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Non-linear amplitude distortion may be the key impairment in some practical multiple-input multiple-output (MIMO) communications systems. However, there are only a few past investigations addressing its impact on the system performance. In this paper, we derive an approximate upper bound on the bit error rate for MIMO zero-forcing receivers with M-ary quadrature amplitude modulation (MQAM) in the presence of channel estimation error, which is valid for any order of non-linearity and arbitrary numbers of transmit and receive antennas. In the absence of channel estimation error, the results derived herein give a true BER upper bound. Comparison with Monte Carlo simulations suggest that the theoretical bound is accurate and useful for practical system design.