1. A system for coherent coupling of topological and conventional qubits, the system comprising: first, second, and third Josephson junctions; a superconducting flux qubit that supports clockwise and counterclockwise supercurrents in a superconducting ring defined by the first, second, and third Josephson junctions, the flux qubit having a first basis state that corresponds to the clockwise supercurrent and a second basis state that corresponds to the counterclockwise supercurrent; a Majorana nanowire network coupled to a first superconducting island formed between the first and second Josephson junctions, wherein the Majorana nanowire network provides a topological qubit; a semiconductor coupled to a second superconducting island formed between the second and third Josephson junctions, wherein the semiconductor provides a conventional double-dot qubit, wherein a joint parity of the topological and conventional qubits is measured via an Aharonov-Casher effect that gives rise to an energy splitting in the flux qubit that depends on a total charge enclosed in a region between the first and third Josephson junctions.