  We propose a thermodynamic refrigeration cycle which uses Indefinite Causal
Orders to achieve non-classical cooling. The cycle cools a cold reservoir while
consuming purity in a control qubit. We first show that the application to an
input state of two identical thermalizing channels of temperature $T$ in an
indefinite causal order can result in an output state with a temperature not
equal to $T$. We investigate the properties of the refrigeration cycle and show
that thermodynamically, the result is compatible with unitary quantum mechanics
in the circuit model but could not be achieved classically. We believe that
this cycle could be implemented experimentally using tabletop photonics. Our
result suggests the development of a new class of thermodynamic resource
theories in which operations are allowed to be performed in an Indefinite
Causal Order.
