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One of the major approaches to spike-based neuromorphic computing is using memristors as analog synapses. We propose unitary quantum gates that exhibit memristive behaviors, including Ohm's law, pinched hysteresis loop and synaptic plasticity. Hysteresis depending on the quantum phase and long-term plasticity that encodes the quantum state are observed. We also propose a three-layer neural network with the capability of universal quantum computing. Quantum state classification on the memristive neural network is demonstrated. These results pave the way towards quantum spiking neural network built on unitary processes. We obtain these results in numerical simulations and experiments on the superconducting quantum computer ibmq_vigo.