  We analyze the effect of environment on the gate operation of flux-biased
phase qubits. We employ the master equation for a reduced density matrix of the
qubit system coupled to an Ohmic environment, described by the Caldeira-Leggett
model. Numerically solving this equation, we evaluate the gate error as a
function of energy splitting between qubit states, junction capacitance, and
temperature. The analysis is presented for single-quadrature microwave
(control) pulses as well as for two-quadrature pulses, which lower the gate
error significantly for idealized systems in the absence of environment. Our
results indicate that two-quadrature pulses outperform single quadrature pulses
even in the presence of environment.
