In this paper, we propose a soft-output Multiple-Input and Multiple-Output (MIMO) detector algorithm, which is based on two complementary techniques: exploration and exploitation. The proposed detector, called List Exploration and Exploitation (L2E), achieves near-optimal performance with low and fixed computational complexity. It has a high parallelism degree, which makes it suitable for efficient practical implementation. The soft-output values are calculated by means of squared Euclidean distances in a max-Log Likelihood Ratio (LLR) approximation. The average Bit Error Rate (BER) performances of the L2E are compared to the well-known List Sphere Decoding (LSD) algorithm and it is shown that our method considerably reduces the computation complexity while maintaining near-optimal performance in comparison to LSD algorithm.