Magnetic tunnel junctions (MTJ) have been successfully applied in various sensing application and digital information storage technologies. Currently, a number of new potential applications of MTJs are being actively studied, including high-frequency electronics, energy harvesting or random number generators. Recently, MTJs have been also proposed in designs of new platforms for unconventional or bio-inspired computing. In the present work, it is shown that serially connected MTJs forming a multi-state memory cell can be used in a hardware implementation of a neural computing device. The main purpose of the multi-cell is the formation of quantized weights in the network, which can be programmed using the proposed electronic circuit. Multi-cells are connected to a CMOS-based summing amplifier and a sigmoid function generator, forming an artificial neuron. The operation of the designed network is tested using a recognition of hand-written digits in 20 × 20 pixels matrix and shows detection ratio comparable to the software algorithm, using weights stored in a multi-cell consisting of four MTJs or more.