We characterize a heterodyne receiver based on a surface-plasmon waveguide quantum cascade laser (QCL) emitting at $2.84\mathrm{THz}$ as a local oscillator, and an $\mathrm{NbN}$ hot electron bolometer as a mixer. We find that the envelope of the far-field pattern of the QCL is diffraction-limited and superimposed onto interference fringes, which are similar to those found in narrow double-metal waveguide QCLs. Compared to the latter, a more directional beam allows for better coupling of the radiation power to the mixer. We obtain a receiver noise temperature of $1050\mathrm{K}$ when the mixer is at $2\mathrm{K}$ , which, to our knowledge, is the highest sensitivity reported at frequencies beyond $2.5\mathrm{THz}$ .