Strategies for solving the Fermi-Hubbard model on near-term quantum computers
Quantum computers have now arrived, such as the 53-qubit Google Sycamore chip, that are beyond the reach of classical simulation. This demonstrates that the era of Noisy Intermediate-Scale Quantum (NISQ) devices is upon us.
To match this hardware progression over the last few years, there has been a tremendous increase in both the interest in and the search for suitable quantum software.
NISQ devices will lack error correction and will have a moderate number of qubits (50-100) and two-qubit gates (on the order of hundreds), giving access to computations just outside of the realm of what is possible with best of today's classical computing.
Due to these restrictions, we expect some of the first applications to be hybrid quantum-classical algorithms.
This Insight piece describes some of our recent work, which was concerned with one of the most prominent ideas in this field - a hybrid quantum-classical algorithm called the Variational Quantum Eigensolver (VQE).
This algorithm is used to determine properties of a quantum system, such as its "ground" (lowest energy) state. There are many types of quantum system that can be studied. Our work studied the Fermi-Hubbard model, which is a famous unsolved problem in many-body physics, despite its simple structure and relevance to fields such as high-temperature superconductivity.
The idea behind the VQE algorithm is to optimise over a family of quantum circuits - an "ansatz" - for producing the ground state of a quantum system. The optimisation loop uses a standard computer, but a quantum computer is used to run the quantum circuit.
The VQE algorithm has many possible details that can be tailored to improve its performance. As the size of the device (number of qubits) as well as the size of the computation (number of quantum gates) will be limited in near-term devices, we first determined the most efficient encodings for the Fermi-Hubbard model for small problem sizes.
We then found a more efficient way to implement the important "Hamiltonian Variational" ansatz introduced in previous work, as well as novel generalisations. We further found a way to optimise the measurements that would need to be carried out on the quantum computer to retrieve the input for the classical optimiser.
We implemented an efficient software tool in C++ to determine the performance of this ansatz, and simulated VQE experiments on a classical computer with three different levels of realism: an error-free exact device with exact (nonrealistic) measurements; an error-free exact device with realistic measurements; and a noisy device with realistic measurements. We simulated systems of up to 24 qubits using GPUs.
This combination of theory and numerical experiments allowed us to put together a very comprehensive study on different strategies for solving the model, finding ones that are most likely to have the best results and impact in the near future.
The results show significant promise for producing the ground state of the Fermi-Hubbard model for problem sizes beyond those that can be solved exactly with classical computational methods.
Chris Cade, Lana Mineh, Ashley Montanaro and Stasja Stanisic