Methods and apparatus are disclosed for cooling superconducting signal lines disposed on an interconnect such as a flexible cable or a rigid substrate. The superconducting signal lines are cooled to a cryogenic temperature lower than the temperature at which at least some superconducting logic devices coupled to the interconnect are operated. In some examples, an airtight conduit, heat pipe, or thermally conduct of strap provided to cool the superconducting interconnect. In one example of the disclosed technology, a system includes at least two sets of superconducting logic devices, cooling apparatus adapted to cool the logic devices to a first operating temperature, and interconnect coupling the superconducting logic devices, and a cooling apparatus in thermal communication with the interconnect. The apparatus is adapted to cool superconducting signal lines on the interconnect to a lower operating temperature than the first operating temperature at which the superconducting logic devices operate.