Write an octal calculator with Qt framework. Use QtHelloWorld project as your base. You should implement four operations for octal numbers, which are addition (+), subtraction (-), multiplication (*), division (/). The calculator should support up to 4 digits octal numbers. You can care about only the integer part for the output and assume the input takes always positive values, but the calculator must display both positive and negative values correctly for the output. In addition, implement the clear function in the calculator to clean the previous output/result.

Also, could the Qt portion also be demonstrated.