With increased interests of neural networks, hardware implementations of neural networks have been investigated. Researchers pursue low hardware cost by using different technologies such as stochastic computing and quantization. For example, the quantization is able to reduce total number of trained weights resulting in low hardware cost. Stochastic computing aims to lower hardware costs substantially by using simple gates instead of complex arithmetic operations. In this paper, we propose a new stochastic multiplier with shifted unary code adders (SUC-Adder) for quantized neural networks. The new design uses the characteristic of quantized weights and tremendously reduces the hardware cost of neural networks. Experimental results indicate our stochastic design achieves about 10x energy reduction compared to its counterpart binary implementation while maintaining slightly higher recognition error rates than the binary implementation.