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Quantum speedup for solving the one-dimensional Hubbard model using quantum annealin

Kunal Vyas, Fengping Jin, Hans A. De Raedt, Kristel F. L. Michielsen

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.34734/fzj-2026-02257

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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.
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.
Quantum annealing is a computational paradigm in which optimisation problems are mapped onto the energy landscape of an interacting quantum system and explored through its dynamical evolution. By continuously transforming a simple initial Hamiltonian into one whose ground state encodes the solution, the system traverses a complex landscape via a combination of quantum fluctuations, tunnelling processes, and dissipative dynamics. Unlike gate-based quantum computing, quantum annealing is a specialised and near-term approach aimed primarily at discrete optimisation and sampling tasks. While it is not expected to provide polynomial-time solutions to NP-hard problems in the worst case, it offers a physically motivated heuristic for navigating rugged energy landscapes that arise across science and engineering. Modern quantum annealers realise programmable spin systems with thousands of qubits, placing them among the largest controllable quantum devices currently available. As a result, their significance extends beyond optimisation: they also function as experimental platforms for studying non-equilibrium many-body quantum dynamics in regimes that are difficult to access using classical simulation. In this review we present an accessible introduction to the principles of quantum annealing, describe the main hardware platforms and algorithmic techniques, and analyse how tunnelling, spectral gaps, and open-system effects shape computational performance. We survey applications ranging from optimisation and machine learning to quantum simulation and many-body physics, and discuss the central challenges in benchmarking, scaling, and control. These perspectives position quantum annealing as a distinctive framework at the interface of optimisation, stochastic sampling, and programmable quantum dynamics, with a role that is complementary to both classical algorithms and gate-based quantum computing.
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 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.