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

TL;DR

This paper compares two tensor network algorithms, one quantum and one classical, for efficiently approximating ground states of 1D quantum many-body systems, showing exponential and logarithmic improvements respectively.

Contribution

It introduces a quantum-inspired algorithm and a classical variational method for ground state approximation, demonstrating their efficiency in 1D models.

Findings

Quantum algorithm's accuracy improves exponentially with alternations.

Classical variational method achieves accurate ground states with logarithmic imaginary time.

Both methods outperform traditional approaches in their respective settings.

Abstract

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 $1/ \Delta$ of $H$. This is much faster than imaginary time evolution by $H$, which would require $\tau \sim 1/ \Delta$.