Exact average error probability of QPSK in Rayleigh fading channel
From Tse and Viswanath, &quot;Fundamentals of Wireless Communication&quot; (which I highly recommend): $$p_e = \frac{1}{2}\left(1-\sqrt{\frac{\text{SNR}}{2+\text{SNR}}}\right)$$ Note that this is the probability of symbol error.