Implementing QPSK Demodulation
Your block diagram is correct. However, it seems to me that your understanding of how to implement it is not. There are no 'bits' in the modulated signal! The input signal is a noisy analog bandpass signal, and the I and Q components are derived from this signal as shown in the block diagram: by demodulating and filtering the incoming signal. Then this (complex) signal is sampled at the symbol rate (you may need timing recovery to know when to sample) and fed to a slicer. The slicer decides which of the four possible symbols was (probably) transmitted in each symbol interval. After the detected symbols are known you simply decode each symbol according to the coding in the transmitter (each symbol carries two bits).

EDIT: I didn't mention matched filters even though they were emphasized in Dilip's comment. For an optimum receiver they are of course necessary, but given the level of the question it seems to me that the block diagram is OK, and that the idea is probably that it should be implemented as is. Even though it will result in a sub-optimum receiver it can perform well under mild channel conditions. The only real difference is the type of low-pass filter used (matched to the transmitted pulses or just eliminating out-of-band noise).