Surface plasmon polariton has attracted more and more attention and has been studied extensively in the recent decades, owing to its ability to confine the electro-magnetic field to a sub-wavelength scale near the metal-dielectric interface. On one hand, the tightly confined surface plasmonic modes can reduce the size of integrated optical device beyond the diffraction limit; on the other hand, it provides an approach to enhancing the interaction between light and matter. With the development of experimental and numerical simulation techniques, its investigation at a quantum level has become possible. In the recent experiments, scientists have realized quantum interference between single plasmons in a nanoscale waveguide circuit and achieved the strong coupling between photons and single molecules by using plasmonic structure, which demonstrates its superiority over the traditional optics. Here, we review the theoretical and experimental researches of surface plasmon polariton in the field of quantum information processing. First, we introduce the experiments on the basic quantum properties of surface plasmons, including the preservation of photonic entanglement, wave-particle duality and quantum statistical property. Second, we review the research work relating to the generation, manipulation and detection of surface plasmons in a quantum plasmonic integrated circuit. Then, we present the research of the interaction between surface plasmons and single quantum emitters and its potential applications. Finally, we make a discussion on how the intrinsic loss affects the quantum interference of single plasmons and the coupling between quantum emitters. The collision and combination of quantum optical and plasmonic fields open up possibilities for investigating the fundamental quantum physical properties of surface plasmons. It can be used to make ultra-compact quantum photonic integrated circuits and enhance the interaction strength between photons and quantum emitters.