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arXiv:2310.16775 · Physical review. B./Physical review. B

From Heisenberg to Hubbard: An initial state for the shallow quantum simulation of correlated electrons

Bruno Murta, Joaquín Fernández-Rossier

Published Jan 16, 2024• 4 citations
arXivHTMLPDF

Abstract

The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-$\frac{1}{2}$ Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-$\frac{1}{2}$ wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin-$\frac{1}{2}$ state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.

DOI
10.1103/physrevb.109.035128

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The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-12\frac{1}{2}21​ Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-12\frac{1}{2}21​ wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin-12\frac{1}{2}21​ state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.
Quantum algorithms for ground-state energy estimation require initial states with non-negligible fidelity to the ground state. Guided by classical simulations, we design shallow circuits for the two-dimensional half-filled Hubbard model by first preparing an approximate Heisenberg ground state and then applying charge-fluctuation gates derived directly from the Schrieffer-Wolff (SW) generator. For the Heisenberg model, we demonstrate effective parameter transfer from a 4×44\times44×4 lattice to lattices up to 10×1010\times1010×10 without further optimization. We obtain promising lower bounds on the ground-state fidelity using tensor-network simulations, variational Monte Carlo reference states, and estimates of the ground-state energy and singlet gap. Exact 4×44\times44×4 calculations show that the SW gates substantially improve the Hubbard ground-state fidelity of the embedded Heisenberg state. The successful Heisenberg parameter transfer supports the use of these small-lattice results to design shallow Hubbard circuits on larger lattices beyond the reach of classical simulations. For a 10×1010\times1010×10 lattice, the estimated preparation cost is on the order of 10510^5105 TTT gates, including Clifford+TTT synthesis, which is well within the megaquop regime. These results offer a route to low-cost initial-state preparation for ground-starongly correlated systems.
It is expected that the simulation of correlated fermions in chemistry and material science will be one of the first practical applications of quantum processors. Given the rapid evolution of quantum hardware, it is increasingly important to develop robust benchmarking techniques to gauge the capacity of quantum hardware specifically for the purpose of fermionic simulation. Here we propose using the one-dimensional Fermi-Hubbard model as an application benchmark for variational quantum simulations on near-term quantum devices. Since the one-dimensional Hubbard model is both strongly correlated and exactly solvable with the Bethe ansatz, it provides a reference ground state energy that a given device with limited coherence will be able to approximate up to a maximal size. The length of the largest chain that can be simulated provides an effective fermionic length. We use variational quantum eigensolver to approximate the ground state energy values of Fermi-Hubbard instances and show how the fermionic length benchmark can be used in practice to assess the performance of bounded-depth devices in a scalable fashion.
Fermionic atoms in optical lattices provide a native implementation of Fermi-Hubbard (FH) models that can be used as analog quantum simulators of many-body fermionic systems. Recent experimental advances include the time-dependent local control of chemical potentials and tunnelings, and thus enable us to operate this platform digitally as a programmable quantum simulator. Here, we explore these opportunities and develop ground-state preparation algorithms for different fermionic models, based on the ability to implement both single-particle and many-body high-fidelity fermionic gates, as provided by the native FH Hamiltonian. In particular, we first design variational precompiled quantum circuits to prepare the ground state of the natively implemented FH model, with significant speedups relative to competing adiabatic protocols. Besides, the versatility of this variational approach enables us to target extended FH models, i.e., including terms that are not natively realized on the platform. As an illustration, we include next-nearest-neighbor tunnelings at finite dopings, relevant in the context of d -wave superconductivity. Furthermore, we discuss how to approximate the imaginary-time evolution using variational fermionic circuits, both as an alternative state-preparation strategy and as a subroutine for the quantum Lanczos algorithm to further improve the energy estimation. We benchmark our protocols for ladder geometries, though they can be readily applied to two-dimensional experimental setups to address regimes beyond the capabilities of current classical methods. These results pave the way for more efficient and comprehensive explorations of relevant many-body phases with existing programmable fermionic quantum simulators.
A Quantum Monte Carlo method of calculating operator expectation values for the ground state of the nearest-neighbor Heisenberg model on large square lattices is presented, along with some comparisons of the results obtained from this method and alternative methods for 2D lattices. In the simulations the ground state is projected out from an arbitrary state and sampled in a valence\nbond basis spanning the spin singlet subspace. The projection method is then extended to be used on a simplified Hubbard model at half-filling, which is\naptly called the T0-less Hubbard model, for arbitrary relative strengths of the electron hopping and electron-electron repulsion energy parameters. Results for calculated ground state energies in 1D are presented. These calculations are performed using a spin-charge separated basis, in which the valence bonds\nare generalized to also carry charge.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids of up to six sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver (VQE) called ADAPT-VQE [H. R. Grimsley et al., Nat. Commun. 10, 3007 (2019)]. In contrast with nonadaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the nonadaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green's functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach [A. Peruzzo et al., Nat. Commun. 5, 4213 (2014); J. R. McLean et al., New J. Phys. 18, 023023 (2016); K. Sharma et al., New J. Phys. 22, 043006 (2020)], and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well suited for noisy intermediate-scale quantum devices.