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We develop a computationally efficient and memory efficient approach to (near) maximum a posteriori probability demodulation for MIMO systems with QPSK signalling, based on semidefinite relaxation. Existing approaches to this problem require either storage of a large list of candidate bit-vectors, or the solution of multiple binary quadratic problems. In contrast, the proposed demodulator does not require the storage of a candidate list, and involves the solution of a single (efficiently solvable) semidefmite program per channel use. Our simulation results show that the resulting computational and memory efficiencies are obtained without incurring a significant degradation in performance.