As usual, it is too soon to make comparisons like this.
The power consumption of a device will depend strongly on the architecture it uses, for one.
However, in principle, there is no reason to suspect that quantum computers would consume more energy than classical devices performing the same operations.
Indeed, one would expect the opposite, the fundamental reason being that quantum computers work (mostly) via unitary operations.
A unitary operation is a reversible operation, or, in other words, an operation during which no information is lost to the environment.
Such an operation is basically "perfectly" energy efficient (for one, it wouldn't produce heat).
So, in principle, the elementary operations performed in a quantum algorithm which uses unitary operations can be ideally energy efficient.
This is in direct contrast with what you have with classical devices, in which the elementary operations are non-reversible, and therefore necessarily "waste" some amount of information for every operation.
Having said this, there are a million caveats to be taken into account.
For example, quantum computers in the real world will have to deal with decoherence, so that the operations are not really unitary. This implies that error-correction protocols are necessary to take this into account, and one should then go and track what is the added energy consumption of this whole process.
Also, while unitary operations are energy efficient, in practice when one acquires the result of the measurement, measurements have to be performed, and this are non-reversible operations which typically destroy the information. After each such measurement, one will need to generate the carriers of information again. Also, many quantum computing protocols rely on repeated measurements during the computation.
One could go on for a long time, as this is very much uncharted territory.
One recent work that discusses in some measure the power consumption problem is 1610.02365, in which the authors present a method for (classical machine learning) information processing using photonic chips.
One claim of the authors is that photonic chips allow to perform operations in an extremely energy efficient way, exploiting the natural evolution of coherent light.
They do not demonstrate any form quantum computation, but their energy efficiency reasonings would not change much when using the same device for quantum information processing.