Abstract: ‍ ‍In this paper， we studied a downlink Intelligent Reflective Surface （IRS）-assisted cell-free Massive multiple-input multiple-output （MIMO） system in which each Access Point （AP） used low-resolution Digital-to-analog Converters（DACs）. To further reduce the hardware cost and power consumption， we combined IRS with cell-free system and equipped the AP with a low-resolution DACs. And then we mathematically modeled the low-resolution DACs using an additive quantization noise model， while establishing expressions for downlink users sum rate. Due to the non-convexity and high complexity of this formulation， an alternating optimization framework was proposed in this paper to solve this problem in order to improve the user sum rate. In particular， we decoupled this problem by fractional programming and used the Lagrange multiplier method as well as semi-definite programming （SDP） method to obtain the expressions of precoding matrix and phase shift matrix. Finally， the simulation results show that the network capacity under this scheme can be significantly improved compared with traditional cell-free network.