What is the minimum sampling rate for QPSK or what is the bandwidth of QPSK signal?
What is the SNR Signal Noise Ratio? 

For perfect synchronization, with I and Q mixers, filtered with "boxcar" sin(x)/x filters that match to NRZ (on, off) data, then only ONE sample per symbol will suffice; you must simultaneously sample both the I and the Q mixer outputs.

And the SNR will not matter, because for certain assumptions about the noise statistics, the matched-filter is optimal.