Direct probe of topology and geometry of quantum states on the IBM Q quantum processor

The concepts of topology and geometry are of critical importance in exploring exotic phases of quantum matter. Although they have been investigated on various experimental platforms, to date a direct probe of the topological and geometric properties on a universal quantum computer even for a minimum model is still in vain. In this work, we first show that a density matrix form of the quantum geometric tensor (QGT) can be explicitly reconstructed from Pauli operator measurements on a quantum circuit. We then propose two algorithms suitable for IBM quantum computers to directly probe QGT. The first algorithm is a variational quantum algorithm particularly suitable for noisy intermediate-scale quantum era devices, whereas the second one is a pure quantum algorithm based on quantum imaginary time evolution. Explicit results obtained from IBM Q simulating a Chern insulator model are presented and analyzed. Our results indicate that transmon qubit-based universal quantum computers have the potential to directly simulate and investigate topological and geometric properties of a quantum system.