Abstract: A superconducting qubit is presented. The qubit is a shaped long Josephson junction with a magnetic fluxon such that, in the presence of an externally applied magnetic field, a fluxon potential energy function indicating a plurality of pinning sites in the qubit is produced. In one embodiment, a heart-shaped Josephson junction is formed where a trapped fluxon has a double-welled potential energy function, indicating two pinning sites, when the junction is placed in an externally applied magnetic field. The qubit is manipulated by preparing an initial state, creating a superposition of the two states by decreasing the magnetic field, evolving of the quantum state with time, freezing in a final state by increasing the magnetic field, and reading out the final state. In other embodiments, qubit exhibiting potential energy functions having any number of pinning sites can be realized.