Observing topological invariants using quantum walks in superconducting circuits

The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here, we motivate a toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schrödinger cat state and highlights a novel refocusing technique, which allows for the direct implementation of a digital version of Bloch oscillations. As the walk is performed in phase space, our scheme can be extended to higher synthetic dimensions by adding additional microwave cavities, whereby superconducting circuit-based simulations can probe topological phases ranging from the quantum-spin Hall effect to the Hopf insulator.