Solving the Hubbard model using the variational quantum eigensolver

Lana Mineh, Ashley Montanaro

Calculating the ground state properties of a Hamiltonian can be mapped to the problem of finding the ground state of a smaller Hamiltonian through the use of embedding methods. These embedding techniques have the ability to drastically reduce the problem size, and hence the number of qubits required when running on a quantum computer. However, the embedding process can produce a relatively complicated Hamiltonian, leading to a more complex quantum algorithm. In this paper we carry out a detailed study into how density matrix embedding theory (DMET) could be implemented on a quantum computer to solve the Hubbard model. We consider the variational quantum eigensolver (VQE) as the solver for the embedded Hamiltonian within the DMET algorithm. We derive the exact form of the embedded Hamiltonian and use it to construct efficient ansatz circuits and measurement schemes. We conduct detailed numerical simulations up to 16 qubits, the largest to date, for a range of Hubbard model parameters and find that the combination of DMET and VQE is effective for reproducing ground state properties of the model.

Alexandre Choquette, Agustín Di Paolo, Panagiotis Kl. Barkoutsos +3 more

Takashi Yanagisawa, Izumi Hase, Yamaji K.

Benjamin D. M. Jones, Lana Mineh, Ashley Montanaro

Data used in the paper "Benchmarking a wide range of optimisers for solving the Fermi-Hubbard model using the variational quantum eigensolver" by Benjamin D.M. Jones, Lana Mineh, and Ashley Montanaro, available here: https://arxiv.org/abs/2411.13742. See the README.md file for details on how the data is organised, and data.zip for the full data (~37GB when uncompressed).

Ashley Montanaro, Stasja Stanisic

The Fermi-Hubbard model is a plausible target to be solved by a quantum computer using the variational quantum eigensolver algorithm. However, problem sizes beyond the reach of classical exact diagonalisation are also beyond the reach of current quantum computing hardware. Here we use a simple method which compresses the first nontrivial subcase of the Hubbard model -- with one spin-up and one spin-down fermion -- enabling larger instances to be addressed using current quantum computing hardware. We implement this method on a superconducting quantum hardware platform for the case of the $2 \times 1$ Hubbard model, including error-mitigation techniques, and show that the ground state is found with relatively high accuracy.

S. N. Coppersmith, J. J. Sohn