  Quantum algorithms provide a potential strategy for solving computational
problems that are intractable by classical means. Computing the topological
invariants of topological matter is one central problem in research on quantum
materials, and a variety of numerical approaches for this purpose have been
developed. However, the complexity of quantum many-body Hamiltonians makes
calculations of topological invariants challenging for interacting systems.
Here, we present two quantum circuits for calculating Chern numbers of
two-dimensional quantum matter on quantum computers. Both circuits combine a
gate-based adiabatic time-evolution over the discretized Brillouin zone with
particular phase estimation techniques. The first algorithm uses many qubits,
and we analyze it using a tensor-network simulator of quantum circuits. The
second circuit uses fewer qubits, and we implement it experimentally on a
quantum computer based on superconducting qubits. Our results establish a
method for computing topological invariants with quantum circuits, taking a
step towards characterizing interacting topological quantum matter using
quantum computers.
