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arXiv:2309.09685 · Quantum

Quantum Computed Green's Functions using a Cumulant Expansion of the Lanczos Method

Gabriel Greene‐Diniz, David Zsolt Manrique, Kentaro Yamamoto, Evgeny A. Plekhanov, Nathan Fitzpatrick, Michał Krompiec, Rei Sakuma, David Muñoz Ramo

Published Jun 20, 2024• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2024-06-20-1383

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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 an algorithm to compute Green's functions on quantum computers for interacting electron systems, which is a challenging task on conventional computers. It uses a continued fraction representation based on the Lanczos method, where the wave functions are expanded as linear combination of basis states within a quantum subspace. While on conventional computers the cost of the computation grows exponentially with system size, limiting the method to small systems, by representing the basis states on a quantum computer one may overcome this exponential scaling barrier. We propose a two-level multigrid Trotter time evolution for an efficient preparation of the basis states in a quantum circuit, which takes advantage of the robustness of the subspace expansion against Trotter errors. Using a quantum emulator we demonstrate the algorithm for the Hubbard model on a Bethe lattice with infinite coordination, which we map to a 16 qubit Anderson impurity model within the dynamical mean field theory. Our algorithm computes the Green's function accurately for both the metallic and Mott insulating regimes, with a circuit depth several orders of magnitude below what has been proposed using time evolution. The two-level multigrid time evolution reduces the number of Trotter steps required to compute the Green's function to about four to six. We therefore expect that the method can be used on near term quantum computers for moderate system sizes, while allowing for scalability to larger circuit depths and qubit numbers on future fault tolerant quantum computers.
We propose an algorithm to obtain Green's functions as a continued fraction on quantum computers, which is based on the construction of the Krylov basis using variational quantum algorithms, and included in a Lanczos iterative scheme. This allows the integration of quantum algorithms with first principles material science simulations, as we demonstrate within the dynamical mean-field theory (DMFT) framework. DMFT enables quantitative predictions for strongly correlated materials, and relies on the calculation of Green's functions. On conventional computers the exponential growth of the Hilbert space with the number of orbitals limits DMFT to small systems. Quantum computers open new avenues and can lead to a significant speedup in the computation of expectation values required to obtain the Green's function. We apply our Krylov variational quantum algorithm combined with DMFT to the charge transfer insulator La2_{2}2​CuO4_44​ using a quantum computing emulator, and show that with 8 qubits it predicts the correct insulating material properties for the paramagnetic phase. We therefore expect that the method is ideally suited to perform simulations for real materials on near term quantum hardware.
We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Green's function at a very low temperature. We compare the Davidson+++Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.
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′r=r'r=r′) and inter-site (r≠r′r\neq r'r=r′) Green's functions Grr′(ω)G_{rr'}(ω)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.
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.
Abstract We use the cumulant Green’s functions method (CGFM) to study the single-band Hubbard model. The starting point of the method is to diagonalize a cluster (‘seed’) containing N correlated sites and employ the cumulants calculated from the cluster solution to obtain the full Green’s functions for the lattice. All calculations are done directly; no variational or self-consistent process is needed. We benchmark the one-dimensional results for the gap, the double occupancy, and the ground-state energy as functions of the electronic correlation at half-filling and the occupation numbers as functions of the chemical potential obtained from the CGFM against the corresponding results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The particle-hole symmetry of the density of states is fulfilled, and the gap, occupation numbers, and ground-state energy tend systematically to the known results as the cluster size increases. We include a straightforward application of the CGFM to simulate the singles occupation of an optical lattice experiment with lithium-6 atoms in an eight-site Fermi-Hubbard chain near half-filling. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and extended to other strongly correlated models, like the Anderson Hamiltonian, the t − J, Kondo, and Coqblin-Schrieffer models.