Quantum speed-up for solving the one-dimensional Hubbard model using quantum annealing

TL;DR

This work demonstrates how to perform a gate-based quantum computer simulation of quantum annealing for the Hubbard Hamiltonian, and finds that for the half-filled cases considered, there is a substantial quantum speed-up over algorithms based on the Bethe-ansatz equations.

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 which 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 substantial quantum speed-up over algorithms based on the Bethe-ansatz equations.

Figures (9)

• Figure 1: Dispersion relation and the ground state occupation of the energy levels of the initial Hamiltonian for $L=5$, $N_\uparrow=3$, $N_\downarrow=2$
• Figure 2: $\Delta E$ as a function of $T_A$ for a linear annealing schedule with $U/t_H=4$.
• Figure 3: $\Delta E$ as a function of $T_A$ for a linear annealing schedule with (a) $U/t_H=8$ and (b) $U/t_H=16$. The crosses indicate onsets $\epsilon(L)$, which are plotted as a function of $L$ in the inset.
• Figure 4: $\alpha(L)$ as a function of $L$ for a linear annealing schedule and $U/t_H=4,8,16$.
• Figure 5: Quantum circuit for implementing a Givens rotation on neighbouring qubits