Bang-bang algorithms for quantum many-body ground states: a tensor network exploration

We use matrix product techniques to investigate the performance of two algorithms for obtaining the ground state of a quantum many-body Hamiltonian $H=H_A+H_B$ in infinite systems. The first algorithm is a generalization of the quantum approximate optimization algorithm (QAOA) and uses a quantum computer to evolve an initial product state into an approximation of the ground state of $H$, by alternating between $H_A$ and $H_B$. We show for the 1D quantum Ising model that the accuracy in representing a gapped ground state improves exponentially with the number of alternations. The second algorithm is the variational imaginary time ansatz (VITA), which uses a classical computer to simulate the ground state via alternating imaginary time steps with $H_A$ and $H_B$. We find for the 1D quantum Ising model that an accurate approximation to the ground state is obtained with a total imaginary time $\tau$ that grows only logarithmically with the inverse energy gap $1\mathrm{/}\mathrm{\Delta }$ of $H$. This is much faster than imaginary time evolution by $H$, which would require $\tau \sim 1\mathrm{/}\mathrm{\Delta }$.