Numerical simulation of leakage effect for quantum NOT operation on   three-Josephson-junction flux qubit
  Superconducting flux qubits with three Josephson junctions are promising candidates for the building blocks of a quantum computer. We have applied the imaginary time evolution method to study the model of this qubit accurately by calculating its wave functions and eigenenergies. Because such qubits are manipulated with magnetic flux microwave pulses they might be irradiated into non-computational states which is called the leakage effect. Through the evolution of the density matrix of the qubit under either hard-shaped pulse or Gaussian-shaped pulse to carry out quantum NOT operation, we show that the leakage effect is not a serious problem for a flux qubit. 
Tao Wu, Jianshe Liu, Zheng Li