for the one- and two-dimensional Fermi-Hubbard model up to $4\ifmmode\times\else\texttimes\fi{}4$ sites lattice (32 qubits) and demonstrate the validity of our protocol compared to a standard method based on the Trotter decomposition. We finally present a detailed estimation of the gate count for the large-scale Fermi-Hubbard model, which also illustrates the advantage of our method over the Trotter decomposition.
Abstract This study introduces a systematic approach for analyzing strongly correlated systems by adapting the conventional quantum cluster method to a quantum circuit model. We have developed a more concise formula for calculating the cluster’s Green’s function, requiring only real-number computations on the quantum circuit instead of complex ones. This approach is inherently more suited to quantum circuits, which primarily yield statistical probabilities. As an illustrative example, we explored the Hubbard model on a 2D lattice. The ground state was determined utilizing Xiaohong, a superconducting quantum processor equipped with 66 qubits, supplied by QuantumCTek Co., Ltd. Subsequently, we employed the circuit model with controllable noise to compute the real-time retarded Green’s function for the cluster, which is then used to determine the lattice Green’s function. We conducted an examination of the band structure in the insulator phase of the lattice system. This preliminary investigation lays the groundwork for exploring a wealth of innovative physics within the field of condensed matter physics.
We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.
We present and benchmark quantum computing approaches for calculating real-time single-particle Green's functions and nonlinear susceptibilities of Hamiltonian systems. The approaches leverage adaptive variational quantum algorithms for state preparation and propagation. Using automatically generated compact circuits, the dynamical evolution is performed over sufficiently long times to achieve adequate frequency resolution of the response functions. We showcase accurate Green's function calculations using a statevector simulator on classical hardware for Fermi-Hubbard chains of 4 and 6 sites, with maximal ansatz circuit depths of 65 and 424 layers, respectively, and for the molecule LiH with a maximal ansatz circuit depth of 81 layers. Additionally, we consider an antiferromagnetic quantum spin-1 model that incorporates the Dzyaloshinskii-Moriya interaction to illustrate calculations of the third-order nonlinear susceptibilities, which can be measured in two-dimensional coherent spectroscopy experiments. These results demonstrate that real-time approaches using adaptive parametrized circuits to evaluate linear and nonlinear response functions can be feasible with near-term quantum processors.
We propose quantum algorithms for projective ground-state preparation and calculations of the many-body Green's functions directly in frequency domain. The algorithms are based on the linear combination of unitary (LCU) operations and essentially only use quantum resources. To prepare the ground state, we construct the operator exp(−τH^2) using Hubbard-Stratonovich transformation by LCU and apply it on an easy-to-prepare initial state. Our projective state preparation procedure saturates the near-optimal scaling, O(γΔ1logγη1), of other currently known algorithms, in terms of the spectral-gap lower bound Δ, the additive error η in the state vector, and the overlap lower bound γ between the initial state and the exact ground state. It is straightforward to combine our algorithm with spectral-gap amplification technique to achieve quadratically improved scaling O(1/Δ) for ground-state preparation of frustration-free Hamiltonians, which we demonstrate with numerical results of the q-deformed XXZ chain. To compute the Green's functions, including the single-particle and other response functions, we act on the prepared ground state with the retarded resolvent operator R(ω+iΓ;H^) in the LCU form derived from the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has O(Γ21logΓε1) complexity scaling for the frequency resolution Γ of the response functions and the targeted error ε, while classical algorithms for FIT usually have polynomial scaling over the error ε. To illustrate the complexity scaling of our algorithms, we provide numerical results for their application to the paradigmatic Fermi-Hubbard model on a one-dimensional lattice with different numbers of sites.