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Finding the ground state of the 1D Hubbard Model using quantum annealing

Fengping Jin

Published Jan 1, 2025• 0 citations
Publisher

Abstract

We want to illustrate how to find the ground state of the one-dimensional Hubbard model. We start by introducing the classical algorithms that can do this, like exact diagonalization, Lanczos algorithm and the Bethe-ansatz equations. Subsequently, we show how to solve the problem on a gate-based quantum computer using the quantum annealing algorithm. The procedure involves transforming the fermionic Hamiltonian using the Jordan-Wigner transformation, preparing the initial state for quantum annealing using Givens rotations, and constructing the circuit for the time evolution according to the Schrodinger equation using the second-order product-formula algorithm. We conclude by presenting some results and analysis of performance for the quantum annealing algorithm.

DOI
10.34734/fzj-2026-00022

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We want to illustrate how to find the ground state of the one-dimensional Hubbard model. We start by introducing the classical algorithms that can do this, like exact diagonalization, Lanczos algorithm and the Bethe-ansatz equations. Subsequently, we show how to solve the problem on a gate-based quantum computer using the quantum annealing algorithm. The procedure involves transforming the fermionic Hamiltonian using the Jordan-Wigner transformation, preparing the initial state for quantum annealing using Givens rotations, and constructing the circuit for the time evolution according to the Schrodinger equation using the second-order product-formula algorithm. We conclude by presenting some results and analysis of performance for the quantum annealing algorithm.
The Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. This model provides insight into a range of phenomena in correlated electron systems. We wish to examine the paradigm of quantum algorithms for solving such many-body problems. The focus of our current work is on the one-dimensional model that is integrable, meaning that there exist analytical results for determining its ground state. In particular, we demonstrate how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian. We perform simulations for systems with up to 40 qubits to study the scaling of required annealing time for obtaining the ground state. We find that for the half-filled cases considered, there is a polynomial quantum speedup over algorithms based on the Bethe-ansatz equations.
The Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. This model provides insight into a range of phenomena in correlated electron systems. We wish to examine the paradigm of quantum algorithms for solving such many-body problems. The focus of our current work is on the one-dimensional model that is integrable, meaning that there exist analytical results for determining its ground state. In particular, we demonstrate how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian. We perform simulations for systems with up to 40 qubits to study the scaling of required annealing time for obtaining the ground state. We find that for the half-filled cases considered, there is a polynomial quantum speedup over algorithms based on the Bethe-ansatz equations.
The Fermi-Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. With a simplistic form of the Hamiltonian, the model can potentially provide explanation to interesting phenomena in correlated electrons. The interest of our current work is the model in one dimension. The one-dimensional Fermi-Hubbard model has been extensively studied and there exist analytical results for determining its ground-state energy in the thermodynamic limit. We want to find the ground-state from the perspective of quantum computing. In particular, we employ the protocol of quantum annealing and perform high-performance simulations of the same using the Jülich Quantum Computer Simulator (JUQCS) for systems with up to 40 qubits. We learn that the time required to find the ground-state would scale sublinearly with system size for the half-filling cases considered.
One of the main applications of future quantum computers will be the simulation of quantum models. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just O(N)\mathcal{O}(N)O(N) gates and O(log⁡N)\mathcal{O}(\log N)O(logN) circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive approaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.
The Fermi-Hubbard model has occupied the minds of condensed matter physicists for most part of the last century. With a simplistic form of the Hamiltonian, the model can potentially provide explanation to interesting phenomena in correlated electrons. The interest of our current work is the model in one dimension. The one-dimensional Fermi-Hubbard model has been extensively studied and there exist analytical results for determining its ground-state energy in the thermodynamic limit. We want to find the ground-state from the perspective of quantum computing. In particular, we employ the protocol of quantum annealing and perform high performance simulations of the same using the Juelich Quantum Computer Simulator (JUQCS) for systems with upto 40 qubits. We learn that the time required to find the ground-state would scale sublinearly with system size for the half-filling cases considered.
Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.