Isn't consciousness more like what emerges out of the neurons firing? If I turn off my calculator, I can't input calculations into it and get an answer, but if I turn it on, the electricity running in the circuitry will react to the button presses, and I can say "the calculator is working".

So the consciousness is the "computing session", and if we can persist the state of the hardware before shutdown, then booting it back up will give us that consciousness.