  We report the amplification of 250~GHz pulses as short as 260~ps without
observation of pulse broadening using a photonic-band-gap circuit gyrotron
traveling-wave-amplifier. The gyrotron amplifier operates with 38~dB of device
gain and 8~GHz of instantaneous bandwidth. The operational bandwidth of the
amplifier can be tuned over 16 GHz by adjusting the operating voltage of the
electron beam and the magnetic field. The amplifier uses a 30~cm long
photonic-band-gap interaction circuit to confine the desired TE$_{03}$-like
operating mode while suppressing lower order modes which can result in
undesired oscillations. The circuit gain is $>$55~dB for a beam voltage of
23~kV and a current of 700~mA. These results demonstrate the wide bandwidths
and high gain achievable with gyrotron amplifiers. The amplification of
picosecond pulses of variable lengths, 260-800~ps, shows good agreement with
theory using the coupled dispersion relation and the gain-spectrum of the
amplifier as measured with quasi-CW input pulses.
