Sagnac effect enhancement can improve optical gyro precision.For a certain input intensity,we suggest that the other input port of beam splitter (BS) should be fed with some quantum light to break through shot noise limit (SNL) to improve Sagnac effect without increasing radiation-pressure noise (NRP).We design a Sagnac effect quantum enhancement criterion (SQEC) to judge whether some quantum light can enhance Sagnac effect and present a Sagnac effect enhancement scheme that utilizing Fock state light and parity measurement technique to extract the output phase.The results of the theoretical analysis show that the maximum sensitivity can be reached at θ =0,and the phase precision can break through SNL and even achieve Heisenberg limit (HL).When the Fock state average photon number n is far less than coherent state,the minimum measurable angular rate is improved with √2n+1 times,which can deduce shot noise and increase NRP little.