Geometrically manipulating photonic Schr\"{o}dinger cat states and realizing two-logical-qubit gates

Geometric phases are of particular importance for manipulation of quantum states and for quantum information processing. Here, we show that a transmon qubit can be forced to traverse a circuit, accumulating a geometric phase, conditional on the photon number of one or two cavities dispersively coupled to the qubit. Based on this geometric effect, we demonstrate the photon-number parity of a Schr\"{o}dinger cat state in a cavity can be controlled, offering the possibility for correcting the parity flip due to single-photon loss. We further geometrically realize phase gates for one and two photonic qubits with logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. More importantly, we use this geometric method to implement a controlled-phase gate between two logical qubits binomially encoded in error-correctable logical space, which represents an important step towards fault-tolerant quantum computation.