A highly superlinear in radiation intensity photoconductance induced by continuous wave terahertz laser radiation with low intensities has been observed in quantum point contacts made of GaAs quantum wells operating in the deep tunneling regime. For very low values of the normalized dark conductance G(dark)/G(0) approximate to 10(-6), with the conductance quantum G(0) = 2e(2)/h, the photoconductance scales exponentially with the radiation intensity, so that already at 100 mW/cm(2), it increases by almost four orders of magnitude. This effect is observed for a radiation electric field oriented along the source drain direction. We provide model considerations of the effect and attribute it to the variation of the tunneling barrier height by the radiation field made possible by local diffraction effects. We also demonstrate that cyclotron resonance due to an external magnetic field manifests itself in the photoconductance, completely suppressing the photoresponse.