Fidelity and leakage of Josephson qubits

The unit of quantum information is the qubit, a vector in a two-dimensional Hilbert space. On the other hand, quantum hardware often operates in two-dimensional subspaces of vector spaces of higher dimensionality. The presence of higher quantum states may affect the accuracy of quantum information processing. In this Letter we show how to cope with quantum leakage in devices based on small Josephson junctions. While the presence of higher charge states of the junction reduces the fidelity during gate operations we demonstrate that errors can be minimized by appropriately designing and operating the gates.