Logical devices based on electrical currents are ubiquitous in modern society. However, digital logic does have some drawbacks, such as a relatively high power consumption. It is, therefore, of great interest to seek alternative means to build logical circuits that can either work as stand-alone devices or in conjunction with more traditional electronic circuits. One direction that holds great promise is the use of heat currents for logical components. In the present paper, we discuss a recent abstract proposal for a quantum thermal transistor and provide a concrete design of such a device using superconducting circuits. Using a circuit quantum electrodynamics Jaynes-Cummings model, we propose a three-terminal device that allows heat transfer from source to drain, depending on the temperature of a bath coupled at the gate modulator and show that it provides similar properties to a conventional semiconductor transistor.