Quantum circuit algorithm for topological invariants of second order topological many-body quantum magnets

Topological quantum matter represents a flexible playground to engineer unconventional excitations. While non-interacting topological single-particle systems have been studied in detail, topology in quantum many-body systems remains an open problem. Specifically, in the quantum many-body limit, one of the challenges lies in the computational complexity of obtaining the many-body ground state and its many-body topological invariant. While algorithms to compute ground states with quantum computers have been heavily investigated, algorithms to compute topological invariants in a quantum computer are still under active development. Here we demonstrate a quantum circuit to compute the many-body topological invariant of a second-order topological quantum magnet encoded in qubits. Our algorithm relies on a quantum circuit adiabatic evolution in transverse paths in parameter space, and we uncover hidden topological invariants depending on the traversed path. Our work puts forward an algorithm to leverage quantum computers to characterize many-body topological quantum matter.