An ultra-fast compact optical quantum random number generator (QRNG) is designed and realized by using the amplified spontaneous emission (ASE) noise of superluminescent light-emitting diodes as the quantum random entropy source. In order to reduce the influences of the classical noise and imperfect devices on the randomness and based on the minimum entropy estimation for the quantum entropy source, the exclusive OR (XOR) operation of the adjacent bits for the each collected sequence is conducted and the 12-least-significant bit is intercepted as the final random sequence in the high speed field programmable gate array in real-time. The proposed QRNG with a real-time random number generation rate of 1.4 Gb/s can directly deliver to a host computer and work stably for a long time, which possesses the practical application potential.